Table of Contents

*bendDangerous Bend

theorem a[thm]Theorem A

ThmMain Theorem

notadefn[thm]Notation and Definition
*notadefn*Notation and Definition

On the geometry of some algebras related to the Weyl groupoid.
Ian M. Musson

Department of Mathematical Sciences

University of Wisconsin-Milwaukee
email<EMAIL_ADDRESS>
Let $\ttk$ be an algebraically closed field of characteristic zero.  Let $\fg$ be a finite dimensional classical simple Lie superalgebra over $\ttk$ or $\fgl(m,n)$. In the case that $\fg$ is a Kac-Moody algebra of finite  type with set of roots $\Gd$, Sergeev and Veselov introduced the
Weyl groupoid $\mathfrak{W}=\mathfrak{W}(\Gd)$, which has significant connections with the representation theory of $\fg$. Let $\fh$, $W$ and $Z(\fg)$  be a Cartan subalgebra of
$\fg_0$, the Weyl group of $\fg_0$ and  the center of $U(\fg)$ respectively. Also let $G$ be a Lie  supergroup  with Lie $G =\fg$. There are several important
commutative algebras related to $\mathfrak{W}$.  Namely

* The image $I(\fh)$ of the injective Harish-Chandra map $Z(\fg)\lra S(\fh)^W$.

* The supercharacter $\Z$-algebras $J(\fg)$ and $J(G)$ of finite dimensional representations of $\fg$ and $G$.
Let $\A = \A(\fg)$ be denote either $I(\fh)$ or $J(G) \ot_{\Z}\ttk$.
The purpose of this  paper
is to investigate the algebraic geometry of $\A.$  In many cases, the algebra $\A$ satisfies
the Nullstellensatz. This gives a bijection between radical ideals in $\A$ and superalgebraic sets (zero loci of such ideals).  Any  superalgebraic set
is uniquely a finite union of irreducible superalgebraic  components. In the

Kac-Moody case, we describe the smallest superalgebraic set containing a given (Zariski) closed set, and show that the superalgebraic sets are exactly the closed sets that are unions of groupoid orbits.

<ref>. Introduction.
<ref>. General Results.
<ref>. Applications.
<ref>. The Nullstellensatz.
<ref>. Weyl Groupoids and their Actions.
<ref>. The Geometry of Superalgebraic Sets.

§ INTRODUCTION

§.§ Background.

§.§.§ The classical Lie superalgebras.

We are interested in the following classical Lie superalgebras $\fg$.  All have $\fg_0$ reductive and $\fg_0\neq \fg$.
The KM Kac-Moody Lie superalgebras of finite type. These algebras are also known as contragredient Lie superalgebras, <cit.>, <cit.> Chapter 5. algebras
$    \osp(r|2n), $
$\fsl(m|n)$ with $m\neq n,$
$    D(2|1;\ga),$
$    F(4),$ $G(3),$
The strange Lie superalgebras $\fp(n)$, $\fq(n)$ and their simple relatives $\sfp(n)$, $\psq(n)$.
There are some conditions on $r, m,n\in\N$ and $\ga\in\ttk$ which we omit.
The last three algebras listed in (a) are called exceptional.
The algebras listed in (a) with $ \fgl(m|n)$ replaced  by  $\fpsl(n|n)$, together with the simple algebras from (b) are  precisely the  finite dimensional simple Lie superalgebras $\fg$ with  $\fg_0$ reductive and $\fg_0\neq \fg$.

§.§.§ Algebras related to the Weyl groupoid.

The Weyl groupoid $\mathfrak{W}(\Gd)$ of Sergeev and Veselov, <cit.>
become a fundamental tool in the  representation theory  of a KM algebra $\fg$. Here $\Gd$ is the root system of  $\fg$. This groupoid was introduced in relation to the algebras
$J(\fg)$ and $J(G)$ listed in the abstract.  The
Lie supergroups $G$ we  consider are defined  in Lemma <ref>.  In particular the supergroups $P(n)$ and $Q(n)$ have
$\Lie P(n)=\fp(n)$ and $\Lie Q(n)=\fq(n)$.
Weyl groupoids for   $ P(n)$ and $Q(n) $ were defined in  <cit.> Section 5.4
and  <cit.>, Section 5

For  a KM algebra $\fg $, an important role is played by a non-degenerate, symmetric  bilinear form   $(\;,\;)$ on  $\fg $.  This can be constructed in a uniform way,  <cit.> Theorem 5.4.1, Remark 5.4.2.
The restriction of this form to $\fh$ remains non-degenerate
and this yields an isomorphism $\fh \lra \fh^*$, denoted  $h_\ga\mapsto \ga$. Thus we obtain a non-degenerate, symmetric  bilinear form on
$\fh^*$  which we also denote by $(\;,\;)$.  This latter form
is invariant

under the Weyl group $W$ of $\fg_0$,
and we  can normalize the whole construction by requiring that its Gram matrix is the (symmetrized) Cartan matrix $(\ga_i,\ga_j)$, where
$\{\ga_i\}$ is  the distinguished basis of simple roots for  $\fg$,
<cit.> Section 3.2.
The issue with $ \fpsl(n|n)$ is
that  any set of simple roots
$\{ \alpha_i  \}$ is linearly dependent, and the  matrix $A= (  \alpha_i,  \alpha_{j})$ is  singular.  Thus there is no bilinear form as above. The usual solution to this problem is to take a minimal realization of $A$, leading  to the construction of $\fgl(n|n),$
which  has $\fpsl(n|n)$ as a subquotient.

For  $\alpha \in
\Gd_{iso}$, the set of isotropic roots of $\fg$,  set   Π_=
{ λ∈^* | (λ,) =0}.
The other algebra of interest $I(\fh)$ is isomorphic to
$Z(\fg) $ via the  Harish-Chandra map.
This  algebra may be defined as
I()= {f∈S(𝔥)^W| f(λ) = f(λ+ t α) α∈_iso, ∈Π_  t ∈}.Several other definitions of $I(\fh)$ can be given based on <cit.>, Lemma 12.1.1.
By work of Gorelik and Kac <cit.>, <cit.>, the  Harish-Chandra map yields an isomorphism $Z(\fg)\lra I(\fh)$, see  also <cit.>, Theorem 13.1.1.
For a description of $Z(\fg)$ when  $\fg=\fp(n)$ and $\fq(n)$, see <cit.>,  <cit.> respectively.

There is a description of $J(G)$ parallel to (<ref>).
Let $\mathbb{T}$ be a maximal torus  in $G_0$,
$X(\mathbb{T}) = \Hom(\mathbb{T},\ttk^*) $ (resp.
$Y(\mathbb{T}) = \Hom(\ttk^*, \mathbb{T})$)  the group of
characters (resp. one-parameter subgroups) of  $\mathbb{T}$.
$D_\ga$ be the derivation of  $\Z[X(\cT)] $ given by
D_(^) = (, ) ^.Then in the KM case, we have by <cit.> Equation (1),
J(G)= {f ∈[X()]^W| D_f ∈(^-1)  α∈_iso}.Various conditions equivalent to that defining $J(G)$ are given in
<cit.> Section 3.

§.§.§ Continuous Weyl groupoids.

In the theory of algebraic groups acting on varieties, a key role is played by closed orbits.  However the orbits of
$\mathfrak{W}$ on  $\fh^*$ or  $ \mathbb{T}$  are not closed unless they are finite, see Remark <ref>, and for  this reason in Section <ref>
we introduce the continuous   Weyl groupoids $\cW^c$  and $\cW_*^c$.
The action of a groupoid $\fG$ on an affine variety $X$ and  the invariant ring $\cO(X)^\fG$ are defined
in Section <ref>.
In Proposition <ref>, we show there are actions of  $\cW^c$  and $\cW_*^c$
on $\fh^* $  and $ \mathbb{T}$ respectively such that

$$I(\fh) =S(\fh)^{\cW^c} \mbox{ and }  J(G) \ot_{\Z}\ttk =\cO(\mathbb{T})^{\cW_*^c}.$$

We assume one of the following holds
$X=\mathbb{\fh^*}, \; \fG = \mathfrak{W}^c,\;  \B = \cO(X)^W= S(\fh)^W$ and  $\A = I(\fh)$.
$X=\mathbb{T}, \; \fG = \mathfrak{W}_*^c,\;  \B = \cO(X)^W$ and  $\A = J(G) \ot_{\Z}\ttk$.

If $\fg=\fgl(m|n)$ the algebras $I(\fh)$ and  $J(G) \ot_{\Z}\ttk$ are the algebras of
supersymmetric polynomials and Laurent supersymmetric polynomials over $\ttk$ respectively.  These algebras have very beautiful combinatorics, see <cit.> Chapter12, <cit.>  and  <cit.>. Suprisingly though, the algebras of (Laurent) supersymmetric polynomials have received little attention from a geometric point of view, perhaps because they are not Noetherian.

In the simplest case where $m=n=1$, $\cW$ is the smallest groupoid that is not a disjoint union of groups.  Thus $\cW$  has two objects, and two non-identity morphisms. To describe the action of $\cW$  on the plane $\fh^*=\ttk^{1|1}$, take the objects to be $\pm a$ where $a \neq 0$ is a point in $\fh^*$.  The functor sends  $\pm a$ to the line $L$ they span, and the non-identity morphisms add or subtract $a$ to a point on $L$. The invariant ring $S(\fh)^{\cW}$ consists of all polynomial functions that are constant on $L$. So $S(\fh)^{\cW}= \ttk + TS(\fh)$ where $S(\fh) =\ttk[S,T]$ and $L$ is the line defined by $T$. The algebra $S(\fh)^{\cW}$ is a well-known example of a non-Noetherian domain of Krull dimension two.
This answers a question of S. Paul Smith.

§.§.§ The Duflo-Serganova functor.

We recall the definition of the Duflo-Serganova functor $DS_x$,   in the case that $x$ is a root vector such that $[x,x]=0.$
For $M$ as above, set
$M_x =\Ker_M x/x M$.  Then
$M_x$ is a module for
$\fg_x$ where
$\fg_x =\Ker \ad x/ \Im \ad x,$ which is a Lie superalgebra of smaller rank  than $\fg$. Although
$\fg_x$ is a quotient of a subalgebra of  $\fg$, it was shown in
<cit.> Lemma 6.3, that it embeds in $\fg$ at least in the KM case.  We follow the description of
section 2C in the case that $x$ is a  root vector corresponding to the isotropic root $\gb$.

Let $x$ be a root vector corresponding to the isotropic root $\gb$.
We can identify $\fg_x$ with a subalgebra of $\fg$ such that the centralizer $\fg^x$ of $x$ in $\fg$ is a semidirect sum $\fg^x= \fg_x \ltimes [x,\fg]$. The subalgebra $\fg_x$ is given by
_x= ()_∈()^,where
$\Gd(\gb) = \{\ga\in \Gd|(\ga,\gb) =0, \ga\neq \pm\gb\}$,
and $\fh(\gb) = \span \{h_\ga|\ga \in \Gd(\gb)\}$.  We have $\fh(\gb) = \fg_x\cap \fh$.
The restriction of $(\;,\;)$ to $\span\Gd(\gb) $ is non-degenerate.
Let $\{\ga_1,\ldots, \ga_r\}$ be a basis for $\span\Gd(\gb) $ and suppose  $\gs\in \fh^*$ with
$(\gs,\gb)=1$.  For $i\in [r],$ let  $\ga_i' = \ga_i - (\gs,\ga_i)$.  The Gram matrix of  $(\;,\;)$ using the basis
$\{\ga_1,\ldots \ga_r,\gb,\gs\}$ for $\fh^*$
has the same determinant as the Gram matrix using the basis where each
$\ga_i $ is replaced by $\ga_i'$.  Since the latter matrix is block diagonal and $(\ga_i, \ga_j) = (\ga_i', \ga_j')$, the result follows.
Under the isomorphism $\fh^* \lra \fh^*$,   $\span \Gd(\gb)$ maps to  $\span \{h_\ga|\ga \in \Gd(\gb)\}$ which we identify with $\fh(\gb)$. We have $\span \Gd(\gb) = \fh(\gb)^*$.
If $(\gs(\gb),(\gb))=1$, we have an orthogonal direct sum

^*  = ()^* { ,  ()}.
In the table below, taken from <cit.> Section 4.5, the first column lists the Lie superalgebras we consider, $x$ is a root vector, and the second column describes $\fg_x$ up  to isomorphism. If $\fg=\fp(n)$, we assume that $x\in \fg^{\gep_1+\gep_2}$ in the notation of <cit.>, 2.4.1.

\[ \begin{tabular}{|c|c|}\hline
$    \fg$ &$\fg_x$
\\ \hline\hline %1
$    \fgl(m|n)$ &$\fgl(m-1|n-1)$ \\ \hline  %2
$    \fsl(m|n), \; m \neq n$ &$\fsl(m-1|n-1)$ \\ \hline %3
$    \osp(m|2n) $ &$\osp(m-2|2n-2)$ \\ \hline
$    D(2|1;\ga)$ &$\ttk$\\ \hline %4
$    F(4)$ &$\fsl(3)$ \\ \hline  %5
$G(3)$ & $\fsl(2)$
\\ \hline%1'
\\ \hline   %2'
\\ \hline %3'
%\\ \hline {}
\end{tabular}  \]

Since we use induction on  the rank as a proof technique, we make some brief remarks about the base cases.  These occur when $\fg_x$
has no isotropic roots.  Then the Weyl groupoid for $\fg_x$ is just the Weyl group. The desired results  hold  trivially in these cases.
By convention $    \fgl(0|k) =    \fgl(k|0) =  \fgl(k)$, and other low dimensional entries in the column  for $\fg_x$ should be interpreted similarly.

The main use we make of the functor
$DS_x$ is to construct certain algebra  maps $ds_x$ in Subsections <ref> and <ref>.
These algebra maps can be realized in two equivalent ways, and both realizations can be found in the literature, at least implicitly.  Both methods can be easily adapted  to obtain
maps on suitable overrings as in the diagram preceding Lemma <ref>.
In the situation of  Lemma <ref>, $\fh^*$ has been  identified with a subspace of $\fh^*$, so we have a resriction map   $\res: S(\fh) \lra S(\fh_x), f \mapsto f|_{\fh^*}$.
This map descends to $S(\fh)^W$  and  $I(\fh)$ and we have   $\res = ds_x: I(\fh) \lra I(\fh_x)$. On the other hand the algebra $I(\fh)$ can be defined in terms of polynomials satisfying certain partial evaluation conditions,  and then it is natural to view $ds_x$ as evaluation $\ev:I(\fh) \lra I(\fh_x)$.  Similar remarks apply to $ds_x: J(G) \lra J(G_x)$.

§.§.§ Harish-Chandra pairs and representations of Lie supergroups.

An (algebraic) Lie supergroup $G$ can be defined using a Hopf superalgebra $\cO(G)$ and the functor of points. Denote  the
category of supercommutative $\ttk$-algebras ${\bf Alg}$.  Then for any supercommutative algebra $A$, the $A$-points of  $G$  are given by

$$G(A) = \mo _{\bf Alg

The Hopf superalgebra structure on $\cO(G)$ induces a natural group structure on $G(A)$. Another  useful approach (due to Kostant <cit.> and  Koszul <cit.>) to $G$ uses Harish-Chandra pairs. Following <cit.> Definition 7.4.1, a Harish-Chandra pair (HCP) is a pair $(G_0,\fg)$
where consisting of an algebraic group $G_0$ and a Lie superalgebra $\fg$
such that
$ \fg_0 = \Lie G_0,$
There is a representation $\gr$ of $G_0$ on $\fg$  such that $\gr(G_0)|_{\fg_0} =\Ad$ and the differential of $\gr$ acts on $\fg$ as the adjoint representation: for $X\in\fg_0, Y\in \fg$,

$$ d\gr(X)Y = [X,Y].$$

We write $(G_0,\fg,\gr)$ if we want to stress the role of $\gr$. Harish-Chandra pairs form a category, for morphisms see <cit.> Definition 7.4.2.
Given a  Lie supergroup  $G$, we obtain a HCP $(G_0, \Lie G, \Ad)$.  Conversely given a HCP, we can put a Hopf superalgebra structure on
(G):=_U(_0)(U(),(G_0)),and thus obtain a Lie supergroup $G$.
In (<ref>), $ U(\fg)$ is naturally a left $U(\fg_0)$-module, and the action of $x\in \fg_0$ on $\cO(G_0)$ is given by a left-invariant differential operator, <cit.> page 125.
This is the basis for the following result, see <cit.> Theorem 7.4.5.
There is an equivalence of categories between the category of Lie supergroups and  the category of  Harish-Chandra pairs.
The connection between representations of  Harish-Chandra pairs and the corresponding Lie supergroup was explained in <cit.>. A representation of a HCP $(G_0,\fg)$ consists of the following data
A representation $\gS: G_0 \lra GL(V),$
A representation $\gs:\fg\lra \fgl(V)$,  such that $d\gS = \gs|_{\fg_0}$ and a further compatabilty condition given in <cit.> Definition 6 (2) holds.
There is a bijection between representations of the HCP
and representations of $G$ on $V$.
See <cit.>  Proposition 6.
Let $\cF_\fg$ (resp. $\cF_G$)
be the category of finite dimensional $\Z_2$-graded $\fg$-modules (resp. $G$-modules), and $\Pi$ the parity change functor on $\cF_\fg$.
The extra condition mentioned above essentially says that the representations of $G$ that arise correspond to representations of $\fg_0$ that are integrable as  $G_0$-modules.  For example, suppose that $\fg_0$ has a non-trivial center.  If $\fa$ is an abelian Lie algebra and $T$ is a torus with $\Lie T = \fa$, then integrable representations of $\fa$ correspond to elements of the character group $X(T)$.
Hence   $\fg_0$ has many more finite dimensional modules than
$G_0$, and these modules can be induced to give   graded modules for $\fg$. Thus the category  $\cF_G$ is more manageable than $\cF_\fg$. In addition, the results we quote from <cit.>, <cit.> and <cit.> work best for supergroups.
Let $\cF=\cF_G$ or  $\cF_\fg$.  The Grothendieck group $K(\cF)$ of $\cF$ is the  free abelian group on the symbols $[M]$, where $M$ is an object in $\cF$ modulo the relations that $[M] = [L] +[N]$ whenever there is an exact sequence
$0\drar L\drar M\drar N \drar 0$
in $\cF$.   Since $\cF$ is closed under tensor product, $K(\cF)$  has a ring structure determined by $[M ][N] =[M\ot N]$.
The supercharacter ring (resp. character ring)
is the factor ring of $K(\cF_\fg)$ by the ideal generated by all $[\Pi(M)]+[M]$ (resp.  $[\Pi(M)]-[M]$) where $M\in \cF_\fg$. For the classical Lie superalgebras $\fg$ listed in Subsection <ref>, we define  $J(\fg)$ to be the   character ring  of  $\fg$ if $\fg$ has type $\fq$ and the  supercharacter ring in all other cases. The $\Z$-algebra $J(G)$  is the factor ring of $K(\cF_G)$ defined in a similar way.  The  Lie superalgebras $\fg$ listed in <ref> such that $\fg_0$ has a non-trivial center  are $\fg= \fgl(m|n),\fsl(m|n)$ and $\fg=\osp(2|2n).$  In these cases $J(\fg)$ is graded by an abelian group, and  $J(G)$ is the identity component of this grading, see <cit.> Propositions 7.1, 7.2, 7.3 and 7.5.

We mention an immediate consequence of
the weak  Nullstellensatz for $Z(\fg)$, Theorem <ref>.
Let $\fg$ be any  classical Lie superalgebra. A central character  of $\fg$ is an algebra map $Z(\fg) \lra \ttk$, <cit.> Section 3.  Any central character $\chi$ determines, and
is determined by the maxmal ideal $\Ker\chi$.  An element   $z\in Z(\fg)$ acts on any
Verma module $M(\gl)$ as a scalar $\chi_\gl (z)$ and  $\chi_\gl$ is called    the central character  afforded by $M(\gl)$, <cit.> 8.2.4.
The next result proves and extends
Conjecture 13.5.1 from <cit.>.
Any central character of $\fg$ is afforded by a Verma module. This paper replaces the unpublished preprint <cit.> which dealt only with the cases
where $\fg=\fgl(m|n)$ or
$    \osp(r|2n)$.

I am grateful to    Maria Gorelik,  Hanspeter Kraft  and Shifra Reif for helpful correspondence.

§ GENERAL RESULTS.

Throughout  $[k]$ denotes  the set of the first $k$ positive integers and iff means if and only if. We start with some general results on commutative rings.

§.§ Certain Ring Extensions.

In this Subsection we assume the following hypothesis on the ring extension $A\subseteq B$.
For some
$T\in A$,  a non-zero divisor in $B$, we have  $TB \subseteq A$. If $I$ is an ideal of $A$, let
$$\rad(I) = \{f\in A| f^n \in I, \mbox{ for some } n>0\}.$$

If $I=\rad(I)$ we say $I$ is a radical ideal.
Suppose $R$ is a  commutative ring  and $T\in  R$, is a non-zero divisor. Denote the localization by $R_T$.
Assume Hypothesis <ref>.
We have an equality of localizations $A_T = B_T.$
If $P$ is a radical ideal
of $A$  with $T\in P$,
then $TB \subseteq P$. We leave (a) as an exercise. Then (b) holds since $(TB)^2% =T^2B \subseteq AT
\subseteq A P = P$.

If $L$ is an ideal of $R$ define the extension of $L$ to $R_T $ to be $L^e = L_T$, the localization of $L$.   If $M$ is an ideal of $R_T$, set $M^c= M \cap R$, the contraction of  $M$ to $R$. Then
extension  and contraction provide order preserving bijections
{P ∈R|  T ∉P } ↔R_T.This applies to $R= A$ or $B$ in Lemma <ref>, so   we have a bijection

{P ∈B| T ∉P} ↔{P ∈A| T ∉P}

,given by $P \mapsto P\cap A$.    From now on we identify the two sets on either side of (<ref>).
Let $\phi: A\lra A/BT= C$ be the natural map.  Then we have a disjoint union
A = B_T ∪ϕ^-1 (C).
As noted above we have $A_T =B_T $.    For a prime ideal $Q$ of $A$ there are two possibilities.  If $T\notin Q$, then $Q_T$ is a prime ideal of $A_T=B_T$ such that $Q_T\cap A=Q.$ If $T\in Q$, then $Q$ is the inverse image of the  prime ideal of $Q/BT$   under $\phi.$
If         $m$ is a maximal ideal of  $A$ and $T\notin m$, the  maximal ideal $M$ of  $B$
given by $M= m_T\cap B$ satisfies  $m = M \cap A$.    Also $mB \neq B$.     This first statement follows from Equation  (<ref>), and the second is an immediate consequence.         In Corollary <ref>, if $B=\cO(X)$ for an affine variety $X$,
with $B$ a finitely generated $\ttk$-algebra,
$m$ and $M$ have the same set of zeroes in $X$, so by the usual Nullstellensatz  $M  = \Rad mB$.  However $M$ can strictly contain $mB$,
see  <cit.> Example 2.2.

In the situation of Theorem <ref>, suppose that the rings
$B_T $ and $ C$ satisfy  the ascending chain condition on radical ideals.  Then so does $A$. Let $R^{1} \subseteq R^{2} \subseteq \ldots $
be an ascending chain of radical ideals in $A$.
Then $R_T^{1} \subseteq R_T^{2}  \ldots $ is an ascending chain of radical ideals in $A_T=B_T$, so by assumption, there is an $m$ such that $R_T^{m} = R_T^{i}$ for all $i\ge m$.
Write $R_T^{m} = p_1 \cap p_2 \cap  \ldots  \cap p_r$ for some prime ideals of $B_T$.
We can assume that this intersection is irredundant, and all the $p_k$ are minimal over $R_T^{m} $.
Then if  $P_k= p_k \cap A$ and $i\ge m$,  the $P_k$ are exactly the minimal primes over $R^{i} $
that do not contain $T$. By <cit.> Corollary 2.12 every radical ideal in a commutative ring is the intersection of (a possibly infinite number of)  prime ideals. So for $i\ge m$, write $R^{i}  $ as an intersection of prime ideals in $A$.  Say

\[ R^{i}  = P_1 \cap \ldots \cap P_r \cap  \bigcap_{j\in \Lambda_i} Q_j.
\]

where the $Q_j$ are prime ideals that contain $TB$. Thus  $D_i =  \bigcap_{j\in \Lambda_i} Q_j$ is a radical ideal in $A$ containing $TB$, and if $\overline{D}_i = D_i/TB$, then $\overline{D}_{1} \subseteq \overline{D}_{2} \subseteq \ldots $ is an ascending chain of radical ideals of $C$.  So there is an $n \ge m$, such that $\overline{D}_{i} = \overline{D}_{n}  $ for all $i>n$ and  then $R^{i} = R^{n}  $.
With the same assumptions as the Lemma,
every radical ideal $I$ of $A$ is a finite intersection
of prime ideals. These prime ideals can be taken to be the prime ideals minimal over $I$.See <cit.> Theorem 87.

§.§ Towards the Weak Nullstellensatz.

Consider a  ring  extension  $A \subseteq B$, and $X\subseteq \Max A$.  We say the Relative Weak Nullstellensatz (RWN) for $X$ and the pair $(A, B)$, if for every $m \in \Max A$ we have $mB \neq B$. If the RWN holds in the case $X=\Max A$, we say  simply that
RWN  holds for the pair $(A, B)$.
If $A, B$ are $\ttk$-algebras, with $B$ finitely generated  and $M \in \Max B $, then $B/M \cong \ttk$, so if  $m=M\cap A$, then   ↪A/m ≅(A +M)/M ⊆B/M ≅. Thus $m \in \Max  A$ and we have a map
ψ:B A,   M ↦M∩A . If RWN  holds for the pair $(A, B)$, and $m \in \Max A$, then  by Zorn's Lemma  there is  a $M \in \Max B $ containing $mB$, we have $M\cap A = m$, so the map in (<ref>)
is surjective.

Consider the following diagram of ring homomorphisms where the vertical maps are inclusions, and
${\phi}$ is the restriction of ${\Phi}$ to $A$.

(100,20)*+A'="b"; (40,0)*+B="c";
(100,0)*+B'="d"; "a";"b"?*!/_2mm/ϕ; "a";"c";

@> "c";"d";?*!/_2mm/Φ;

Suppose $\Phi$ and $\phi$ are onto and  RWN holds
for the pair $(A', B')$. Then the RWN holds for the pair $(A, B)$
and the set
$X= \phi^{-1}(\Max A)$.
$m \in  X $, that is $m$ is a maximal ideal of $A$ containing $\Ker \phi$.
$\phi (m)$ is  a maximal ideal of $A' $, so by assumption  $J=\phi(m)B'$ is a proper ideal of $B' $. Since $\Phi(mB )=J,$ it follows that $mB $
is a proper ideal of  $B$.
Suppose the finite group  $G$  acts on $S$ and  assume $|G|$ is a unit in $S$.  If RWN holds
for the pair $(R, S)$, then the RWN holds for the pair $(R^G, S^G)$.In this situation we have a Reynolds operator $R\lra  R^G$.  Regarded as a $G$-module $R^G$ is the isotypic component of $ R$ corresponding to the trivial module.  Let $R^+$ be the sum of the remaining isotypic components.  Thus
$ R=R^G\op  R^+$
is a decomposition into $R^G$-modules. If $m \in
\Max  R^G$, then $mR= m \op mR^+$ is a proper ideal of $R$, so  by assumption  $mS$ is a proper ideal of $S$. The result follows since  $ m S^G= S^G$ implies   $mS=S $.
RWN does not hold always when $S =\ttk[x,y]$ is a polynomial algebra in 2 variables: if $R = \ttk[x] + (xy-1)S$, then $m=x\ttk[x] + (xy-1)S$ is a maximal ideal of $R$ such that $mS=S$.

§ APPLICATIONS.

In this Section  if $\A=I(\fh)$ or   $J(G) \ot_{\Z}\ttk$ we consider  a ring extension $\A\subset \B$ and an element $T\in \A$ such that $T\B \subset \A$. Thus Hypothesis <ref> holds. Because we will need to change the Lie superalgebra, we sometimes   write $\A=\A(\fg)$.
The algebra $\B$ is either $S(\fh)^W$ or $\cO(\mathbb{T})^W$,
so is in particular a finitely generated  $\ttk$-algebra.  To prove the weak Nullstellensatz we consider a map  with kernel $T\B$,
()(_x),where $\fg_x$ is as in the table from Subsection <ref>, and use induction on the rank of $\fg$.

§.§ Application to $I(\fh)$.

By <cit.> Section 3, the functor
$DS_x$  induces an algebra map $Z(\fg) \lra Z(\fg_x)$. Using the Harish-Chandra map, this gives rise to a map     $ds_x:I(\fh) \lra I(\fh_x)$.  Because of (<ref>) and the above remarks, this map can be thought of as restriction of functions
$\res(f) = f|_{\fh_x^*}$.

§.§.§ The Kac-Moody  case.

In the KM case, if $\fg_1$ is a simple $\fg_0$-module, let $\gO=\Gd_{iso}$.  Otherwise
$\fg_1= \fg_1^+ \op \fg_1^-$ is a direct sum of two simple $\fg_0$-modules,
and  we
let $\gO=\Gd_{1}^+$ be the set of roots of $\fg_1^+$.  Then $W$ acts transitively on $ \gO$.  Define
$T=\prod_{\beta\in\gO} h_{\beta}$. Then  $T$ is $W$-invariant.
We describe  an automorphism  $\gs$ on  $\fg_x$
which we are
going to use below.
For $\fg=\osp(2m|2n)$,  then  $\gs$ is the diagram  automorphism of $\fg_x$ described in <cit.> Lemma 5.5.12. This automorphism preserves $\fh$.
For $\fg=F(4)$, then
$\gs$ is induced by an involution of the Dynkin diagram for $\fg_x=\fsl(3)$.
for $\fg=D(2|1;\ga)$ we have $\fg_x=\ttk$ and  $\gs=-\Id$.
In all other cases  $\gs=\Id$.
We have
$T\cS(\fh)^W\subset I(\fh)$ and the
kernel of $\res: I(\fh)\to I(\fh_x)$ equals
We have $\res(I(\fh)= I(\fh_x)^{\gs_x}$.
By construction $T$ is $W$-invariant and $T$ vanishes on all hyperplanes $\Pi_{\ga}$. Thus by (<ref>),
$T\cS(\fh)^W\subset \Ker\res$. Suppose $x =e _\gb$ is a root vector and  $f \in\Ker\res$.
Then $h _\gb$ divides $f$ in $\cS(\fh)$, so by  $W$-invariance, so does
$T$.  Write $f=gT$ with
$g\in \cS(\fh) $.  Since $f,T$ are  $W$-invariant,  so is $g$
proving (a).  For (b) see <cit.> Theorem 6.4.

§.§.§ Algebras of type $\fq$.

There are four related Lie superalgebras of type  $\fq$. We refer to <cit.> 2.1, <cit.>  for full details, but mention that $\fq(n)$ is constructed as a subalgebra of $\fgl(n|n)$, and has derived
subalgebra $\sfq(n)$. We have $\fq(n)_0 \cong \fgl(n)$ and  $\sfq(n)_0\cong \fsl(n)$.  The algebras $\fq(n)$,  $\sfq(n)$ have factor algebras
and $\psq(n)$ respectively.  The algebra $\psq(n)$ is simple for $n\ge 3$.
<cit.> Theorem 13.1.

Z((n))≅{ f ∈S()^W|  f|_x_n =- x_n-1=t  t} := I_n.
Now (<ref>)   shows that if $T=\prod_{i<j} (x_i + x_j)$, then $T\in \A = I_n$. Set  $\B =  S(\fh)^W$.  Define
f $\ev: I_{n}\to I_{n-2}$ by $\ev(f) = f|_{x_n =- x_{n-1}=0}.$
We have
$T\B\subset I_{n} $ and the
kernel of $\ev: I_{n}\to I_{n-2}$ equals
We have $\ev(I_{n}) =I_{n-2}$
(a) follows as in  the proof of Theorem <ref>  (a).  For (b) see <cit.>  Proposition 5.8.3.
To unify notation, we set $I(\fh) =I_{n}$ if $\fg=\fq(n)$. In this case, by
<cit.> Lemma 12.1.1.
I()= {f∈S(𝔥)^W| f(λ) = f(λ+ t α) ∈^+, ∈Π_  t ∈}

§.§.§ Algebras of type $\fp$.

We are interested in
two Lie superalgebras of type  $\fp$. We refer to <cit.>

for details, but recall  that $\fp(n)$ is constructed as a subalgebra of $\fgl(n|n)$, and has derived
subalgebra $\sfp(n)$. We have $\fp(n)_0 \cong \fgl(n)$ and  $\sfp(n)_0\cong \fsl(n)$.
The algebra $\psq(n)$ is simple for $n\ge 3$.
If  $\fg =\fp(n)$, then by <cit.> Theorem 4.1, $ Z(\fg)$ contains
an element $t$ such that  $t^2=0$ and there is a linear isomorphism  $Z(\fg) \cong  \ttk \op tS(\mathfrak{h})^W$.  This means that $Z(\fg)$ has a unique prime ideal and the methods of Subsection <ref> do not apply to the study of $Z(\fg)$.  For this reason $\fp(n)$ is  excluded in some results. Corollary <ref> holds when  $\fg=\fp(n)$, since there is only one central character.  The ring
$J(G)$ when $G$ is the Lie supergroup $P(n)$ remains of interest.

§.§ Application to $J(G) \ot_{\Z}\ttk$.

§.§.§ Lie supergroups

We make a  basic hypothesis, compare <cit.> Section 8.
Assume one of the following holds
If  $ \fg = \fgl(m|n) $, set $G_0=GL(m)\ti GL(n) $.
If  $ \fg = \osp(r|2n)$, set  $G_0=SO (r)\ti SP(2n) $, where $r=2m+1$ or $2m$.

In  cases (i) and (ii) set
P= ∑_i=1^m _i ∑_i=1^n _i.If  $ \fg = \fsl(m|n) $ $m\neq n$, set $G_0=\{(A,B)\in GL(m)\ti Gl(n)| \det A = \det B \} $ and
let $P$ be the root lattice of $\fg$.

If  $ \fg = \fp( n) $, set $G_0=\{(A,A^{-t})\in GL(n)\ti GL(n)\} $.
If  $ \fg = \fq(n) $, set $G_0=\{(A,A)\in GL(n)\ti GL(n) \} $.

In (iv) $A^{-t}$ is the transpose of $A^{-1}$. Suppose  (iv) or  (v) holds.  The    simple roots are
$\ga_i =\gep_i - \gep_{i+1}, i \in [n]$.  We use the bilinear form on
$ \fh^*$ given by
$(\gep_i,  \gep_{j}) =\gd_{i,j}$ and set $P = \sum_{i=1}^{n}\Z\gep_i$.
Recall that  $\mathbb{T}$ is a maximal torus in $G_0$.
There is a
Harish-Chandra  pair $(G_0,\fg)$ and the Lie supergroup determined
by (<ref>) is
$G=GL(m|n),  OSP(r|2n),  SL(m|n), P(n) $  or $Q(n)$ in  cases $($i$)$-$($v$)$ respectively.
There is an isomorphism  $P \lra X(\mathbb{T})$, which we write as $\ga\lra \tte^{\ga}$.
Here $\tte$ is a formal symbol used to convert to multiplicative notation.
The possible weights of

$G_0$-modules (and hence also $G$-modules) are determined by $P$. Thus the result follows from the explicit description of the characters or supercharacters of these modules given in <cit.>, <cit.> and <cit.>.
If $ \fg = \fp( n) $ or   $  \fq(n) $, and  $\ga=\gep_i - \gep_{j} \in \Gd_0^+,$ the set of  positive even roots, set  $\overline{\ga}=\gep_i + \gep_{j}$ and
define in place of (<ref>),
{ λ∈^* | (λ,) =0}.
In the KM case, an iso-set is a linearly independent set of mutually orthogonal isotropic roots. If $ \fg = \fp( n) $ or   $  \fq(n) $, an  iso-set is  a linearly independent subset $A$ of $\Gd_0^+ $ such that $(\ga,\overline{\gb}) =0$ for all $\ga, \gb\in A$.
The non-standard convention that $\Gd_{iso} = \Gd_0^+ $ in types p and q, allows us to give unified statements of certain results.
Let $P\subset \fh^*$ be as in Hypothesis <ref>. For all $\ga \in P$, define the subtorus $\ttT_\ga $
of  $\mathbb{T}$ by

_= ^    _= ^  Composition gives a non-degenerate pairing
$X(\mathbb{T}) \ti Y(\mathbb{T}) \lra \Hom(\ttk^*,\ttk^*) \cong \Z.$  The next result relates this pairing to the bilinear form on $P$.

For each  root $\gb$
there is a unique $c_\gb \in Y(\mathbb{T})$ such that
^c_(t)= t^(,)   t ∈^*.First suppose   $G=GL(m|n) $ or
$    OSP(r|2n), $ with $r = 2m+1$ or $2m$ and
let $p=m+n$. We show (<ref>) holds for all $\gb\in P$.
It is convenient to relabel the ordered basis of $P$ as follows

$$(\ga_1,\ldots,\ga_p):= (\gep_1,\ldots,\gep_m, \gd_1,\ldots,\gd_n).$$
Then the  dual basis is

$$(\varpi_1,\ldots,\varpi_p):= (\gep_1,\ldots,\gep_m, -\gd_1,\ldots,-\gd_n).$$

There are unique $c_1,\ldots,c_p  \in Y(T)$ such that $\tte^{\ga_i}c_j(t)= t^{\gd_{i,j}}$.
If $ \gb \in P$, write $\gb= \sum_{i=1}^p a_j
\varpi_j$ with $a_j\in\Z$.  Then set $ c_\gb = \prod_{i=1}^p c_j ^{a_j}$. It is enough to show (<ref>) for $\ga=\ga_i$, but in this case we have, for $t \in \ttk^*$ we have

$$\tte^{\ga_i}c_\gb(t)= t^{a_i} =t^{(\ga_i,\gb)}.$$

If   $G=SL(m|n) $ the distinguished set of simple roots is

\begin{eqnarray*}
\alpha_i & = &  \epsilon_i - \epsilon_{i + 1}, \quad\quad i \in [m - 1]\\
\alpha_{m} & = & \epsilon_{m}  - \delta_{1}, \quad\quad   \\
%\alpha_{n-1} & = & \delta_{n-1} - \epsilon_1 \\
\alpha_{m+i} & = & \delta_i - \delta_{i+1}, \quad\quad   i \in   [m - 1]
\end{eqnarray*}

Set $x_{i} = \tte^{\epsilon_i}, y_{i} = \tte^{\delta_i}$. Then for a simple root $\ga$,
$\tte^\ga =x_{i}x_{i+1}^{-1}, x_{m}y_{ 1}^{-1}$
and $y_{i}y_{i+1}^{-1}$ in the three cases listed above. For a root $\gb$ define
$c_\gb \in Y(\mathbb{T})$, by

$$c_\gb(t) = (t^{(\gb, \gep_1)}, \ldots , t^{(\gb, \gep_m)}, t^{(\gb, \gd_1)}
, \ldots ,  t^{(\gb, \gd_n)}).$$
It suffices to show (<ref>) for $\ga, \gb$ for simple roots and this is done with a short computation.
$G=P(n) $  or $Q(n)$, then for all  $\gb\in \Gd_0^+$
there is a unique $c_\gb \in Y(\mathbb{T})$ such that
^c_(t)= t^(,)  ∈_0^+
t ∈^*.Define $c_\gb \in Y(\cT)$ by $c_\gb(t)=(1, \ldots, t, \ldots , t^{-1}, \ldots ,1)  $, where $t, t^{-1}$ are in positions $i$ and $j$.  The result holds by a short computation.
In the KM case, it was shown by Hoyt and Reif, <cit.> Proposition 8, that the functor
$DS_x$ induces a ring homomorphism $ds_x:J(G)\lra J(G_x).$
Our next aim is to describe the image and kernel of the map $ds_x$, and deduce that the inclusion
$\A=J(G) \ot_{\Z}\ttk \subseteq \B=\Z[X(\mathbb{T})]^W \ot_{\Z}\ttk
$ satisfy Hypothesis <ref>.  Though not strictly necessary for all the results we cite, we extend scalars to $\ttk$. This brings out the analogy with $I(\fh)$. Note that
$\Z[ X(\cT)]\ot_\Z \ttk=\cO(\cT).$

§.§.§ Properties of  $ds_x$.
In the the KM case, set
${\gr_\iso}=\frac{1}{2}\sum_{\ga \in \Gd^+_\iso} \ga$.
There is an element $T\in\B$ such that
$\Ker ds_x =T\B\subset \A$. In particular  Hypothesis <ref> holds.
Let $G$ be one of the Lie supergroups $SL(m | n), m \neq n, GL(m | n), $ or
$OSP(r| 2n)$. Then $ds_x: J(G)\lra J(G_x)$ is surjective.(a)  Define $T$ by

$$T = \tte^{\gr_\iso}\prod_{\ga\in \Gd^+_\iso}(1-\tte^{-\ga}).$$
In the notation of <cit.> Lemma 16  we have $T = k({\gr_\iso})$.  So the result follows from the cited Lemma and <cit.>  Theorem 17. Also (b)
is <cit.>  Theorem 20.

§.§.§ The case of  $Q(n)$.

The case   $G=Q(n)$ is studied in <cit.>.
Here are some relevant results. In this and the following Subsection, let  $\mathbb{T}_n$ be a maximal torus in $Q(n)_0=GL(n)$. Then  <cit.>
Proposition 3, gives an explicit description of a ring $J_{n}$  isomorphic to  $J(Q(n))$.  For our purposes it is convenient to extend scalars to $\Z[\frac{1}{2}]$. Thus we define

J_n = { f ∈[1/2][X(𝕋_n)]^W|  f|_x_n =- x_n-1=t  t}.
With this definition $J_n \cong J(Q(n)) \ot_\Z \Z[\frac{1}{2}].$
Using this isomorphism,
$ds_x$ identifies with  an evaluation map $\A = J_{n}\lra J_{n-2}=\A'$, given by $\ev(f) = f|_{x_n = -x_{n-1}=1}$.
Note that  $\A$ is contained in $\B = \Z[\frac{1}{2}][X(\mathbb{T}_n)]^W$.
So we have a diagram as in Subsection <ref>, where the map  $\phi=\ev$, and the map $\Phi:
\B \lra \B' = \Z[\frac{1}{2}][X(\mathbb{T}_{n-2})]^{W'}%\Z[\frac{1}{2}][X(\mathbb{T}_n)]^W
$ is defined by
Φ(f) = f|_x_n = - x_n-1 =1.
The map $ \ev:J_{n}\lra J_{n-2}$ is surjective.
$T \in \B=\Z[\frac{1}{2}][X(\mathbb{T}_n)]^W$, by $T= \prod_{i<j} (x_i + x_j)$.  Then
$ T\B \subset \A$. That is Hypothesis <ref> holds. Furthermore $\Ker \ev = T\B.$
For (a) see <cit.> Proposition 7.  Since clearly $T$ is fixed by $W$ and $ \ev(T)=0,$ we have
$ T\B \subset \A$ and $ T\B \subseteq \Ker \ev.$
The other inclusion in (b) follows from the proof of <cit.> Proposition 6.
Without  extending  scalars, we would not have $ T\B \subset \A$.

§.§.§ The case of  $P(n)$.

The case of  $G=P(n)$ is studied in <cit.>. We define

J_n = { f ∈[X(𝕋_n)]^W|  f|_x_n =x_n-1^-1=t  t}.
By  <cit.> Theorem 1.0.1 $J_n \cong J(P(n)),$ the super character ring of $P(n)$. As before this allows us to identify
$ds_x$  with  an evaluation map $\A = J_{n}\lra J_{n-2}=\A'$, given by $\ev(f) = f|_{x_n = x_{n-1}^{-1}=1}$.
Note that  $\A$ is contained in $\B = \Z [X(T_n)]^W$. So we have a diagram as in Subsection <ref>, where the map  $\phi=\ev$, and the map $\Phi:
\B \lra \B' = \Z[X(\mathbb{T}_{n-2})]^{W'}
$ is defined
by $\Phi(f) = f|_{x_n = x_{n-1}^{-1}=1}.$

The map $ \ev:J_{n}\lra J_{n-2}$ is surjective.
Define $T \in \B=\Z [X(\mathbb{T}_n)]^W$, by $T= \prod_{i<j} (1-x_i x_j)$.  Then
such that $ T\B \subset \A$. That is Hypothesis <ref> holds. Furthermore $\Ker \ev = T\B.$
For (a) see <cit.> Theorem 4.0.1.   Statement (b) follows from the proof of <cit.> Proposition 3.2.1.

§ THE  NULLSTELLENSATZ.

In this Section we assume one of the following holds, compare Notation <ref>
$\fg\neq\fp(n)$, $\A= I(\fh), \;
X = \fh^* \mbox{ and } \B= \cO(X)^W =S(\fh)^W.$
$\A= J(G) \ot_{\Z}\ttk, \;  X = \mathbb{T}, %\mbox{ and }
\B= \cO(X)^W$ and $G$ is not exceptional KM.

§.§ Maximal Ideals.

First we prove the analog of the weak Nullstellensatz.

If $m$ is a maximal ideal of  $\A$, then there is a maximal ideal $M$ of  $\B$ such that $m = M \cap \A$.We give the details in case (i).
If $T\notin m$ this follows from Corollary <ref>. Assume is KM   and let ${W_x}$ be the Weyl group of ${\fg_x}$.
By induction the conclusion holds with $(\A,\B)$ replaced by the pair $(I(\fh_x), S(\fh_x)^{W_x}).$  Thus by
Lemma <ref> it holds also for  $(I(\fh_x)^{\gs}, S(\fh_x)^{W_x,\gs})$ where $\gs=\gs_x$ is as in Theorem <ref>.

Thus if  $T\in m$, the result holds by Lemma <ref>, taking  $M$ to be any maximal ideal containing $m\B$. The same argument works for  $\fg=\fq(n)$ using Theorem <ref>.  The proof in case (ii) is similar.

If  $\gl \in X$, set

$$m_\gl = \{ f\in \cA| f(\gl)=0\}.$$

If  $m$ is a maximal ideal of $\A$, then $m = m_\gl$ for some   $\gl\in X$.
This follows from Theorem <ref>.   Note that  maximal ideals in $\B$ correspond to $W$-orbits in $X$.
For $\gl \in X,$ let $M_\gl\in \Max \cO(X)$ be the  ideal of functions  vanishing at $\gl$.

We have There are surjective maps
$$\Max\cO(X) \lra \Max \B  \lra \Max \A.$$

Denote the composite of the maps in $(a)$ by $\gs$ and suppose that $\sim$ is
the smallest equivalence relation on $X$ such that
$w\lambda \sim \lambda$ for all $w \in
If $X=\fh^*$ and  $(\lambda , \alpha) = 0$ for $\ga\in \Gd_{iso}$, then $\lambda \sim \lambda + t\alpha$ for all $t \in \ttk.$
If $X=\cT$ and  $\lambda \in \ttT_\alpha $ for $\ga\in \Gd_{iso}$,  then $\lambda \sim
\lambda  $ for all $t \in \ttk.$

Then $ M_{\mu } \in\gs^{-1}(m_\gl)$ iff $\lambda \sim \mu.$ In (a) the map $\Max \B\lra \Max \A$ is  given by $M\lra  M \cap \A$, and the other map is defined similarly.  The first map is surjective since $
\B$ is the fixed ring of  $\cO(X)$ under the action of the finite group $W$.  The second map is surjective  by Theorem <ref>. If $ X = \fh^* $, part (b) is a reformulation of <cit.> Theorem 13.5.4, and the proof when $ X = \cT $ is similar.

§.§ The Strong Nullstellensatz.

We deduce the strong Nullstellensatz from Theorem <ref>. There is a small problem because we need the analogous result when the pair $(\A, \B) $ is replaced by $(\A \ot_\ttk {\ttk[z]}, \B\ot_\ttk {\ttk[z]})$. This is easily taken care of by noting that the description of
$\A$  given by  (<ref>) and (<ref>) is independent of the algebraically closed field, and hence holds also for any field extension $K$ of $\ttk$.
If  $m\in \Max \A \ot_\ttk {\ttk[z]}$, then $m\B\ot_\ttk {\ttk[z]}$  is a proper ideal of $\B\ot_\ttk {\ttk[z]}$. Let $S = {\ttk[z]} \bsk 0$.  If $m \cap S= \emptyset$, then $m_S \in \Max \A \ot_\ttk  {K}$,
$K={\ttk(z)}$, and the result   follows by the above remarks.  Otherwise, we have $z-\gl\in m$ for some $\gl\in \ttk$.  Now if $C= \A$ or  $\B$, set $C_\gl = C\ot_\ttk {\ttk[z]}/(z-\gl).$ Then $\overline{m}:= m/(z-\gl)   \in \Max \A_\gl .$  Now

$$ \A \cong \A_\gl \subseteq \B_\gl \cong \B,$$
so by Theorem <ref>
$\overline{m} \B_\gl$ is a proper ideal of $\B_\gl$.

If $I$ is a subset of $\A$, let
$\cV(I) = \{x\in X| f(x) =0 \mbox{ for all } f \in I\}$.
Such a set is called an superalgebraic set. If instead
$I$ is a subset of $\B$, we say that $\cV(I)$ is an algebraic set
(or closed) in $X$.
Thus any superalgebraic set is algebraic.
In addition if
$V$ is a subset of $X$,
$$\cI_\cA(V) = \{f\in \cA| f(x) =0 \mbox{ for all } x \in V\}.$$
We will also need

$$\cI(V) = \{f\in \cO(X) | f(x) =0 \mbox{ for all } x \in V\}.$$

The maps $\cI_\cA$ and $\cV$ are inverse bijections between the set of superalgebraic sets in $X$, and the set of radical ideals in $\A$. Both maps are order reversing.

The key point is that $\cI_\cA(\cV(I)) \subseteq \rad(I)$. To show  $I$ is not  assumed finitely generated, we  repeat the well-known  “Rabinowitsch trick",
<cit.> Theorem 4.19,
<cit.> Chapter 1. Equation (<ref>) below uses only finitely many elements from $I.$
Suppose $G\in \cI_\cA(\cV(I))$, and let $J$ be the ideal of $\A  \ot \ttk [z]$ generated by $I$ and $zG-1$. Then $\cV(J)$ is empty, since $G$ vanishes wherever all polynomials in $I$ vanish. Therefore by Corollary <ref>, $1\in J$, and we can write
1 = ∑_i=1^r A_i F_i + B(zG-1),where the $F_i $ are in $I$ and $B, A_i \in \cA  \ot \ttk [z].$
Now set $Y=1/z$ and multiply by a large power of $Y$ to obtain

\[Y^N = \sum_{i=1}^r C_i F_i + D(G-Y)\]
where $D, C_i \in \cA \ot \ttk [Y].$
The result follows by setting $Y=G$.

The maps $\cV,$ and $\cI_\cA$ satisfy  the following properties. Suppose that $E_\gl, V_\gl$ are subsets of  $\A$, and
$X$ respectively and that
$\fa, \fb$ are ideals of $\cA.$
$\cV(\cup_{\gl\in \Gl} E_\gl) = \cap_{\gl\in \Gl}\cV( E_\gl)$.
$\cV(\fa \cap \fb) = \cV(\fa \fb) = \cV(\fa) \cup \cV(\fb)$.
$\cI_\cA(\cup_{\gl\in \Gl}V_\gl) = \cap_{\gl\in \Gl}\cI_\cA( V_\gl)$.
Left to the reader.

§.§ Prime ideals and irreducible components.

We say that  a superalgebraic set
is irreducible
if it cannot be written as the union of two proper superalgebraic sets.
If $I$ is a radical ideal of $\A$ and $V = \cV(I)$ is the corresponding superalgebraic set, then $I$ is prime iff $V$ is irreducible as a superalgebraic set.The proof is completely analagous to the classical case <cit.>
Chapter 1.5, Proposition 1, page 15.In general if
$I$ is a radical ideal of $\cA$, then using Corollary <ref>, we can write $I$ uniquely in the form $I =  P_1 \cap \ldots \cap P_r$,
where the $P_i$ are the prime ideals of $\cA$ which are minimal over $I$.
Then $\cV(I) =  \bcu_{i =1 }^{r}  \cV(P_i)$ by Proposition <ref>.
We call the superalgebraic  sets $\cV(P_i)$ the  irreducible superalgebraic  components of $\cV(I)$.  They are irreducible.

Every superalgebraic set is uniquely a finite union of  irreducible superalgebraic components.

This follows  from  the Nullstellensatz and Proposition <ref> (b).

Our approach to the Nullstellensatz was to relate ideals of $\A$ and $\B = \cO(X)^W =\cO(X/W).$ Therefore it might seem more naural to relate radical ideals in  $\A$  to their zero loci in $X/W$. However the connection between closed sets in $X$ and $X/W$ is well understood, when $\cO(X)$ is finitely generated,  <cit.> Section 2.3.1.
If $U$ is a closed $W$-invariant subset of $X$, then $  \cI(U)$ is a  $W$-stable  ideal  of $\cO(X)$. Let $\pi:X\lra X/W$ be the natural map, corresponding to the inclusion of rings $\cO(X)^W \subseteq \cO(X)$.
If $V $ is  closed  in $ X/W$, and $U$ is an irreducible component of $\pi^{-1}(V),$ then so is $wU$ for $w \in W$, and
$\pi(U) = \pi (wU)$. In Section <ref> it will be more convenient to work with closed sets in $X$, rather than  $X/W$. Several  closed subsets of $X $  we consider are identified under $\pi $, see Lemma <ref>.

§ WEYL GROUPOIDS AND THEIR ACTIONS.

§.§ Actions of groupoids on varieties.
A groupoid $\fG$  can be
defined as a small category with all morphisms  invertible.
We denote the set of objects by $\mathfrak B$ which we call the base.
As in <cit.>  we use the same notation $\fG$ for the set of morphisms  as for the groupoid itself.
For a variety  $X$ consider  the  groupoid
$\mathfrak A(X)$ with  base all  subvarieties of $X$, and  morphisms all
isomorphisms between subvarieties. We say the groupoid $\fG$ acts on   $X$
if there is a functor
F:𝔄(X).If $g:b\lra b'$ is a morphism in $\fG$, set $s(g) = b$.
Then for $\gl\in X,$ set $\fB_\gl =\{b\in \fB| \gl\in F(b) \}$ and $\fG_{(\gl)} =\{g\in \fG|s(g) \in \fB_\gl\}.$ If $g\in \fG_{(\gl)}$, we say $g$ is defined at $\gl $ and often write $g\gl=F(g)\gl$.
Set $\fG_{(0)}\gl = \{\gl\}$ and for $i \ge 0$, define
$\fG_{(i+1)}\gl  = \{g\mu|\mu\in \fG_{(i)}\gl,
g\in \fG_{(\mu)}
\}$.  The $\fG$-orbit of $\gl$  is
$\fG \gl =\bcu_{i\ge 0} \fG_{(i)}\gl$. Informally, $\fG_{(i)}\gl$ is the set of points in $X$ we can reach from $ \gl$ using $i$ morphisms from $\fG.$
If $X$ is affine, the invariant ring for the action of $\fG$ on $X$ is

\[ \cO(X)^\fG =\{f \in \cO(X)| f \mbox{  is constant on } \fG \mbox{-orbits}\}. \]
Unlike the  case of a group action on $X$, the  groupoid $\fG $ does not in general act on functions.
The comorphism $\pi: X \lra \Spec \cO(X)^\fG =Y$ is constant on $\fG$-orbits.This follows from the definition of $\cO(X)^\fG$.
For continuous Weyl groupoids the map $\pi$ has some  pleasant properties, see Corollary <ref>.

§.§ Continuous Weyl groupoids and their actions.

In <cit.> Sergeev and Veselov associated a certain groupoid $\mathfrak{W}= \mathfrak{W}(\Gd)$,  which they the call Weyl groupoid,  to the  root system
$\Gd $ of a KM  algebra. They also defined a functor $\mathfrak W \lra \mathfrak A(\fh^*)$. We introduce two closely related  groupoids $\mathfrak{W}^c= \mathfrak{W}^c(\Gd)$ and $\mathfrak{W}^c_*= \mathfrak{W}^c_*(\Gd)$.

We need a preliminary construction, namely
the semi-direct product groupoid $\Gc \ltimes \mathfrak G$.   Let $\mathfrak G$ be a groupoid and  $\Gamma$ a group  acting on $\mathfrak G$  by automorphisms of the corresponding category. In particular, $\Gamma$ acts on the base $\mathfrak B$ of $\mathfrak G$.

Then the  semi-direct product groupoid
$\Gamma \ltimes \mathfrak G$ has the same base $\mathfrak B$, and
is defined in <cit.>.
To define the continuous Weyl groupoids, we first introduce some auxilliary groupoids  $\cG^c$  and $\cG^c_*$.

§.§.§ An action on $\fh^*$.

In the KM case consider the following groupoid $\mathfrak T^c_{iso}$ with base
$\Gd_{iso}^c=\Gd_{iso}\ti \ttk.$

The non-identity  morphisms  are  $\tau_{\alpha, t}:(\alpha, t) \rightarrow (-\ga, t)$. The group $W$ acts on $\mathfrak T_{iso}^c$ in a natural way: $\alpha \rightarrow w(\alpha),\,
\tau_{\alpha,t} \rightarrow \tau_{w(\alpha),t}$.  Let $\cG^c$  be the semi-direct product groupoid $\cG^c =W \ltimes \mathfrak T_{iso}^c$ with base
$\Gd_{iso}^c.$  The continuous Weyl groupoid $\cW^c$ is defined

𝔚^c = W ∐^c,the disjoint union of the group $W$ considered as a groupoid with a single point base $[W]$ and $\cG^c$.
There is an action of $\mathfrak{W}^c$ on  $\fh^*$, given by a functor from $\mathfrak{W}^c$ to $\mathfrak A(\fh^*)$. Here we adapt <cit.>, Section 9.
The base point $[W]$ maps to the whole space $\fh^*$.
The base element  $(\alpha,t)$ maps to the hyperplane $\Pi_{\alpha}=\Pi_{-\alpha}$.
The morphism $\tau_{\alpha,t}$ acts via

τ_α,t(x) = x + tα,   x ∈Π_α.If $ \fg = \fp( n) $ or   $  \fq(n) $, define the groupoid $ \mathfrak T^c$
with base
$\fB^c=\{(\pm\overline{\ga},t)\mid \ga \in \Gd_0^+, t\in \ttk\} $ and  non-identity morphisms
$\gt_{\ga,t}:(\overline{\ga} ,t) \lra (-\overline{\ga} ,t)$ and
$\gt_{-\ga,t}:(-\overline{\ga} ,t) \lra (\overline{\ga} ,t)$ for $\ga \in \Gd_0^+$. Next form the semi-direct product $\cG^c =W \ltimes \mathfrak T^c$ with base $\fB^c$, and define $ \mathfrak{W}^c $ as in (<ref>).
There is an action of
$\mathfrak{W}^c$ on  $ \fh^*$  defined as before, except that in  (<ref>)
for $\ga \in \Gd_0^+$,
$\Pi_{\alpha}$ is given   by
(<ref>). The orbits of
$\mathfrak{W}^c$ on  $ \fh^*$   are described in all cases in Theorem <ref>.  However, we make no use of this action when $ \fg = \fp( n) $, see Subsection  <ref>.

§.§.§ An action on $\mathbb{T}$.

We need a variant   $\mathfrak{W}_*^c$  of $\mathfrak{W}^c$. In the KM case define $\cG_*^c$  in exactly the same as  $\cG^c$, except that the base of $\cG_*^c$  is
$\Gd_{iso}\ti \ttk^*.$ Then set
$\mathfrak{W}_*^c = W \coprod  \cG_*^c$.  There is an  action of $\mathfrak{W}_*^c $ on
corresponding to a functor from $\mathfrak{W}_*^c$ to $\mathfrak A(\mathbb{T})$. As before, the base point $[W]$ maps to the whole space $\mathbb{T}$.   The
base element  $(\alpha,t)$ maps to the subtorus  $\ttT_\ga$ defined in (<ref>).
The morphism $\tau_{\alpha,t}:\ttT_\ga  \lra \ttT_\ga$ acts via
τ_α,t(x) = c_α(t)x,   x ∈_.
$G=P(n) $  or $Q(n)$, The groupoids
$\cG_*^c $ and  $ \mathfrak{W}_*^c $ are defined by analogy with the groupoids for
$ \fp( n) $ or   $  \fq(n) $
except that the base of
$\cG_*^c$ is $\fB_*^c=
\{(\pm\overline{\ga},t)\mid \ga \in \Gd^+, t\in \ttk^\ti \} .$
There is a functor  $\cW_*^c\lra \fA (\cT)$, such that $W=N_{G_0}(\cT)/\cT$ acts in the obvious  way.   The base elements $(\pm\overline{\ga},t)$ map to $\ttT_\ga = \Ker \tte^{\overline{\ga}}$,  and for  $\ga \in \Gd_0^+$
acts via (<ref>).
If $(\ga,\overline{\gb}) =0$, then $(\gb,\overline{\ga}) =0$.
$\ga=\gep_i - \gep_{j},$ and
$ {\gb}=\gep_k - \gep_{\ell}$ where $i<j, k<\ell$.
If  $(\ga,\overline{\gb}) =0$, then
$0=\gd_{i,k} + \gd_{i,\ell}-\gd_{j,k}-\gd_{j,\ell}.$  If $i=k$, then $j=\ell$ and hence
$(\gb,\overline{\ga})=\gd_{i,k} - \gd_{i,\ell}+\gd_{j,k}-\gd_{j,\ell}=0.$ Similar arguments for the cases $i=\ell$ and $i\neq k,\ell$ complete the proof.
Equation  (<ref>) is a multiplicative  analog of (<ref>).
If $\ga\in\Gd_{iso}$ is isotropic,  $ \ttT_\ga$ is a codimension one subtorus in $\ttT_\ga$, and by
(<ref>)  $\Im \; c_\ga \subseteq \ttT_\ga$. Then $\tau_{\alpha,t}$ translates  $x\in \ttT_\ga$ using the one-parameter subgroup $c_\ga.$ In types P and Q, this holds  since $(\ga,\overline{\ga})=0.$

§.§.§ Orbits.
If $\fG $ is the coutinuous Weyl groupoid as in  Notation <ref>, we describe the orbit   $\fG\gl$ of $\gl\in X$.

Define the  degree of atypicality of $\gl$ to be

$$\atyp \; \gl =  \max\{s|(\gl,A)=0 \mbox{ for some iso-set } A \mbox{ with } |A|  = s \} \quad
\mbox{ if  }
$$\atyp \; \gl =  \max\{s|\gl\in \bca_{\ga \in A}\ttT_\ga \mbox{ for some iso-set } A \mbox{ with } |A|  = s \} \quad
\mbox{ if  } X=\cT.$$

We say $\gl $ is typical if  $\atyp \; \gl =  0.$  Next set

$$E(\gl) = \{\ga\in \Gd_{iso}|\gl\in \Pi_\ga\}, \quad \ttE_\gl=\bcu_{w\in W} w[\bcu_{\ga\in E(\gl)}(\gl + \ttk\ga)]$$

$$E(\gl) = \{\ga\in \Gd_{iso}|\gl\in \ttT_\ga \}, \quad \ttE_\gl=\bcu_{w\in W} w[\bcu_{\ga\in E(\gl)}(\Im c_\ga\gl )],$$
$X=\fh^*$ or $X=\cT$ respectively.
We need a preliminary result.
We have

Clearly $\ttE_\gl \subseteq \fG{\gl}.$ We show
that  for $\mu\in \fG_{(i+1)} \gl,$
we have $\ttE_\mu =\ttE_\gl $.
$\mu  = g\nu$ where $\nu\in \fG_{(i)}\gl$ and
$g\in \fG_{(\nu)}$. By induction $ \ttE_\nu =\ttE_\gl$. Thus we can assume that $\nu =\gl$.
Since $\ttE_{\gl}$ is $W$-invariant, we  may  further assume $g=\gt_{\ga,t} $.
The action of $g$ is given by (<ref>) or
(<ref>), so since  $g$ is defined at $\gl $, $\gl \in \Pi_\ga$ or $\gl \in \ttT_\ga$.
$\ga\in E(\gl)$ and this   gives the result.
If $\gl\in\fh^*$, the  proof shows that $\cW{\gl} =\bcu_{w\in W} w[\bcu_{\ga\in E(\gl)}(\gl + \Z\ga)],$  which has closure $\cW^c {\gl}$.
This explains why  our primary interest is in orbits of
continuous Weyl  groupoids.
To give a definitive description of these  orbits we need two more Lemmas.
Let $F(\gl)\subseteq   E(\gl)$  be an iso-set   with $|F(\gl)| =\atyp\;\gl$, and set $G(\gl)
If $\gb\in E(\gl)\bsk F(\gl)$, then for some  reflection $u\in W$ we have
$u\gl = \gl$ and  $u\gb  = \pm\ga  \in G(\gl).$

See <cit.> Equation (14). For convenience we give a short proof.
In the KM case, we can  argue as follows. There is some $\ga\in F(\gl)$ such that $(\ga,\gb)\neq 0$.  The Jacobi identity applied to the three root vectors $e_{\pm\ga}\in\fg^{\pm\ga}$ and
$ e_\gb\in\fg^{\gb}$ implies that one of $[e_\gb,e_{\pm\ga}]$ is  non-zero.  Replacing ${\ga}$ by ${-\ga}$ if necessary, we can assume ${\gc=\gb}-{\ga}$ is an (even) root.
from the rank two case, <cit.> Table 3.4.1  we see that $\gc$-string through $-\gb$ consists of $- \ga$ and $-\gb$. Thus if $u=s_\gc $ is the reflection corresponding to $ \gc$, we have $u \gb=\ga$.  Since $ (\gl,\gc)=0$,  $u$ fixes $\gl .$
For types $P$ and $Q$ we adapt <cit.>. We have
$(\ga,\overline{\gb})\neq 0$ for some $\ga\in F(\gl)$.
We can assume
$\ga = \gep_i - \gep_j, \gb=\pm(\gep_i - \gep_k) \in \Gd_0^+$, where $i<j\neq i, k$. Then

$$( {\gl},\overline{\gb} ) =   0= ({\gl},\overline{\ga}),$$

and the result holds with $\gc = \gep_j - \gep_k$.
Next set

$$\ttF_\gl=\bcu_{w\in W} w(\gl +\sum_{\ga\in F(\gl)} \ttk\ga) \mbox{ or } \ttF_\gl  = \bcu_{w\in W} w\left(\prod_{\ga\in F(\gl)} \Im c_\ga\right)\gl$$
$X=\fh^*$ or $X=\cT$ respectively. The definitions do not change if $F(\gl)$ is replaced by $G(\gl)$.
We have $\ttF_\gl\subseteq \fG\gl$.By (<ref>) and   Lemma <ref>, if $\alpha,\gb \in F(\gl)$, $s,t \in \ttk$ and
$\gl \in \ttT_\ga \cap \ttT_\gb$ then $c_\alpha(t)\gl \in \ttT_\gb$.
Similar remarks apply to (<ref>) if $\gl \in \Pi_\ga \cap \Pi_\gb$. This implies the statement.  We have

$\fG{\gl} =\ttF_\gl $,
$\dim \fG  \gl=  \atyp\;\gl,$
Every $\fG$-orbit
is closed.
Let $\sim$ be the relation   from Corollary <ref> $($b$)$.
Then ${\mu\sim\gl}$ iff
$\fG \mu= \fG\gl$.
If  $m_\gl =M_\gl \cap \A$ as in Corollary <ref>, then
$\cV(m_\gl)=\fG  \gl$.
By  Lemmas <ref>  and <ref>
$ \ttF_\gl\subseteq  \ttE_\gl =\fG  \gl$.
We  show   $ \ttE_\gl\subseteq  \ttF_\gl.$
If $\gb\in E(\gl)\bsk F(\gl)$, let $u $ be as in Lemma <ref>.  Then   $u(\gl +\ttk\gb) =
(\gl + \ttk\ga) $ and $u (c_\gb(t) \gl) = c_\ga( t)\gl$ in the two cases under consideration. This  shows   (a)
and  (b), (c) are immediate consequences.
By definition  $ \sim$ is the smallest equivalence relation satisfying
the conditions in Corollary <ref> (b).  Hence if ${\mu\sim\gl}$ we have
$\fG\mu= \fG\gl$. Conversely if $\fG\mu= \fG\gl$, then
${\mu\sim\gl} $ by Lemma <ref>,  proving (d).
From Corollary <ref> and  (d) we have
μ∼  μ= .
Since $\gl\in\cV(m_\gl)$ this implies $\cV(m_\gl) \supseteq\fG  \gl$.
For the other inclusion,  if $\mu\in \cV(m_\gl)$, then
$m_\mu = \cI_\A(\mu ) \supseteq   m_\gl$, and equality must hold. Thus
$\mu\in\fG  \mu=\fG  \gl,$ proving (e).  We interpret some of our results geometrically.
The map $\pi$ from Lemma <ref> restricts to the  map

ψ:(X) A. as in (<ref>).
The map in (<ref>)  is surjective.
Identifying $X$ with
$\Max \cO(X)$, the fiber over  $m_\gl$ is   $\fG \gl.$
(a) is a restatement of the Weak Nullstellensatz, Corollary <ref> (a) and (b) follows from (<ref>).

Assume Hypothesis <ref> holds. Then
$\cO(\cT)^{\cW_*^c} =J(G)\ot_\Z \ttk.$
$S(\fh)^{\cW}=S(\fh)^{\cW^c} = I(\fh) $ if $\fg  \neq \fp(n)$.
By (<ref>), (<ref>), (<ref>),   and
<cit.> Lemma 3.1,

if $f\in \cO(\cT) $, then $f \in J(G)\ot_\Z \ttk$ iff $f(\lambda) = f(c_\ga({t})\lambda)$ for all relevant roots $\ga$, $t\in\ttk$ and all $\gl \in \ttT_\ga$. Equivalently $f\in \cO(\cT)^{\cW_*^c}.$ This shows (a) and the proof of (b) is similar using (<ref>).
The first
equality in (b) holds because any polynomial that is constant on infinitely many points on a  line is constant on the whole   line.

§ THE GEOMETRY OF SUPERALGEBRAIC SETS.

§.§ Outline

Let   $ \fg  $ be a KM Lie superalgebra and let $G$ be the corresponding supergroup as in Lemma <ref>.

We use Notation <ref>.
We define the S-topology on $\Spec \cO(X) $ by declaring that the S-closed sets are the $\fG$-stable  (Zariski) closed sets. To give an idea of the  geometry of superalgebraic sets, we show that they are  are precisely the the S-closed sets.
We also compute the $S$-closure $V^S$ of an arbitrary closed set $V=V_0=\cV(I)$ with $I$ a $W$-invariant radical ideal in $\cO(X) $. From $V$ we obtain a superalgebraic set which we call  the heart  and another closed set $V_1=\cV(I_1)$ with $I_1$ a $W$-invariant  radical ideal of $\cO(X) $. The process can be repeated giving new closed sets $V_i$ and new hearts. Then  $V^S$ is the union of all the hearts and also the union  of all the  $V_i$. The hearts are determined by the minimal degrees of atypicality in the Zariski  irreducible components of the $V_i$. A somewhat simplified version of this procedure and some examples are given in <cit.>.
If the defining equations for $V$ are known, it can be done algorithmically using Groebner bases, see

<cit.> Example 2.11. The heart is defined in the next subsection, and then $V^S$ is analyzed using projections onto intersections of hyperplanes corresponding to iso-sets.

$$\atyp \;V = \max\{s| \mbox{ for some } \gl\in V, \atyp \; \gl = s \}.$$

Now suppose $r=\atyp \;V$ and choose an iso-set  $S=\{\gb =\gb_1, \gb_2, \ldots, \gb_r\}$ with  $(\gl,S)=0$ for some $\gl\in V$. Since we can replace $\ga \in S$ by $-\ga$ if necessary, we can assume  $S\subseteq \gO.$

§.§ The heart

Suppose $I = \bca_{i=1}^\ell P_i$, an irredundant intersection  of prime ideals of $\B$,  where $T\notin P_i$ iff $i\in [k]$. Equivalently $i\in [k]$ iff  $\cV(  P_i)$ contains typical points.  For
$i\in [k]$, set $Q_i = P_i\cap \A$. Then set
V=_0 V = (_i=1^k Q_i),   V_= (_i=1^k P_i). If $k =0$, set $\heart \; V  = \emptyset.$
Note that $\heart \; V$ is superalgebraic, since $ \bca_{i=1}^k Q_i$ is an ideal of $\A$.
Set $\Vc = \cV(I,T)$.
For $i \in [k]$,
$\cV(P_i) \subseteq \cV(Q_i)$ and
$V_\reg   \subseteq \heart \; V$.
If  $\gl \in \cV(Q_i)$, and $T(\gl)\neq 0$, then $\gl \in \cV(P_i)$.
If  $\gl \in \heart \; V$, and $T(\gl)\neq 0$, then $\gl \in V_\reg $.
$\heart \; V\bsk V_\reg  \subseteq \Vc$.
If $i>k,$ then  $\cV(P_i)  \subseteq \Vc.$
(a) is clear since the
$\cV(P_i)$ and $ \cV(Q_i)$
are the irreducible components of
$V_\reg $ and $\heart \; V$ respectively.  Next if  $f \in P_i$ we have $T^bf \in  Q_i$ for some $b\ge 0$. So if $\gl \in \cV(Q_i)$, and $T(\gl)\neq 0$ we obtain  $f(\gl)=0,$ proving (b)  and (c) is proved similarly.   By (c), if $\gl\in \heart \; V\bsk V_\reg$, then $T(\gl)=0$ and (d) follows from this. Finally (e) follows since $T\in P_i$ for $i>k$.Note that
V = V_∪⋃_i>k (P_i).

\[ V = \heart \; V \cup V^c.\]

The inclusion $\subseteq$  follows from (a), (e) in the Lemma and (<ref>).  The
opposite inclusion results from  $V^c\subseteq V$, (d)  in the Lemma and (<ref>).

§.§ Analysis using projections

If  $q:X\lra Y$ is a morphism of affine varieties, define the comorphism $q^*:\cO(Y) \lra \cO(X)$ by $(q^*f)(x) =(fq)(x)$ for all $f\in \cO(Y) $ and $x\in X$. It is easy to check the following.

Let $i:Y\lra X$ be an embedding of affine varieties and suppose $q:X\lra Y$ is a morphism such that $qi= \id_Y$.  If $I=\ker i^*$, then
$$I \cap q^*\cO(Y) =0 \mbox{ and }
I +q^*\cO(Y) =\cO(X). $$
Furthermore, if $V$ is the closed subset of $Y$ defined by the radical ideal $K$ of $\cO(Y)$, then as a closed subset of $X$ we have $V =\cV(q^*K). $

Let $X= \fh^*$ and $X(\gb)=\fh(\gb)^*$.
Define $q_\gb: \fh^*  \lra \fh(\gb)^*$
by restriction $q_\gb(\gl)= \gl |_{\fh(\gb)}$,  and let  $i_\gb: \fh(\gb)^* \lra \fh^*  $ be the map induced by the inclusion $\Gd(\gb)\subset \Gd$.  The map  $q_\gb$ can also be described as follows.  IAN pink notebook
Let $\{\gep_i\}_{i=1}^p$ and $\{h_i\}_{i=1}^p$ be dual bases for $\fh(\gb)^*, \fh(\gb)$.
Extend the $h_i$ to functionals on $\fh^*$ that vanish on the orthogonal complement of
$ \fh(\gb)^*$ in $\fh^*$, see (<ref>).
Then for all $\gl\in \fh^*$ we have
$q_\gb(\gl)= \sum_{i=1}^{p}h_i(\gl) \gep_{i}$

If $X=\cT$,  let $X(\gb)=\cT(\gb)$ be the subtorus of $X$ generated by $\Im c_\ga$ for $\ga\in \Gd(\gb)$.
Let $\{\chi_i\}_{i=1}^p$ and $\{c_i\}_{i=1}^p$ be bases for $X(\cT(\gb)), Y(\cT(\gb))$ respectively such that ${\chi_i}c_j(t)= t^{\gd_{i,j}}$.  Extend the $c_i$ to $Y(\cT)$ by analogy with the previous example.  Then define $q_\gb: \cT \lra \cT(\gb) $ by $q_\gb(t)= \prod_{i=1}^p   c_i(\chi_i(t))$.
Now for $X= \fh^*$ or $\cT$,
and $\gb$ an isotropic root, define $X(\gb)$ as in the Examples. Define  $p_\gb=i_\gb \ci q_\gb$.  We have

Π_=()^* +  _=()c_.For  the torus case, $\cT(\gb)\subset \ttT_\gb$
by Lemma <ref>
and equality holds because
$\ttT_\gb$ is  the subtorus generated by $\Im c_\ga$ for $\ga\in \Gd$ with $(\ga,\gb)=0$.

If $\gl\in \Pi_\gb$, then $\gl-p_\gb(\gl)\in \ttk \gb.$
If $t\in \ttT_\gb$, then $tp_\gb(t)^{-1}\in \Im c_\gb.$
We prove (a).  The proof of (b) is similar.  By (<ref>)
we can write $\gl=  \gl_0 + c\gb$ with
$\gl_0 \in \fh(\gb)^*$, $c\in \ttk.$ Since
$i_\gb\ci q_\gb$ is the identity on $\fh(\gb)^*$ and $q_\gb(\gb)=0$, we obtain the result.
The action of $W$ on $\cO(X)$ is given by $(wf)(x) = f(w^{-1}x)$ for $w\in W, f\in \cO(X)$ and $x  \in X$.  Then $w  \Gd(\gb)= \Gd(w\gb)$, and we have
$w X(\gb)= X(w\gb).$ Since $W$ acts transitively
on $\gO$ we may assume $\gs(w\gb)=w\gs(\gb)$ in (<ref>).  Then

wq_()=q_w(w).We have an embedding $q_\gb^*:\cO(X(\gb))\lra \cO(X)$
as in Lemma <ref> with $Y=X(\gb)$ and $q=q_\gb.$ In what follows we suppress

Set $X_\gb= \Pi_\gb$ or $\Pi_\gb=\ttT_\gb$
in  case (i) or (ii) of <ref> holds

For $\gb\in \Gd_{iso}$, set  $V_\gb =V\cap X_\gb$.

Since $T(\gl)=0$ for $\gl \in\Vc $, we have
= ⋃_∈
V_.Next we study  the sets $ V_{\gb}$ in more detail.
If $\gb = \gb_1$, set $ U_{\gb}= q_{\gb}(V_{\gb})\subseteq X({\gb})$ and

K_   =     (V_) ∩(X()), We denote the closure of $U\subseteq X$ by $\overline{U}$.

For all ${\gb\in\gO } $, in case $($i$)$ $V^S$ contains the sets
Y_:= U_ + =U_ +.
If $\gl\in V_{{\gb }}$, then by Lemma <ref>, $\gl-p_{\gb}(\gl) =a{\gb }$ for some $a\in \ttk$.
By $\cW^c$-invariance, $V^S$ contains $ \gl +(t-a){\gb } = p_{\gb}(\gl) + t{\gb }$ for all $t\in \ttk.$ For the second equality in (<ref>), note that $\overline{U_{\gb}} \cap \ttk\gb=0.$
Similarly in case (ii) $V^S$ contains the sets

$$ Y_{\gb}:= \overline{U_{\gb} \ti \Im c_\gb} =\overline{U_{\gb}} \ti \Im c_\gb.$$

Next if $X= \fh^*$ set $z_{\gb} = h_{\gs(\gb)}$, and   if $X=\cT$
set $z_{\gb}= \tte^{\gs(\gb)}-1$.  Then
let $ L_\gb$ be the ideal of $\cO(X)$ generated by $K_\gb$ and $z_{\gb}$.

We have $\overline{U}_{\gb}=\cV(K_\gb)$ and $Y_{\gb}  =\cV(L_{\gb})$.
The first statement follows  by <cit.>, Theorem 3.2.3, page 122 and the second is an immediate consequence.
The ideal $K_{\gb}$ is called an elimination ideal.
Next set  $V_1=\bcu_{\gb\in\gO} Y_\gb$. Then  $V_1=\cV (I_1)$ where $I_1=\bca_{\gb\in\gO} L_\gb$.  Because  $I=\cI(V)$  is a radical ideal, so too are $K_{\gb}, L_{\gb}$  and $I_1$.  By Lemma <ref> we have

The set $V^S$ contains $\heart \; V\cup   V_1$.  So, $V^S=\heart \; V\cup   V_1^S=  V\cup   V_1^S.$The next result shows that $V_1$ and $I_1$ are $W$-invariant.
If $V$ is $W$-invariant, then  for $w\in W,$ we have
$w(U_{\gb})= U_{w\gb} $ and $w(Y_\gb)=Y_{w\gb}$
$w(K_\gb\cO(X))=K_{w\gb}\cO(X)$ and  $w(L_\gb)=L_{w\gb}.$ The first statement in (a) follows from $W$-invariance of the bilinear form $(\;,\;)$, the second from (<ref>) and the third is an immediate consequence.

By Lemma <ref>, the closed sets defined by  $w(K_\gb\cO(X))$ and $K_{w\gb}\cO(X)$ are $w(\overline{U}_{\gb})$ and $ \overline{U}_{w\gb} $.  Since these are equal, the defining ideals are equal.  This gives the first part in (b) and the second also follows.
Now we repeat this process.
Set  $A=A(q)=\{\gb_1,\ldots,\gb_q \} $ and
$V_A= V\cap \bca_{\gb\in A} \Pi_\gb$ in case (i) or
$V_A= V\cap \bca_{\gb\in A} \ttT_\gb$  in case (ii).

Define  $p_{A}: X\lra X$ by $p_{A} = p_{\gb_q}\ci \ldots \ci p_{\gb_1}.$
Next, if  $X_A=\bca_{\gb\in A} X(\gb)$, then
$\cO(X_A)=\bca_{\gb\in A} \cO(X(\gb)).$
Consider the elimination ideal

K_A   =     (V_A) ∩(X_A).Let $ L_A$ be the ideal of $\cO(X )$ generated by $K_A$ and $z_{\gb_1},\ldots, z_{\gb_q}.$

If $ U_{A}= p_{\ga}(V_{A})$, then  $\overline{U}_{A}=\cV(K_A)$.

For all $ q\le r$, $V^S$ contains  the sets
{ p_A(q)() + ∑_∈A(q)|∈V_A(q) }. { p_A(q)()
∏_∈A(q) c_
|∈V_A(q) }  (ii).Hence  $V^S$ also contains  the closures $  Y_{A(q)}=\cV(L_{A(q)})$ of these sets.
This follows as in the proof of Lemmas <ref> and <ref>, by $\fG$-invariance.
We define $V_q = \bcu_{w\in W} wY_{A(q)}$ and
$I_q = \bca_{w\in W} wL_{A(q)}$. Then
$V_q = \cV(I_q)$.
By construction if $\gl \in V_q$ then  $\atyp \; \gl \ge q.$
Suppose $I_q = \bca_{i=1}^\ell P_i$, an irredundant intersection  of prime ideals,  and suppose  $i\in [k]$ iff  $\cV(  P_i)$ contain a point $\gl$ with $\atyp \; \gl = q.$  For
$i\in [k]$, set $Q_i = P_i\cap A$. Then set
$ \heart_q V = \cV(\bca_{i=1}^k Q_i).$
If $k =0$, set $\heart_q V  = \emptyset.$

The S-closure of $V$ is
$V^S = \bcu_{i=0}^r\heart_i V= \bcu_{i=0}^r V_i.$  The set $V^S$ is superalgebraic.
As in Proposition <ref> if $q<r$, we have $V_{q}^S=\heart_q  V\cup   V_{q+1}^S =V_q  \cup   V_{q+1}^S$ and $V_{r}^S =\heart_r  V$. This gives the equalities. Each $\heart_i V$ is superalgebraic, so the result follows.
The superalgebraic sets are exactly the closed sets that are invariant under the Weyl groupoid  $\fG$.

If  $V$ is closed and
$\fG$-invariant, then
$V= V^S$ is superalgebraic.  Conversely a superalgebraic set has the form $V = \cV(I)$, with $I$ an ideal of $\A=\cO(X)^\fG$. Since elements of $\A $ are constant on orbits, $V$ is a union of $\fG$-orbits.

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# ISOMETRIC MT: NEURAL MACHINE TRANSLATION FOR AUTOMATIC DUBBING

###### Abstract

Automatic dubbing (AD) is among the machine translation (MT) use cases where
translations should match a given length to allow for synchronicity between
source and target speech. For neural MT, generating translations of length
close to the source length (e.g. within $\pm 10\%$ in character count), while
preserving quality is a challenging task. Controlling MT output length comes
at a cost to translation quality, which is usually mitigated with a two step
approach of generating N-best hypotheses and then re-ranking based on length
and quality. This work introduces a self-learning approach that allows a
transformer model to directly learn to generate outputs that closely match the
source length, in short Isometric MT. In particular, our approach does not
require to generate multiple hypotheses nor any auxiliary ranking function. We
report results on four language pairs (English $\rightarrow$ French, Italian,
German, Spanish) with a publicly available benchmark. Automatic and manual
evaluations show that our method for Isometric MT outperforms more complex
approaches proposed in the literature.

Index Terms—  Machine Translation, Isometric Translation, Automatic Dubbing

##  1 Introduction

Reaching the global audience is a primary factor for audio-visual content
localization. Automating the task of localization requires translation of
source language speech and a seamless integration of the target language
speech with the original visual content [1]. Recent developments in AD [2, 1,
3] have focused on achieving isochrony, a form of synchronization at the level
of speech utterances [4]. The AD architecture proposed in [1], includes an MT
model that translates source transcriptions to a target language, followed by
a prosodic alignment (PA) module which segments the translation into phrases
and pauses according to the prosody pattern in the source speech. Finally, a
text-to-speech (TTS) module synthesizes the target language speech for a final
rendering with the original audio background and visual content.

For a TTS module, to generate a natural sounding speech in synchrony with the
source utterances, it is important that the length of the translated script
should match that of the source.111Following [1], length of input text in
characters is directly proportional to the duration of the TTS utterance. With
a goal of generating translations that match the source length, [5, 6] have
proposed approaches that bias the output length of MT. In a subsequent work
[7], the authors proposed a better approach, where the main idea is to
generate translations that fall within a $\pm 10\%$ range of the source length
in character count. They also confirmed that generating translations within
this range easily allow the PA and the TTS modules to adjust the target speech
to a natural sounding speaking rate. [7] is state of the art approach for MT
with length control, but it depends on a two step process i) $N$-best
hypotheses (where $N$=50) generation and ii) a re-ranking step which
interpolates the model scores and length ratios (translation to source) to
find the best hypothesis in the $N$-best space. In this work, we propose a
self-learning based approach whose improvements are orthogonal to the re-
ranker’s improvements and when combined we achieve state of the art results on
a public benchmark.

In statistical MT, self-learning [8] has been investigated to augment training
data with pseudo bi-text. In neural MT, the most commonly used self-learning
approach is back-translation [9], that leverages a reverse (target
$\rightarrow$ source) model to generate pseudo bi-text from a target language
monolingual data, to train the desired source $\rightarrow$ target direction.
Subsequent works [10, 11, 12], have shown variants of self-learning using a
backward and forward translation settings.

This work proposes a self-learning approach that applies a controlled
generation of pseudo bi-text with a length constraint, to model isometric MT.
To assess the effectiveness of the proposed approach, we perform evaluation on
a speech translation data [13], in four language directions exhibiting
different degree of target to source length ratio: English $\rightarrow$
French, Italian, German, Spanish. Specifically, our contributions are:

  * •

We propose a self-learning based approaches to learn MT model that can
generate isometric translations.

  * •

We compare our proposed approach and show it is on par or better than previous
state of the art for controlling MT output length, without using multiple
hypothesis generation and re-ranking function.

  * •

We introduce a new MT evaluation mechanism leveraging TTS (i.e, generating
audio from translations for rating by subjects), and metrics to measure the
acceptability rate of isometric translations, particularly for an AD use case.

The rest of this work, discusses MT and output length control approaches
$\S$2, followed by a description of our proposed isometric MT $\S$3,
experiment and evaluation settings $\S$4, and finally results and discussion
of our findings $\S$5.

##  2 Background

Fig. 1: MT output length control approaches; (left) current SOTA using length
token+$N$-best hypotheses re-ranking [7], (center) our proposed offline self-
learning using a reverse MT (bottom) trained in isolation to improve the
desired (top) forward MT direction, and (right) the online self-learning
variant using a single bi-directional MT. Broken arrows show test time
hypotheses generation (left), and self-learning training data generation
(center, right). For simplicity, we show embedding lookup function $e()$ for
input tokens, where $e(v)$ for length token, $e(l)$ for language token, and
$e(vl)$ for combination of length and language token.

###  2.1 Neural Machine Translation

For a language pair with parallel data $\mathcal{D}=\\{(s_{i},t_{i})$:
$i=1,\ldots,N\\}$, an MT model parameterized with $\theta$, trains to maximize
likelihood on the training sample pairs as,

$L(\theta)=\underset{\theta}{\operatorname{arg}\,\operatorname{max}}\;\sum_{i=1}^{N}log\
p(t_{i}|s_{i},\theta)$ (1)

The model considers the source context ($s$) and previously generated target
language tokens ($t_{<k}$), when forming the prediction ($t$),

$p(t|s)=\prod_{k=1}^{K+1}\ p(t_{k}|t_{<k},s,\theta)$ (2)

$K$+1 is the length of $t$ with reserved tokens $\langle bos\rangle$ at $k=0$
and $\langle eos\rangle$ at $k$=$|K|$+1 indicating the beginning and end of
the prediction.

###  2.2 Towards Controlling Output Length of MT

Prior to recent studies on MT output length control [6, 14, 7], several
attempts have been made for controlling output attributes: text length in
summarization [15, 16], text complexity and formality [17, 18, 19], domain
information [20], and languages in multilingual MT [21]. Specific to MT output
length control, [6, 14] proposed variants of length-encoding motivated by the
positional-encoding of self-attention [22]. In closely related works, [23, 24]
proposed approaches to respectively control verbosity of speech transcription
and speech translation for subtitling. We will focus on approaches that are
particularly suited for the AD use case.

####  2.2.1 Length Token Embedding

The MT length control approach in [6] first proposed to classify the bi-text
$\mathcal{D}$ into three classes, based on the target to source character
count length ratio ($LR$) of each sample ($s,t$) pair. The class labels are
defined on $LR$ thresholds: short $<0.95\geq$ normal $\leq 1.05>$ long. Then
model training is performed by prepending the length token
$v\in\\{short,normal,long\\}$, at the beginning of the source sentence. At
time of inference, the desired $v$ is prepended on the input sentence.222Both
[6] and [7] stated $v=normal$ as the best setting for generating more suitable
sentence for AD.

In comparison with a baseline length penalty approach to generate shorter
translations [25], and the length-encoding variants of [6, 14], length token
has been shown to generate more suitable translations for AD [7].

####  2.2.2 Hypotheses Re-Ranking

To maximize the suitability of translations for AD, [5] proposed MT hypotheses
re-ranking. The approach works by first generating $N$-best hypotheses ($t$)
for a source $s$, followed by a re-ranking step based on a scoring function
that combines the likelihood of each hypothesis with a so called synchrony
score ($S_{p}$), as follows:

$S_{d}(t,s)=(1-\alpha)\log P(t\mid s)+\alpha S_{p}(t,s),$ (3)

Where $S_{p}$ is computed as $S_{p}(t,s)=(1+|len(t)-len(s)|)^{-1}$, $\alpha$
is a parameter tuned on a validation set.

####  2.2.3 Length Token + Re-Ranking

Recently, [7] proposed to combine the length token of [6] and a variant of Eq.
3 by modifying the synchrony sub-score as,
$S_{p}(t,s)=(1+\frac{len(t)}{len(s)})^{-1}$, which aligns with the overall
objective of reducing target to source $LR$ in MT output length control. Fig.
1 (left), illustrates the approach of [7] which reported a higher $\%$ of
dubbing suitable translations and human preference of dubbed videos.

##  3 Isometric MT with Self-Learning

In this section, we describe our self-learning based output length control for
MT. To show the impact of our approach, we take AD architecture proposed in
[1].

###  3.1 Self Learning for Isometric MT

In MT, the main idea of self-learning is to learn a better model using the
predictions of the model itself or an auxiliary model output. Self-learning
has shown impressive results in unsupervised [26], zero-shot [27, 28], and
semi-supervised [9, 10] neural MT modeling. Despite the success of self-
learning in MT, existing approaches rely on using external monolingual data
and do not evaluate the impact of the pseudo generated data quality before
incorporating it into the training stage.

In this work, the self-learning approach is different in the following
aspects, i) we propose a new synthetic data generation technique based on an
output length control criteria, ii) we then filter and classify the pseudo
data based on predefined length classes. In the sections below, we discuss in
detail two type of self-learning approaches.

###  3.2 Offline Self-Learning

As shown in Fig. 1 (center), the offline self-learning approach considers two
independent MT models. The first one is a reverse target $\rightarrow$ source
model (MTR) trained with the length token approach [7], optimizing the
objective in Eq. 1. The second model is the desired source $\rightarrow$
target forward model (MTF) which is used to generate isometric translations.

Specifically, for parallel training data $\mathcal{D}$ we implement the
offline self-learning by first generating pseudo source $s^{\prime}$ from the
target $t$, using MTR. For inference we prepend $v=long$ on $t$ with the goal
of generating more samples with a lower $LR$. Ultimately, we aim to address
the imbalance of the three length classes in the original bi-text
$\mathcal{D}$.

After performing inference for all $t$, we then construct the pseudo bi-text
$\mathcal{D^{\prime}}=\\{(s^{{}^{\prime}}_{i},t_{i})$: $i=1,\ldots,M\\}$, with
length class label for each sample following the LR and thresholds defined in
[7]. We then concatenate the original $\mathcal{D}$ and the pseudo
$\mathcal{D^{\prime}}$ bi-text to fine-tune the desired MTF model by
optimizing Eq. 1.

###  3.3 Online Self-Learning

Our second approach primarily avoids learning MTR in isolation, as shown in
Fig. 1 (right). We implement a bi-directional model with length classes in
each direction ($v\in\\{short,normal,long\\}$). We prepend a token ($l$)
indicating the source language on the encoder, and target language+length
($lv$) token on the decoder side. For model training, the objective in Eq. 1
is formalized as,

$\displaystyle
L(\theta)=\underset{\theta}{\operatorname{arg}\,\operatorname{max}}\;log\
p(t|s,\theta)+\underset{\theta}{\operatorname{arg}\,\operatorname{max}}\;log\
p(s|t,\theta)$ (4)

During training, we follow a similar three step procedure of the offline self-
learning: length controlled inference, classification of pseudo bi-text into
classes, and model training. Differently from the offline, the steps are
executed on-the-fly at time of model training and the learning procedure is
applied both for the reverse and the forward directions.333Although, we can
evaluate online isometric MT both for source $\leftrightarrow$ target
directions, for a fair comparison with previous work and the offline setting
we focus on the forward direction, and leave the rest for future work. We
hypothesize modeling a bi-directional isometric MT not only able to perform
what normally requires two separate MT models, it also positively re-enforce
output length control in both directions, with the addition of the pseudo
$\mathcal{D^{\prime}}$ bi-text to balance the length classes in $\mathcal{D}$.

For self-learning approaches, based on how the original ($\mathcal{D}$) and
pseudo bi-text ($\mathcal{D^{\prime}}$) is merged, we have two data
configurations, i) Union = $\mathcal{D}\cup\mathcal{D^{\prime}}$, and ii)
Filter = $\mathcal{D}\cup\mathcal{D^{\prime}}$ if LR($t/s^{\prime}$) $\leq$
1.05. In other word, for Filter we remove the pseudo bi-text data portion
labeled $v=long$.

##  4 Experiments

###  4.1 Language and Data Processing

We conduct experiments for English (En) $\rightarrow$ Italian (It), French
(Fr), German (De), Spanish (Es) pairs. To avoid the effect of training with
low-resource data, we pre-train strong MT models on internal data in the
magnitude of $10^{7}$ samples. We then apply our approaches as a fine-tuning
step using public data of Ted Talks speeches of $\approx 200k$ samples per
pair from MuSTC corpus [13]. Across all the approaches compared we use the
same pre-training and fine-tuning data configuration. Moses scripts are used
to pre-process the raw data, followed by token segmentation using
SentencePiece with $32k$ merge operations.444Moses: https://github.com/moses-
smt/mosesdecoder, SentencePiece: https://github.com/google/sentencepiece

###  4.2 Model and Training Configuration

We use a transformer architecture [22] with 6 encoder and decoder layers,
self-attention dimension $1024$ and $16$ heads, and feed-forward sublayers of
dimension $4096$. Model is optimized with Adam [29], with an initial learning
rate of $1\times 10^{-7}$. Dropout is uniformly set to $0.1$ for all model
training. For the online self-learning we share all model parameters between
the forward and reverse directions. For model pre-training the best checkpoint
is selected based on the lowest loss on the validation set after the training
converges. For fine-tuning we train for 5 epochs and take the best checkpoint
for evaluation. At inference time we use beam size of $5$, except for the re-
ranking approach [7] where beam size $=50$.

###  4.3 Model Evaluation

To evaluate both self-learning approaches, we use the human annotated
benchmark from [30, 7]. The test set includes $620$ samples per language pair,
where references are post-edited to match the source length.

####  4.3.1 Automatic Metrics

We report Sacrebleu scores [31] on the de-tokenized translated segments and
reference pair, target to source length ratio (LR) in character count, and the
$\%$ of length compliant ($LC$) translations satisfying the $\pm 10\%$ range.
We also report a single score that takes into account both translation quality
and length compliance, by simply multiplying the BLEU with LC by which we name
LCB.

####  4.3.2 Human Evaluation

To align MT human evaluation with AD, we devise an audio based MT evaluation
scheme. We conduct the evaluation by generating audio for a translated segment
and its reference using a TTS module. For a fair comparison, a uniform
speaking rate is used to generate audio. We then ask annotators to listen to
the audio of the translation and the reference and classify them in three
categories:

Acceptable ($A$) | Meaning is similar, fluency is good.
---|---
Fixable ($F$) | Meaning is similar, fluency is poor.
Wrong ($W$) | Meaning is different.

To assess overall MT quality the $\%$ of $A$, $F$, and $W$ are computed,
whereas to identify length compliant translation we report $LC$ for each
rating category. As a final MT human evaluation score we compute,

$HE_{MT}=LC(A)+\frac{1}{2}LC(F)$ (5)

$HE_{MT}$ disregards the wrong translations and considers the $\%$ of $A$ and
$F$ translations that are isometric. We assume that acceptable translations
will not require any post-editing, while fixable will require minimal post-
editing to fix the fluency as such we weigh it with $0.5$.

##  5 Result and Discussion

We compare proposed approaches against a strong Baseline, and two MT length
control approaches ($v$Tok [6] and $v$Tok+Rank [7]). As described in Sec. 2.2,
$v$Tok+Rank combines length token ($v$Tok) and hypotheses re-ranking to
generate a higher $\%$ of translations suitable for AD [7].

###  5.1 Automatic Evaluation

Pair | Method | SL Data | BLEU | LR | LC | LCB
---|---|---|---|---|---|---
| Baseline | - | 46.1 | 1.16 | 33.1 | 15.2
|  $v$Tok | - | 48.4 | 1.06 | 72.7 | 35.2
En-Fr |  $v$Tok+Rank | - | 47.7 | 1.01 | 91.5 | 43.6
| offlineSL | Union | 47.9 | 1.05 | 81.0 | 38.8
|  | Filter | 47.3 | 1.03 | 83.4 | 39.4
| onlineSL | Union | 48.3 | 1.04 | 88.9 | 42.9
|  | Filter | 48.4 | 1.04 | 88.5 | 42.9
| Baseline | - | 37.1 | 1.06 | 56.5 | 20.9
|  $v$Tok | - | 37.5 | 1.05 | 74.7 | 28.0
En-It |  $v$Tok+Rank | - | 38.5 | 1.00 | 90.0 | 34.7
| offlineSL | Union | 38.0 | 1.04 | 80.0 | 30.4
|  | Filter | 38.1 | 1.03 | 81.6 | 31.1
| onlineSL | Union | 36.2 | 1.04 | 85.2 | 30.8
|  | Filter | 36.5 | 1.03 | 84.7 | 30.9
| Baseline | - | 33.2 | 1.15 | 29.8 | 9.9
|  $v$Tok | - | 34.3 | 1.05 | 79.2 | 27.2
En-De |  $v$Tok+Rank | - | 33.9 | 1.00 | 93.1 | 31.5
| offlineSL | Union | 34.3 | 1.01 | 86.5 | 29.7
|  | Filter | 34.7 | 1.03 | 92.4 | 32.1
| onlineSL | Union | 34.0 | 1.03 | 90.8 | 30.9
|  | Filter | 34.1 | 1.03 | 90.5 | 30.9
| Baseline | - | 50.1 | 1.07 | 56.0 | 28.0
|  $v$Tok | - | 49.9 | 1.04 | 87.6 | 43.7
En-Es |  $v$Tok+Rank | - | 49.3 | 1.00 | 95.5 | 47.1
| offlineSL | Union | 50.0 | 1.03 | 90.8 | 45.4
|  | Filter | 49.9 | 1.03 | 93.1 | 46.4
| onlineSL | Union | 49.9 | 1.03 | 95.6 | 47.7
|  | Filter | 49.8 | 1.04 | 93.9 | 46.7
Table 1: Results of self-learning isometric MT approaches (offlineSL,
onlineSL) in comparison with previously proposed mt output length control
mechanisms ($v$Tok [6], and $v$Tok+Rank [7] with N-best hypotheses re-
ranking). We measure translation quality (BLEU), length ratio (LR), $\%$ of
length compliant (LC), and overall metric LCB. Metric in bold shows best
performing approach.

MT quality and verbosity control are measured using the automatic metrics in
Sec. 4.3. We observe that length ratio of the Baseline models for En-Fr and
En-De is significantly higher than that of En-It and En-Es. This is due to the
fact that original MuST-C training data is already well balanced for En-It and
En-Es in the ratio of 30/35/35 on an average for short/normal/long classes
whereas the same ratio is 15/31/54 for En-Fr and En-De. Given that $v$Tok+Rank
optimizes over N-best candidate translations, it shows better length control
over the Baseline and $v$Tok approach.

Self-learning approaches (offlineSL and onlineSL), provides consistent
improvements over the baseline and $v$Tok in terms of LCB metric for both data
configuration (Union, Filter). We attribute the performance gains to training
models using the additional pseudo parallel data. In fact, performance of
self-learning approaches are similar to $v$Tok+Rank for En-De and En-Es which
shows how effective controlled self-learning could be, without a hypotheses
re-ranking module as in $v$Tok+Rank. Although, there’s no clear winning system
in terms of LCB, we prefer onlineSL with Filter simply because it has a higher
length compliance (LC) over offlineSL system and Filter has higher LCB on an
average when compared to Union approach.

###  5.2 Human Evaluation

Following the evaluation criteria in Sec. 4.3, Table 2 show results of MT
human evaluation. We use 100 randomly selected samples of the test set graded
by 40 subjects per language pair and head to head comparison. We compare our
proposed approaches against $v$Tok and $v$Tok+Rank. As it turns out onlineSL
is better than $v$Tok in all languages except En-Fr. This is a very promising
result because in one of the language direction (En-Es), we even outperform
the $v$Tok+Rank system.

###  5.3 Ablation Study

We also ran an ablation study to check if hypotheses re-ranking complements
the self-learning approach. We observe that onlineSL+Rank is up to 9% better
than the onlineSL system in terms of relative LCB improvements on En-It as
shown in Table 3. In fact, onlineSL+Rank beats the state of the art
$v$Tok+Rank on three language pairs (except En-It) in terms of LCB. Concerning
the specific metrics there are only two instances where $v$Tok+Rank
outperformed onlineSL+Rank, BLEU for En-It (38.5 vs. 37.0) and LC for En-De
(93.1 vs. 92.1). Overall our analysis shows that re-ranking adds on top of the
self-learning approach.

Pair |  $v$Tok | offlineSL | onlineSL |  $v$Tok |  $v$Tok+Rank
---|---|---|---|---|---
En-Fr | 55.1 | 45.8 | 52.0 | 60.9 | 67.9
En-It | 58.0 | 60.9 | 66.1 | 66.6 | 73.0
En-De | 67.6 | 66.5 | 74.0 | 65.8 | 75.9
En-Es | 69.5 | 67.4 | 72.9 | 66.9 | 70.9
Table 2: Results of head to head MT human evaluation for, $v$Tok Vs. offlineSL Vs. onlineSL, and $v$Tok Vs. $v$Tok+Rank. $HE_{MT}$ score is computed using Eq. 5 that considers acceptable and fixable length compliant translations. Pair | Method | BLEU | LC | LCB
---|---|---|---|---
|  $v$Tok+Rank | 47.7 | 91.5 | 43.6
En-Fr | onlineSL | 48.4 | 88.5 | 42.9
|  $+$Rank | 48.4 | 93.0 | 45.0
|  $v$Tok+Rank | 38.5 | 90.0 | 34.7
En-It | onlineSL | 36.5 | 84.7 | 30.9
|  $+$Rank | 37.0 | 91.1 | 33.7
|  $v$Tok+Rank | 33.9 | 93.1 | 31.5
En-De | onlineSL | 34.1 | 90.5 | 30.9
|  $+$Rank | 34.4 | 92.1 | 31.7
|  $v$Tok+Rank | 49.3 | 95.5 | 47.1
En-Es | onlineSL | 49.8 | 93.9 | 46.7
|  $+$Rank | 49.3 | 98.4 | 48.5
Table 3: Ablation study showing improvement for onlineSL in all language
pairs. The results show re-ranking is additive and these approaches are
orthogonal.

##  6 Conclusion

In this work, we propose self-learning based approaches to learn an MT model
that can generate isometric translations (i.e., translations matching the
source in character count). We evaluate the proposed approach on an automatic
dubbing use case, where script translation is expected to match the source
length to achieve a synchronicity between source and synthetic target language
utterances. Our findings both from automatic and subjective human evaluations
show the proposed approach can perform better than a strong model with MT
output length control and is on par with the current state of the art that
requires generating multiple hypothesis and re-ranking.

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