# Second cohomology space of $\mathfrak{sl}(2)$ acting on the space of $n$-ary
differential operators on $\mathbb{R}$

Mabrouk Ben Ammar  Université de Sfax, Faculté des Sciences, Département de
Mathématiques, Laboratoire d’Algèbre, Géométrie et Théorie Spectrale (AGTS)
LR11ES53, BP 802, 3038 Sfax, Tunisie. E.mail<EMAIL_ADDRESS>
###### Abstract

We consider the spaces $\mathcal{F}_{\mu}$ of polynomial $\mu$-densities on
the line as $\mathfrak{sl}(2)$-modules and then we compute the cohomological
spaces
$\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$,
where $\mu\in\mathbb{R}$,
$\bar{\lambda}=(\lambda_{1},\dots,\lambda_{n})\in\mathbb{R}^{n}$ and
$\mathcal{D}_{\bar{\lambda},\mu}$ is the space of $n$-ary differential
operators from
$\mathcal{F}_{\lambda_{1}}\otimes\cdots\otimes\mathcal{F}_{\lambda_{n}}$ to
$\mathcal{F}_{\mu}$.

Mathematics Subject Classification (2010). 17B56

Key words  : Cohomology, Weighted Densities.

##  1 Introduction

Consider the space of polynomial $\mu$-densities:

$\mathcal{F}_{\mu}=\big{\\{}fdx^{\mu},~{}f\in\mathbb{R}[x]\big{\\}},\quad\mu\in\mathbb{R}\big{\\}}.$

The Lie algebra $\mathrm{Vect}(\mathbb{R})$ of polynomial vector fields
$X_{h}=h{d\over dx}$, where $h\in\mathbb{R}[x]$, acts on $\mathcal{F}_{\mu}$
by the Lie derivative $L^{\mu}$:

$X_{h}\cdot(fdx^{\mu})=L_{X_{h}}^{\mu}(fdx^{\mu}):=(hf^{\prime}+\mu
h^{\prime}f)dx^{\mu}.$ (1.1)

For $\bar{\lambda}=(\lambda_{1},\dots,\lambda_{n})\in\mathbb{R}^{n}$ and
$\mu\in\mathbb{R}$ we denote by $\mathcal{D}_{\bar{\lambda},\mu}$ the space of
$n$-ary differential operators $A$ from
$\mathcal{F}_{\lambda_{1}}\otimes\cdots\otimes\mathcal{F}_{\lambda_{n}}$ to
$\mathcal{F}_{\mu}$. The Lie algebra $\mathrm{Vect}(\mathbb{R})$ acts on the
space $\mathcal{D}_{\bar{\lambda},\mu}$ of these differential operators by:

$X_{h}\cdot A:=L_{X_{h}}^{\bar{\lambda},\mu}(A)=L_{X_{h}}^{\mu}\circ A-A\circ
L_{X_{h}}^{\bar{\lambda}}$ (1.2)

where $L_{X_{h}}^{\bar{\lambda}}$ is the Lie derivative on
$\mathcal{F}_{\lambda_{1}}\otimes\cdots\otimes\mathcal{F}_{\lambda_{n}}$
defined by the Leibnitz rule. The spaces $\mathcal{F}_{\mu}$ and
$\mathcal{D}_{\bar{\lambda},\mu}$ can be also viewed as
$\mathfrak{sl}(2)$-modules, where $\mathfrak{sl}(2)$ is realized as a
subalgebra of $\mathrm{Vect}(\mathbb{R})$:

$\mathfrak{sl}(2)=\mathrm{Span}(X_{1},\,X_{x},\,X_{x^{2}}).$

According to Nijenhuis-Richardson [9], the space
$\mathrm{H}^{1}\left(\mathfrak{g},\mathrm{End}(V)\right)$ classifies the
infinitesimal deformations of a $\mathfrak{g}$-module $V$ and the obstructions
to integrability of a given infinitesimal deformation of $V$ are elements of
$\mathrm{H}^{2}\left(\mathfrak{g},\mathrm{End}(V)\right)$.

For $\bar{\lambda}\in\mathbb{R}$ the spaces
$\mathrm{H}^{1}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$
are computed by Gargoubi [7] and Lecomte [8], the spaces
$\mathrm{H}^{1}_{\mathrm{diff}}(\mathrm{Vect}(\mathbb{R}),\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$
are computed by Bouarroudj and Ovsienko [4] and the spaces
$\mathrm{H}^{1}_{\mathrm{diff}}(\mathrm{Vect}(\mathbb{R}),\mathcal{D}_{\bar{\lambda},\mu})$
are computed by Feigen and Fuchs [5]. For $\bar{\lambda}\in\mathbb{R}^{2}$ the
spaces
$\mathrm{H}^{1}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$
are computed by Bouarroudj [3]. For $\bar{\lambda}\in\mathbb{R}^{3}$ the
spaces
$\mathrm{H}^{1}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$
are computed by O. Basdouri and N. Elamine [1]. In this paper we are
interested to compute the spaces
$\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$
for $\bar{\lambda}\in\mathbb{R}^{n}$.

##  2 Cohomology

Let us first recall some fundamental concepts from cohomology theory (see,
e.g., [6]). Let $\mathfrak{g}$ be a Lie algebra acting on a space $V$ and let
$\mathfrak{h}$ be a subalgebra of $\mathfrak{g}$. (If $\mathfrak{h}$ is
omitted it assumed to be $\\{0\\}$). The space of $\mathfrak{h}$-relative
$n$-cochains of $\mathfrak{g}$ with values in $V$ is the $\mathfrak{g}$-module

$C^{n}(\mathfrak{g},\mathfrak{h},V):=\mathrm{Hom}_{\mathfrak{h}}(\Lambda^{n}(\mathfrak{g}/\mathfrak{h}),V).$

The coboundary operator
$\partial:C^{n}(\mathfrak{g},\mathfrak{h},V)\longrightarrow
C^{n+1}(\mathfrak{g},\mathfrak{h},V)$ is a $\mathfrak{g}$-map satisfying
$\partial^{2}=0$. The operator $\partial$ is defined by

$\displaystyle(\partial
f)(u_{0},\dots,u_{n})=\sum_{i=0}^{n}(-1)^{i}u_{i}f(u_{0},\dots,\hat{\imath},\dots,u_{n})+$
$\displaystyle~{}~{}+\sum_{0\leq i<j\leq
n}(-1)^{i+j}f([u_{i},u_{j}],u_{0},\dots,\hat{\imath},\dots,\hat{\jmath},\dots,u_{n}).$

The kernel of $\partial|_{C^{n}}$, denoted
$Z^{n}(\mathfrak{g},\mathfrak{h},V)$, is the space of $\mathfrak{h}$-relative
$n$-cocycles, among them, the elements of
$\partial(C^{n-1}(\mathfrak{g},\mathfrak{h},V))$ are called
$\mathfrak{h}$-relative $n$-coboundaries. We denote
$B^{n}(\mathfrak{g},\mathfrak{h},V)$ the space of $n$-coboundaries.

By definition, the $n^{th}$ $\mathfrak{h}$-relative cohomology space is the
quotient space

$\mathrm{H}^{n}(\mathfrak{g},\mathfrak{h},V)=Z^{n}(\mathfrak{g},\mathfrak{h},V)/B^{n}(\mathfrak{g},\mathfrak{h},V).$

Here we consider $\mathrm{H}^{n}(\mathfrak{g},V)$ where
$\mathfrak{g}=\mathfrak{sl}(2)$ and $V=\mathcal{D}_{\bar{\lambda},\mu}$. In
this paper we are interested to the differential cohomology
$\mathrm{H}^{1}_{\mathrm{diff}}$ (i.e., we consider only cochains that are
given by differential operators).

##  3 The spaces
$\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$

Consider $\mu\in\mathbb{R}$,
$\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}$ and
$\bar{\lambda}=(\lambda_{1},\dots,\lambda_{n})\in\mathbb{R}^{n}$, we define

$|\bar{\lambda}|=\sum_{i=1}^{n}\lambda_{i},\quad\delta=\mu-|\bar{\lambda}|\quad\text{and}\quad|\alpha|=\sum_{i=1}^{n}\alpha_{i}.$

We also consider the canonical basis
$(\varepsilon_{1},\,\dots,\,\varepsilon_{n})$ of $\mathbb{R}^{n}$ where
$\varepsilon_{i}$ is the $n$-tuple $(0,\dots,0,1,0,\dots,0)$ ($1$ in the
$i^{th}$ position). The space $\mathcal{D}_{\bar{\lambda},\mu}$ is spanned by
the operators $\Omega^{\alpha}$ defined by

$\Omega^{\alpha}(f_{1}dx^{\lambda_{1}}\otimes\cdots\otimes
f_{n}dx^{\lambda_{n}})=f_{1}^{(\alpha_{1})}\dots
f_{n}^{(\alpha_{n})}dx^{\mu}.$

Consider the elementary operator $A_{\alpha}\Omega^{\alpha}$ where
$A_{\alpha}\in\mathbb{R}[x]$, then we have

$X_{h}\cdot
A_{\alpha}\Omega^{\alpha}=A^{\prime}_{\alpha}h\Omega^{\alpha}+(\delta-|\alpha|)A_{\alpha}h^{\prime}\Omega^{\alpha}-{1\over
2}\displaystyle\sum_{i=1}^{n}\alpha_{i}(\alpha_{i}+2\lambda_{i}-1)A_{\alpha}h^{\prime\prime}\Omega^{\alpha-\varepsilon_{i}}.$
(3.1)

The operators $\Omega^{\alpha}$ are eigenvectors of $X_{x}$:

$X_{x}\cdot\Omega^{\alpha}=(\mu-|\bar{\lambda}|-|\alpha|)\Omega^{\alpha}=(\delta-|\alpha|)\Omega^{\alpha}.$

######  Lemma 3.1.

Any 2-cocycle
$f\in\mathrm{Z}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$
has the following general form

$\begin{array}[]{lll}f(X_{h_{1}},X_{h_{2}})&=&\displaystyle\sum_{\alpha}A_{\alpha}(h_{1}h^{\prime}_{2}-h_{2}h^{\prime}_{1})\Omega^{\alpha}+\displaystyle\sum_{\alpha}B_{\alpha}(h_{1}h^{\prime\prime}_{2}-h_{2}h^{\prime\prime}_{1})\Omega^{\alpha}\\\\[10.0pt]
&{}&+\displaystyle\sum_{\alpha}C_{\alpha}(h^{\prime}_{1}h^{\prime\prime}_{2}-h^{\prime\prime}_{1}h^{\prime}_{2})\Omega^{\alpha},\end{array}$
(3.2)

where $A_{\alpha},\,B_{\alpha}$ and $C_{\alpha}$ are, a priori polynomial,
functions satisfying the following cocycle condition:

$C^{\prime}_{\alpha}+(\delta-|\alpha|-1)B_{\alpha}-\frac{1}{2}\sum_{i}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})A_{\alpha+\varepsilon_{i}}=0.$
(3.3)

Proof. 1) Any 2-cocycle on $\mathfrak{sl}(2)$ should retains the general form
expressed in (3.2). Consider the 2-cocycle condition:

$\begin{array}[]{lll}\partial
f(X_{h_{0}},X_{h_{1}},X_{h_{2}})&=&X_{h_{0}}\cdot
f(X_{h_{1}},X_{h_{2}})-X_{h_{1}}\cdot f(X_{h_{0}},X_{h_{2}})+X_{h_{2}}\cdot
f(X_{h_{0}},X_{h_{1}})\\\\[10.0pt]
&-&f([X_{h_{0}},X_{h_{1}}],X_{h_{2}})+f([X_{h_{0}},X_{h_{2}}],X_{h_{1}})-f([X_{h_{1}},X_{h_{2}}],X_{h_{0}})=0.\end{array}$

which can be expressed as follows:

$\begin{array}[]{lll}\partial
f(X_{h_{0}},X_{h_{1}},X_{h_{2}})={\displaystyle\sum_{\alpha}}\Big{(}C^{\prime}_{\alpha}+(|\alpha|-\delta+1)B_{\alpha}-{1\over
2}\sum_{i=1}^{n}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})A_{\alpha+\varepsilon_{i}}\Big{)}\varphi(h_{1},h_{2},h_{3})\Omega^{\alpha},\end{array}$

where
$\varphi(h_{1},h_{2},h_{3})=(h_{1}h^{\prime\prime}_{2}-h^{\prime\prime}_{1}h_{2})h^{\prime}_{0}+(h^{\prime\prime}_{0}h_{2}-h_{0}h^{\prime\prime}_{2})h^{\prime}_{1}+(h_{0}h^{\prime\prime}_{1}-h^{\prime\prime}_{0}h_{1})h^{\prime}_{2}$.
Thus, the cocycle condition is

$C^{\prime}_{\alpha}+(|\alpha|-\delta+1)B_{\alpha}-{1\over
2}\sum_{i=1}^{n}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})A_{\alpha+\varepsilon_{i}}.$

$\Box$

######  Theorem 3.2.

  * 1)

If $\delta\notin\mathbb{N}$ then ${\rm
H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=0$.

  * 2)

If $\delta=k\in\mathbb{N}$ then, up to a coboundary, any 2-cocycle
$f\in\mathrm{Z}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$
can be expressed as follows:

$\begin{array}[]{lll}f(X_{h_{1}},X_{h_{2}})&=&\displaystyle\sum_{|\alpha|=k}A_{\alpha}(h_{1}h^{\prime}_{2}-h_{2}h^{\prime}_{1})\Omega^{\alpha}+\displaystyle\sum_{|\alpha|=k-1}B_{\alpha}(h_{1}h^{\prime\prime}_{2}-h_{2}h^{\prime\prime}_{1})\Omega^{\alpha}\\\\[10.0pt]
&{}&+\displaystyle\sum_{|\alpha|=k-1}C_{\alpha}(h^{\prime}_{1}h^{\prime\prime}_{2}-h^{\prime\prime}_{1}h^{\prime}_{2})\Omega^{\alpha},\end{array}$

where $A_{\alpha},\,B_{\alpha}$ and $C_{\alpha}$ are a priori polynomial
functions satisfying (3.3).

  * 3)

If $\delta=k\in\mathbb{N}$ and
$-2\bar{\lambda}\notin\\{0,\,\dots,\,k-1\\}^{n}$ then, up to a coboundary, any
2-cocycle
$f\in\mathrm{Z}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$
can be expressed as follows:

$f(X_{h_{1}},X_{h_{2}})=\sum_{|\alpha|=k}A_{\alpha}(h_{1}h^{\prime}_{2}-h^{\prime}_{1}h_{2})\Omega^{\alpha},\qquad
X_{h_{1}},\,X_{h_{2}}\in\mathfrak{sl}(2),$ (3.4)

where the coefficients $A_{\alpha+\varepsilon_{i}}$, with $|\alpha|=k-1$,
satisfy:

$\sum_{i}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})A_{\alpha+\varepsilon_{i}}=0.$
(3.5)

Proof. 1) Consider a 1-cochain

$b(X_{h_{1}})=\displaystyle\sum_{\alpha}U_{\alpha}h_{1}\Omega^{\alpha}+\displaystyle\sum_{\alpha}V_{\alpha}h^{\prime}_{1}\Omega^{\alpha}+\displaystyle\sum_{\alpha}W_{\alpha}h^{\prime\prime}_{1}\Omega^{\alpha},$

where $U_{\alpha}$, $V_{\alpha}$ and $W_{\alpha}$ are, a priori polynomial
functions. Then we have

$\partial b(X_{h_{1}},X_{h_{2}})=X_{h_{1}}\cdot b(X_{h_{2}})-X_{h_{2}}\cdot
b(X_{h_{1}})-b([X_{h_{1}},X_{h_{2}}])$

So, according to (3.1), we can write

$\begin{array}[]{lll}\partial
b(X_{h_{1}},X_{h_{2}})&=&\displaystyle\sum_{\alpha}\big{(}(|\alpha|-\delta)U_{\alpha}+V^{\prime}_{\alpha}\big{)}(h^{\prime}_{1}h_{2}-h_{1}h^{\prime}_{2})\Omega^{\alpha}\\\\[10.0pt]
&+&\frac{1}{2}\displaystyle\sum_{\alpha}\Big{(}\displaystyle\sum_{i}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})U_{\alpha+\varepsilon_{i}}+W^{\prime}_{\alpha}\Big{)}(h_{1}h^{\prime\prime}_{2}-h_{2}h^{\prime\prime}_{1})\Omega^{\alpha}\\\\[10.0pt]
&+&\frac{1}{2}\displaystyle\sum_{\alpha}\Big{(}\displaystyle\sum_{i}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})V_{\alpha+\varepsilon_{i}}+(\delta-|\alpha|-1)W_{\alpha}\Big{)}(h^{\prime}_{1}h^{\prime\prime}_{2}-h^{\prime\prime}_{1}h^{\prime}_{2})\Omega^{\alpha}.\end{array}$
(3.6)

Since $\delta\notin\mathbb{N}$ we have $(\delta-|\alpha|)\neq 0$ and
$(\delta-|\alpha|-1)\neq 0$. So, we can choose

$V_{\alpha}=0,\quad
U_{\alpha}=\displaystyle\frac{A_{\alpha}}{\delta-|\alpha|}\quad\text{ and
}\quad.W_{\alpha}=\displaystyle\frac{C_{\alpha}}{\delta-|\alpha|-1}.$

Therefore, according to the cocycle conditions (3.3), we have

$\displaystyle\sum_{i}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})U_{\alpha+\varepsilon_{i}}+W^{\prime}_{\alpha}=B_{\alpha}$

and therefore we have

$f=\partial b$

and then

${\rm
H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=0.$

2) If $\delta=k\in\mathbb{N}$ then, as in the previous case, using (3.6), we
can annihilate the coefficients $A_{\alpha}$ for $|\alpha|\neq k$ and the
coefficients $B_{\alpha}$ and $C_{\alpha}$ for $|\alpha|\neq k-1$ by adding to
$f$ a suitable coboundary.
3) If $\delta=k\in\mathbb{N}$ and
$-2\bar{\lambda}\notin\\{0,\,\dots,\,k-1\\}^{n}$ then, in (3.6) we choose
$W_{\alpha}=0$ and we consider the two following equations:

$\frac{1}{2}\displaystyle\sum_{|\alpha|=k-1}\sum_{i}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})U_{\alpha+\varepsilon_{i}}=B_{\alpha}$
(3.7)
$\frac{1}{2}\displaystyle\sum_{|\alpha|=k-1}\displaystyle\sum_{i}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})V_{\alpha+\varepsilon_{i}}=C_{\alpha}$
(3.8)

Let us solve (3.7). Without loss of generality, assume that
$-2\lambda_{1}\notin\\{0,\,\dots,\,k-1\\}$. Choose arbitrarily the
coefficients $U_{(0,\alpha_{2},\dots,\alpha_{n})}$. Since
$-2\lambda_{1}\notin\\{0,\,\dots,\,k-1\\}$ then the coefficients
$U_{(1,\alpha_{2},\dots,\alpha_{n})}$ are uniquely defined by (3.7), in
function of the corresponding $B_{(0,\alpha_{2},\dots,\alpha_{n})}$ and the
$U_{(0,\alpha_{2},\dots,\alpha_{n})+\varepsilon_{i}}$, for $i\geq 2$.
Similarly, by (3.7), we define the coefficients $U_{\alpha}$ with
$\alpha_{1}=2$, in function of those with $\alpha_{1}=1$ and the corresponding
coefficients $B_{(1,\alpha_{2},\dots,\alpha_{n})}$. So, step by step, we
define all the coefficients $U_{\alpha}$ so that $(f-\partial
b)(X_{h_{1}},X_{h_{2}})$ does not contain terms in
$(h_{1}h^{\prime\prime}_{2}-h_{2}h^{\prime\prime}_{1})\Omega^{\alpha}$.
Similarly we solve (3.8) so that $(f-\partial b)(X_{h_{1}},X_{h_{2}})$ does
not contain terms in
$(h^{\prime}_{1}h^{\prime\prime}_{2}-h^{\prime}_{2}h^{\prime\prime}_{1})\Omega^{\alpha}$.

$\Box$

######  Theorem 3.3.

If $\delta=k\in\mathbb{N}$ and
$-2\bar{\lambda}\notin\\{0,\,\dots,\,k-1\\}^{n}$ then, we have

$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}.$

These spaces are spanned by the cocycles:

$f(X_{h_{1}},X_{h_{2}})=\displaystyle\sum_{|\alpha|=k}A_{\alpha}(h_{1}h^{\prime}_{2}-h^{\prime}_{1}h_{2})\Omega^{\alpha},$
(3.9)

where the coefficients $A_{\alpha+\varepsilon_{i}}$, with $|\alpha|=k-1$,
satisfy:

$\sum_{i}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})A_{\alpha+\varepsilon_{i}}=0.$
(3.10)

Proof. The expression of the 2-cocycles and the cocycle condition (3.10) are
defined by Theorem 3.2. So, here, we have only to define the dimension of
$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$.
Without loss of generality, assume that
$-2\lambda_{1}\notin\\{0,\,\dots,\,k-1\\}$. Choose arbitrarily the
$A_{(0,\,\alpha_{2},\,\dots,\,\alpha_{n})}$. Then the coefficients
$A_{(1,\,\alpha_{2},\,\dots,\,\alpha_{n})}$ are uniquely defined by (3.10) in
function of $A_{(0,\,\alpha_{2},\,\dots,\,\alpha_{n})+\varepsilon_{i}}$ for
$i\geq 2$. Similarly, by (3.10), we define the coefficients $A_{\alpha}$ with
$\alpha_{1}=2$, in function of those with $\alpha_{1}=1$. So, step by step, we
prove that the space of solutions of the system of linear equations (3.10) is
generated by the arbitrary coefficients
$A_{(0,\,\alpha_{2},\dots,\alpha_{n})}$. The number of such $A_{\alpha}$ is
the well known binomial coefficient with repetition

$\Gamma_{n-1}^{k}=\begin{pmatrix}n+k-2\\\ k\end{pmatrix},$

which is equal to the dimension of
$\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu}).$

$\Box$

##  4 Singular cases

Now assume that

$\delta=k\quad\text{ and
}\quad-2\bar{\lambda}=(t_{1},\,t_{2},\,\dots,\,t_{n})\in\\{0,\,1,\,\dots,\,k-1\\}^{n},$

and let

$\sigma_{n}=t_{1}+\dots+t_{n}.$

In this section we assume that $C^{\prime}_{\alpha}=0$, then for
$|\alpha|=k-1$, the condition of cocycle (3.3) becomes

$\sum_{i}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})A_{\alpha+\varepsilon_{i}}=0.$
(4.1)

Therefore the coefficients $A_{\alpha}$, $B_{\alpha}$ and $C_{\alpha}$ are
independent.
Obviously the dimension of
$\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$
depends only of the rank of the system (4.1).

######  Lemma 4.1.

$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}+3\ell,$

where $(\Gamma_{n}^{k-1}-\ell)$ is the rank of the system (4.1) .

Proof. The system (4.1) is with $\Gamma_{n}^{k}$ unknowns and
$\Gamma_{n}^{k-1}$ equations. Any equation is coming from a given
$\alpha=(\alpha_{1},\dots,\alpha_{n})$ with $|\alpha|=k-1$. The rank of (4.1)
is then less or equal to $\Gamma_{n}^{k-1}$. If the system (4.1) is of rank
$(\Gamma_{n}^{k-1}-\ell)$ then, by the cocycle condition (4.1), we see that
the space of solutions $A_{\alpha+\varepsilon_{i}}$ is of dimension:

$\Gamma_{n}^{k}-\Gamma_{n}^{k-1}+\ell=\Gamma_{n-1}^{k}+\ell.$

On the other hand, by the equations (3.7) and (3.8) we can annihilate
$(\Gamma_{n}^{k-1}-\ell)$ parameters $B_{\alpha}$ and
$(\Gamma_{n}^{k-1}-\ell)$ parameters $C_{\alpha}$ by adding suitable
coboundary. Therefore, each of the spaces spanned by the coefficients
$B_{\alpha}$ and the one spanned by the coefficients $C_{\alpha}$, with
$|\alpha|=k-1$, is of dimension:

$\Gamma_{n}^{k-1}-(\Gamma_{n}^{k-1}-\ell)=\ell.$

$\Box$
Thus, in this section we have only to compute the rank $\Gamma_{n}^{k-1}-\ell$
of the system (4.1). We cut the system (4.1) into the two following subsystems
$(S_{1})$ and $(S_{2})$:

$(S_{1}):\quad\sum_{i=1}^{n}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})A_{\alpha+\varepsilon_{i}}=0,\quad\alpha_{1}\neq
t_{1}.$ (4.2)
$(S_{2}):\quad\sum_{i=1}^{n}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})A_{\alpha+\varepsilon_{i}}=0,\quad\alpha_{1}=t_{1}.$
(4.3)

From $(S_{1})$ we extract the following system $(S^{\prime}_{1})$

$(S^{\prime}_{1}):\quad\sum_{i=2}^{n}(\alpha_{i}+1)(\alpha_{i}+2\lambda_{i})A_{\alpha+\varepsilon_{i}}=0,\quad\alpha_{1}=t_{1}-1.$
(4.4)

######  Theorem 4.1.

If $\sigma_{n}<k-1$, then

$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}.$

Proof. This is equivalent to the fact that, in this case, the system (3.5) is
of maximal rank. We proceed by recurrence to prove this result. It is true for
$n=2$ (see [3]). Assume that the result is true for $n-1$. Consider the
subsystems $(S_{1})$ and $(S_{2})$, it is easy to prove that $(S_{1})$ is of
maximal rank, while, by the recurrence hypothesis, the subsystem $(S_{2})$ is
also of maximal rank. Indeed, the subsystem $(S_{2})$ can be considered as a
system of equations related to the $(n-1)$-tuple
$(\alpha_{2},\dots,\alpha_{n})$. So, we are in the case $n-1$ with

$\alpha_{2}+\cdots+\alpha_{n}=k-1-t_{1}=k^{\prime}-1\quad\text{and}\quad\sigma_{n-1}=t_{2}+\cdots+t_{n}<k-t_{1}-1=k^{\prime}-1.$

If $t_{1}=0$ then, combining the two subsystems $(S_{1})$ and $(S_{2})$, we
get a system with maximal rank, since there are no common unknowns. Indeed,
the unknowns $A_{(\alpha_{1},\,\dots,\,\alpha_{n})}$ of $(S_{1})$ are all with
$\alpha_{1}\neq 0$, while those of $(S_{2})$ are all with $\alpha_{1}=0$.

If $t_{1}>0$ then the unknowns $A_{(\alpha_{1},\,\dots,\,\alpha_{n})}$ of
$(S_{2})$ are all with $\alpha_{1}=t_{1}$. The unknowns
$A_{(\alpha_{1},\,\dots,\,\alpha_{n})}$ with $\alpha_{1}=t_{1}$ can also
appear in $(S_{1})$, but they appear only in the equations corresponding to
$(\alpha_{1},\,\dots,\,\alpha_{n})$ with $\alpha_{1}=t_{1}-1$. Therefore, we
consider the system $(S^{\prime}_{1})$ which can be also considered as a
system of equations related to $(\alpha_{2},\dots,\alpha_{n})$, but, here we
have:

$\alpha_{2}+\cdots+\alpha_{n}=k-t_{1}=k^{\prime}-1\quad\text{and}\quad\sigma_{n-1}=t_{2}+\cdots+t_{n}<k-t_{1}-1=k^{\prime}-2<k^{\prime}-1.$

For $(S^{\prime}_{1})$ we are in the case $n-1$, therefore, by the recurrence
hypothesis, the system $(S^{\prime}_{1})$ is of maximal rank, therefore, there
are no nontrivial combination of some equations of $(S_{1})$ belonging to
$(S_{2})$, since there are no nontrivial combination of some equations of
$(S_{1})$ killing the $A_{\alpha}$ with $\alpha_{1}=t_{1}-1$. That is, the
spaces of the equations respectively of $(S_{1})$ and $(S_{2})$ are
supplementary. Thus, combining the two subsystems $(S_{1})$ and $(S_{2})$, we
get a system of maximal rank which is the system (3.5). Theorem 4.1 is proved.

$\Box$

######  Theorem 4.2.

If $\sigma_{n}=k-1$ then

$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}+3.$

Proof. This is equivalent to the fact that, in this case, the system (3.5) is
of rank $\Gamma_{n}^{k-1}-1$. We proceed by recurrence to prove this fact.
This is true for $n=2$ (see [3]). Assume that the result is true for $n-1$.
Consider the subsystems $(S_{1})$ and $(S_{2})$. As before $(S_{1})$ is of
maximal rank, it is of rank $\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}$. The
subsystem $(S_{2})$ is of rank $\Gamma_{n-1}^{k-t_{1}-1}-1$ (by the recurrence
hypothesis).

If $t_{1}=0$ then, as before, we see that the subsystems $(S_{1})$ and
$(S_{2})$ are independent of each other. Therefore, combining the two
subsystems $(S_{1})$ and $(S_{2})$, we get a system of rank

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}-1=\Gamma_{n}^{k-1}-1.$

Indeed, the unknowns $A_{(\alpha_{1},\,\dots,\,\alpha_{n})}$ of $(S_{1})$ are
all with $\alpha_{1}\neq 0$, while those of $(S_{2})$ are all with
$\alpha_{1}=0$ (there are common unknowns).

If $t_{1}>0$ then the unknowns $A_{(\alpha_{1},\,\dots,\,\alpha_{n})}$ of
$(S_{2})$ are all with $\alpha_{1}=t_{1}$. The unknowns
$A_{(\alpha_{1},\,\dots,\,\alpha_{n})}$ with $\alpha_{1}=t_{1}$ appear in
$(S_{1})$ only in the equations corresponding to
${(\alpha_{1},\dots,\alpha_{n})}$ with $\alpha_{1}=t_{1}-1$. As before, we
consider the system $(S^{\prime}_{1})$. We are in the case $n-1$ with

$\alpha_{2}+\cdots+\alpha_{n}=k-t_{1}=k^{\prime}-1\quad\text{and}\quad\sigma_{n-1}=t_{2}+\cdots+t_{n}=k-t_{1}-1=k^{\prime}-2<k^{\prime}-1.$

By the recurrence hypothesis, the system $(S^{\prime}_{1})$ is of maximal
rank, therefore, there are no nontrivial combination of some equations of
$(S_{1})$ killing the $A_{\alpha}$ with $\alpha_{1}=t_{1}-1$, therefore,
combining the two subsystems $(S_{1})$ and $(S_{2})$, we get a system with
rank

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}-1=\Gamma_{n}^{k-1}-1.$

Theorem 4.2 is proved.

$\Box$

######  Remark 4.3.

If $\sigma_{n}=k-1$, we see that the equation corresponding to
${(t_{1},\dots,t_{n})}$ is trivial, so, it’s enough to prove that, if we
subtract this equation from (3.5), we get a maximal rank system.

######  Theorem 4.4.

If $\sigma_{n}=k$ then

$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}+3(s-1),$

where $s$ is the number of $t_{i}\geq 1$.

Proof. We proceed by recurrence to prove that the rank of (3.5) is

$\Gamma_{n}^{k-1}-(s-1)=\begin{pmatrix}n+k-2\\\ k-1\end{pmatrix}-(s-1).$

This is true for $n=2$ (see [3]), indeed, for $n=2$ we have necessarily $s=2$.
Assume that the result is true for $n-1$. As before, if $t_{1}=0$ then the
subsystems $(S_{1})$ and $(S_{2})$ are independent of each other. The
subsystem $(S_{1})$ is of rank $\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-1}$, while,
according to the recurrence hypothesis, the subsystem $(S_{2})$ is of rank
$\Gamma_{n-1}^{k-1}-(s-1).$ Therefore, the rank of (3.5) is

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-1}+\Gamma_{n-1}^{k-1}-(s-1)=\Gamma_{n}^{k-1}-(s-1)=\begin{pmatrix}n+k-2\\\
k-1\end{pmatrix}-(s-1).$

Now, for $t_{1}>0$, the subsystem $(S_{1})$ is of rank
$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}$, while, according to the
recurrence hypothesis, the subsystem $(S_{2})$ is of rank

$\Gamma_{n-1}^{k-t_{1}-1}-(s-2).$

Consider the subsystem $(S^{\prime}_{1})$. We are in the case $n-1$ with

$\alpha_{2}+\cdots+\alpha_{n}=k-t_{1}=k^{\prime}-1\quad\text{and}\quad\sigma_{n-1}=t_{2}+\cdots+t_{n}=k-t_{1}=k^{\prime}-1.$

Therefore, the subsystem $(S^{\prime}_{1})$ is of rank

$\Gamma_{n-1}^{k-t_{1}}-1.$

The equation corresponding to ${(t_{1}-1,t_{2},\dots,t_{n})}$ appear as a
trivial equation in $(S^{\prime}_{1})$, corresponding in $(S_{1})$ to the
equation

$A_{(t_{1},t_{2},\dots,t_{n})}=0.$ (4.5)

The equation (4.5) appears also in $(S_{2})$ corresponding to any
${(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i}}$ for $i\geq 2$.

Obviously, if we subtract the trivial equation from $(S^{\prime}_{1})$ we
obtain a system of maximal rank. Thus, the system (3.5) is of rank

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}-(s-2)-1=\Gamma_{n}^{k-1}-(s-1).$

Theorem 4.4 is proved.

$\Box$

######  Remark 4.5.

The result of the previous theorem can be explained by the fact that the
equations (3.5) corresponding to $(t_{1},\dots,t_{n})-\varepsilon_{i}$, for
$t_{i}>0$, are the same. They are all equivalent to (4.5). So, it’s enough to
prove that, if we subtract $(s-1)$ equations from these $s$ equivalent
equations we get a maximal rank system.

######  Theorem 4.6.

If $\sigma_{n}=k+1$, then

$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=\left\\{\begin{array}[]{ll}\Gamma_{n}^{k-1}+{3\over
2}s(s-1)-3r&\text{if}\quad\max{t_{i}}\geq 2\\\\[8.0pt]
\Gamma_{n}^{k-1}&\text{if}\quad\max{t_{i}}=1.\end{array}\right.$

where $s$ is the number of $t_{i}\geq 1$ and $r$ is the number of $t_{i}=1$.

Proof. Assume that $t_{1}\geq 1$ and consider the system $(S^{\prime}_{1})$.
We are in the case $n-1$ with

$\alpha_{2}+\cdots+\alpha_{n}=k-t_{1}=k^{\prime}-1\quad\text{and}\quad\sigma_{n-1}=t_{2}+\cdots+t_{n}=k+1-t_{1}=k^{\prime}.$

Therefore, it was proved in the proof of Theorem 4.4 that the system
$(S^{\prime}_{1})$ is of rank

$\Gamma_{n-1}^{k-t_{1}}-(s-2).$

In $(S^{\prime}_{1})$ the equations corresponding to
${(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i}}$, for $t_{i}\geq 1$, are
equivalent to

$A_{(t_{1}-1,t_{2},\dots,t_{n})}=0.$

But, the correspondent equations in $(S_{1})$ are

$A_{(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i}}+A_{(t_{1}-1,t_{2},\dots,t_{n})}=0,\quad
t_{i}\geq 1.$ (4.6)

Case 1: $\max t_{i}\geq 2$. Assume that $t_{2}\geq 2$. We proceed by
recurrence to prove that the rank of (3.5) is

$\Gamma_{n}^{k-1}-\frac{1}{2}s(s-1)+r.$

This is true for $n=2$ (see [3]), indeed, for $n=2$ we have necessarily $s=2$
and $r=0$. Assume that the result is true for $n-1$.

Assume that $t_{1}=1$. By the recurrence hypothesis, the subsystem $(S_{2})$
is of rank

$\Gamma_{n-1}^{k-t_{1}-1}-\frac{1}{2}(s-2)(s-1)+r-1.$

We have $t_{2}\geq 2$, then, in $(S_{2})$ the equation corresponding to
${(t_{1},t_{2}-2,\dots,t_{n})}$ gives

$A_{(t_{1},t_{2}-1,\dots,t_{n})}=0.$

Therefore, the equation corresponding to
${(t_{1},t_{2}-1,\dots,t_{n})-\varepsilon_{i}}$, for $t_{i}\geq 1$, gives

$A_{(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i}}=0,\quad t_{i}\geq 1,$ (4.7)

and the $(s-2)$ correspondent equations (4.6) in $(S_{1})$, for $i\geq 3$,
become trivial (we can also say that the equations (4.6) of $(S_{1})$, for
$i\geq 3$, are combination of some equations of $(S_{2})$). If we subtract the
equations (4.7), for $i\geq 3$, from $(S^{\prime}_{1})$, we obtain a maximal
rank system. Therefore, the rank of (3.5) is

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}-\frac{1}{2}(s-2)(s-1)+r-1-(s-2)=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}-\frac{1}{2}s(s-1)+r.$

If $t_{1}\geq 2$, then the subsystem $(S_{2})$ is of rank

$\Gamma_{n-1}^{k-t_{1}-1}-\frac{1}{2}(s-2)(s-1)+r.$

But, here we have $A_{(t_{1}-1,t_{2},\dots,t_{n})}=0$ as equation
corresponding to $(t_{1}-2,t_{2},\dots,t_{n})$. Therefore, the equation
corresponding to $(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i}$ gives
$A_{(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i}}=0$. So, the $(s-2)$
correspondent equations (4.6) in $(S_{1})$ (for $i\geq 3$) become trivial,
but, the equation in $(S_{1})$: $A_{(t_{1},t_{2}-1,\dots,t_{n})}=0$, appear
also in $(S_{2})$ as equation corresponding to
${(t_{1},t_{2}-2,\dots,t_{n})}$. Thus, the rank of (3.5) is

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}-\frac{1}{2}(s-2)(s-1)+r-(s-2)-1=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}-\frac{1}{2}s(s-1)+r.$

Case 2: $\max t_{i}=1$. In this case we prove the rank of (3.5) is

$\Gamma_{n}^{k-1}=\begin{pmatrix}n+k-2\\\ k\end{pmatrix}.$

By the recurrence hypothesis, the subsystem $(S_{2})$ is of rank

$\Gamma_{n-1}^{k-t_{1}-1}.$

Assume that $t_{1}=t_{2}=1$. Therefore, the $(s-2)$ equations, in
$(S^{\prime}_{1})$, corresponding to
${(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i}}$, $t_{i}=1$ for $i\geq 3$, are
equivalent to $A_{(t_{1}-1,t_{2},\dots,t_{n})}=0$. Obviously, if we subtract
these equations from $(S^{\prime}_{1})$, we get a maximal rank system.
Therefore, the rank of (3.5) is

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}.$

Theorem 4.6 is proved.

$\Box$

######  Remark 4.7.

In the previous theorem, if there exist some $t_{i}\geq 2$, then, for any
$t_{i}\geq 2$, (3.5) applied to ${(t_{1},t_{2},\dots,t_{n})-2\varepsilon_{i}}$
gives $A_{(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i}}=0$. We use one of
$t_{i}\geq 2$ to prove that $A_{(t_{1},t_{2},\dots,t_{n})-\varepsilon_{j}}=0$
for $t_{j}=1$. That is, we have $r$ equations:
$A_{(t_{1},t_{2},\dots,t_{n})-\varepsilon_{j}}=0$, corresponding to
${(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i_{0}}-\varepsilon_{j}}$ for a fixed
$t_{i_{0}}\geq 2$ and $t_{j}=1$. Therefore, all other equations corresponding
to ${(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i}-\varepsilon_{j}}$, for $i<j$
and $t_{i},\,t_{j}\geq 1$, become trivial. We prove that if we subtract these
trivial equations we get a maximal rank system. Thus, the rank of (3.5) is

$\Gamma_{n}^{k-1}-\frac{1}{2}s(s-1)+r=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}-\frac{1}{2}s(s-1)+r.$

######  Theorem 4.8.

If $\sigma_{n}=k+2$, then

$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}+{3\over 2}s(s-1),$

where $s$ is the number of $t_{i}\geq 3$.

Proof. We proceed by recurrence to prove that the rank of (3.5) is

$\Gamma_{n}^{k-1}-\frac{1}{2}s(s-1)=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}-\frac{1}{2}s(s-1).$

This is true for $n=2$. Assume that it is true for $n-1$. As usually, the
result is true if $t_{1}=0$.

If $\max t_{i}=1$ then $(S_{2})$ is of rank

$\Gamma_{n-1}^{k-t_{1}-1},$

and the system $(S^{\prime}_{1})$ is of maximal rank. Therefore, the rank of
(3.5) is

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}.$

If $\max t_{i}=2$ then we can assume that $t_{1}=2$. In this case $(S_{2})$ is
of rank

$\Gamma_{n-1}^{k-t_{1}-1}.$

For $(S^{\prime}_{1})$ we distinguish two cases. $\max_{i>1}t_{i}=1$ or
$\max_{i>1}t_{i}=2$. In the first case, the system $(S^{\prime}_{1})$ is of
maximal rank. Therefore, the rank of (3.5) is

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}.$

In the second case, the system $(S^{\prime}_{1})$ is of rank

$\Gamma_{n-1}^{k-t_{1}}-\frac{1}{2}(s^{\prime}-2)(s^{\prime}-1)+r^{\prime},$

where $s^{\prime}$ is the number of $t_{i}\geq 1$ and $r^{\prime}$ is the
number of $t_{i}=1$. Indeed, for the system $(S^{\prime}_{1})$, we are in the
case $n-1$ with

$\alpha_{2}+\cdots+\alpha_{n}=k-t_{1}=k^{\prime}-1\quad\text{and}\quad\sigma_{n-1}=t_{2}+\cdots+t_{n}=k+2-t_{1}=k^{\prime}+1.$

Assume that $t_{2}=2$. In the system $(S^{\prime}_{1})$, for $i\geq 2$ such
that $t_{i}\geq 2$, the equations corresponding to
${(t_{1}-1,t_{2},\dots,t_{n})-2\varepsilon_{i}}$ are

$A_{(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i}}=0,\quad t_{i}\geq 2,\quad
i\geq 2.$ (4.8)

In particular, we have $A_{(t_{1}-1,t_{2}-1,\dots,t_{n})}=0.$ Therefore, there
are $r^{\prime}$ equations

$A_{(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i}}=0,\quad t_{i}=1,\quad i\geq
3,$ (4.9)

corresponding to ${(t_{1}-1,t_{2}-1,\dots,t_{n})-\varepsilon_{i}}$, for $i\geq
3$ and $t_{i}=1$. Therefore, all other equations corresponding to
${(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i}-\varepsilon_{j}}$, for $i<j$ and
$t_{i},\,t_{j}\geq 1$, become trivial in $(S^{\prime}_{1})$. The number of
these trivial equations is
$\frac{1}{2}(s^{\prime}-2)(s^{\prime}-1)-r^{\prime}$. Of course, if we
subtract these trivial equations from $(S^{\prime}_{1})$ we obtain a maximal
rank system. Therefore, the rank of (3.5) is

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}.$

Indeed, these trivial equations of $(S^{\prime}_{1})$ are associated in
$(S_{1})$ to

$A_{(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i}-\varepsilon_{j}}=0.$ (4.10)

But, (4.10) appears also in $(S_{2})$ as equation corresponding to
${(t_{1},t_{2}-1,\dots,t_{n})-\varepsilon_{i}-\varepsilon_{j}}$, since we have
also $A_{(t_{1},t_{2}-1,\dots,t_{n})-\varepsilon_{i}}=0$, for $t_{i}\geq 2$.

Now, if $t_{1}\geq 3$, then $(S_{2})$ is of rank

$\Gamma_{n-1}^{k-t_{1}-1}-\frac{1}{2}(s-2)(s-1).$

The rank of $(S^{\prime}_{1})$ is

$\Gamma_{n-1}^{k-t_{1}}-\frac{1}{2}(s^{\prime}-2)(s^{\prime}-1)+r^{\prime}.$

In $(S^{\prime}_{1})$, for $i\geq 2$ such that $t_{i}\geq 2$, the equations
corresponding to ${(t_{1}-1,t_{2},\dots,t_{n})-2\varepsilon_{i}}$ are

$A_{(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i}}=0,\quad t_{i}\geq 2,\quad
i\geq 2.$ (4.11)

Moreover, there are $r^{\prime}$ equations

$A_{(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i}}=0,\quad t_{i}=1,\quad i\geq
2,$ (4.12)

corresponding to
${(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i_{0}}-\varepsilon_{i}}$, for a
fixed $i_{0}\geq 2$ such that $t_{i_{0}}\geq 2$ and $i\geq 2$ such that
$t_{i}=1$. All other equations corresponding to
${(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i}-\varepsilon_{j}}$, for $i<j$
such that $t_{i},\,t_{j}\geq 1$, become trivial in $(S^{\prime}_{1})$. The
number of these trivial equations is
$\frac{1}{2}(s^{\prime}-2)(s^{\prime}-1)-r^{\prime}$. These trivial equations
in $(S^{\prime}_{1})$ appear in $(S_{1})$ as

$A_{(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i}-\varepsilon_{j}}=0.$ (4.13)

For any $t_{i}\geq 3$, the equation corresponding to
$(t_{1},t_{2},\dots,t_{n})-3\varepsilon_{i}$ gives
$A_{(t_{1},t_{2},\dots,t_{n})-2\varepsilon_{i}}=0$. Therefore, the equation
(4.13) appear in $(S_{2})$ as equation corresponding to
$(t_{1},t_{2},\dots,t_{n})-3\varepsilon_{i}$ only for $i\geq 2$ such that
$t_{i}\geq 3$. Thus, the rank of (3.5) is

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}-\frac{1}{2}(s-2)(s-1)-(s-1)=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}-\frac{1}{2}s(s-1).$

Theorem 4.8 is proved.

$\Box$

######  Theorem 4.9.

If $\sigma_{n}=k+m$, with $m\geq 2$, then

$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}+{3\over 2}s(s-1),$

where $s$ is the number of $t_{i}>m$.

Proof. Assume that $m\geq 3$, since the case $m=2$ was treated in the previous
theorem. We proceed by recurrence to prove that the rank of (3.5) is

$\Gamma_{n}^{k-1}-\frac{1}{2}s(s-1).$

This is true for $n=2$. Assume that it is true for $n-1$.

###  4.1 If $t_{1}\leq m$.

In this case the system $(S_{2})$ is of rank

$\Gamma_{n-1}^{k-t_{1}-1}-\frac{1}{2}s(s-1).$

The system $(S^{\prime}_{1})$ is of rank

$\Gamma_{n-1}^{k-t_{1}}-\frac{1}{2}s^{\prime}(s^{\prime}-1)$

where $s^{\prime}$ is the number of $t_{i}\geq m$ for $i\geq 2$. Indeed, for
the system $(S^{\prime}_{1})$, we are in the case $n-1$ with

$\alpha_{2}+\cdots+\alpha_{n}=k-t_{1}=k^{\prime}-1\quad\text{and}\quad\sigma_{n-1}=t_{2}+\cdots+t_{n}=k+m-t_{1}=k^{\prime}+m-1.$

In $(S^{\prime}_{1})$, for any $t_{i}\geq m$, the equation corresponding to
$\alpha=(t_{1}-1,t_{2},\dots,t_{n})-m\varepsilon_{i}$ is

$A_{(t_{1}-1,t_{2},\dots,t_{n})-(m-1)\varepsilon_{i}}=0.$ (4.14)

Case 1: $m=2h+1$ with $h\geq 1$. In $(S^{\prime}_{1})$, for $i\neq j$,
$t_{i}\geq 2h+1$ and $t_{j}\geq 1$, according to (4.14), the equation
corresponding to
$(t_{1}-1,t_{2},\dots,t_{n})-2h\varepsilon_{i}-\varepsilon_{j}$ gives

$A_{(t_{1}-1,t_{2},\dots,t_{n})-(2h-1)\varepsilon_{i}-\varepsilon_{j}}=0.$

Step by step, for $t_{i}\geq(2h+1)$ and $t_{j}\geq(2h+1)$, $i\neq j$, the
equations corresponding respectively to
$(t_{1}-1,t_{2},\dots,t_{n})-(h+1)\varepsilon_{i}-h\varepsilon_{j}$ and
$(t_{1}-1,t_{2},\dots,t_{n})-h\varepsilon_{i}-(h+1)\varepsilon_{j}$ are
equivalent to the same equation which is

$A_{(t_{1}-1,t_{2},\dots,t_{n})-h\varepsilon_{i}-h\varepsilon_{j}}=0.$ (4.15)

Therefore, if we subtract from $(S^{\prime}_{1})$ the
$\frac{1}{2}s^{\prime}(s^{\prime}-1)$ equations corresponding to
$(t_{1}-1,t_{2},\dots,t_{n})-(h+1)\varepsilon_{i}-h\varepsilon_{j}$, where
$t_{i},\,t_{j}\geq(2h+1)$ and $i<j$, we get a maximal rank. Thus, the rank of
(3.5) is

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}-\frac{1}{2}s(s-1)=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}-\frac{1}{2}s(s-1).$

Case 2: $m=2h$. In $(S^{\prime}_{1})$, for $i\neq j$, $t_{i}\geq m$ and
$t_{j}\geq 1$, according to (4.14), the equation corresponding to
$(t_{1}-1,t_{2},\dots,t_{n})-(2h-1)\varepsilon_{i}-\varepsilon_{j}$ gives

$A_{(t_{1}-1,t_{2},\dots,t_{n})-(2h-2)\varepsilon_{i}-\varepsilon_{j}}=0.$

Consider a fixed $t_{i_{0}}>1$. The equation corresponding to
$(t_{1}-1,t_{2},\dots,t_{n})-(2h-2)\varepsilon_{i}-\varepsilon_{i_{0}}-\varepsilon_{j}$,
for $t_{i}\geq m$, gives

$A_{(t_{1}-1,t_{2},\dots,t_{n})-(2h-3)\varepsilon_{i}-\varepsilon_{i_{0}}-\varepsilon_{j}}=0.$

Step by step, for $t_{i},\,t_{j}\geq m$, $i\neq j$, the equations
corresponding respectively to
$(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i_{0}}-(h-1)\varepsilon_{i}-h\varepsilon_{j}$
and
$(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i_{0}}-h\varepsilon_{i}-(h-1)\varepsilon_{j}$
are equivalent to the same equation which is

$A_{(t_{1}-1,t_{2},\dots,t_{n})-\varepsilon_{i_{0}}-(h-1)\varepsilon_{i}-(h-1)\varepsilon_{j}}=0.$
(4.16)

Thus, we have the same result as in the previous case.

###  4.2 If $t_{1}>m$

In this case the system $(S_{2})$ is of rank

$\Gamma_{n-1}^{k-t_{1}-1}-\frac{1}{2}(s-2)(s-1).$

The system $(S^{\prime}_{1})$ is of rank

$\Gamma_{n-1}^{k-t_{1}}-\frac{1}{2}s^{\prime}(s^{\prime}-1)$

where $s^{\prime}$ is the number of $t_{i}\geq m$ for $i\geq 2$. We proceed as
in the previous case, but here, since $t_{1}>m$, we prove that in
$(S^{\prime}_{1})$ the equations (4.15) and (4.16) become trivial for $i=j$
and $t_{i}>m$. The corresponding $(s-1)$ equations in $(S_{1})$ are
respectively

$A_{(t_{1},t_{2},\dots,t_{n})-(2h+1)\varepsilon_{i}}=0\quad\text{and}\quad
A_{(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i_{0}}-(2h-1)\varepsilon_{i}}=0.$

But, these equations appear also in $(S_{2})$ as equations corresponding
respectively to $(t_{1},t_{2},\dots,t_{n})-(2h+2)\varepsilon_{i}$ and
$(t_{1},t_{2},\dots,t_{n})-\varepsilon_{i_{0}}-2h\varepsilon_{i}$ for any
$i\geq 2$ such that $t_{i}>m$. Thus, the rank of (3.5) is

$\Gamma_{n}^{k-1}-\Gamma_{n-1}^{k-t_{1}-1}+\Gamma_{n-1}^{k-t_{1}-1}-\frac{1}{2}(s-2)(s-1)-(s-1)=\begin{pmatrix}n+k-2\\\
k\end{pmatrix}-\frac{1}{2}s(s-1).$

Theorem 4.9 is proved

$\Box$

Note that for $n=2$ and $\sigma_{2}\geq k-1$, we have always

${3\over 2}s(s-1)=3(s-1)={3\over 2}s(s-1)-3r={3\over
2}(s+r)(s+r-1)-3r=3\quad(s=2,\,r=0).$

So, in this case, we have

$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=\begin{pmatrix}2+k-2\\\
k\end{pmatrix}+3=4.$

Otherwise,
$\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})$
is one dimensional. This confirms the results of Basdouri et all in [2].

Now, we summarize our results in the following theorem

######  Theorem 4.10.

If

$\delta=k,\quad-2\bar{\lambda}=(t_{1},\,t_{2},\,\dots,\,t_{n})\in\\{0,\,1,\,\dots,\,k-1\\}^{n}\quad\text{
and }\quad\sigma_{n}=t_{1}+\dots+t_{n},$

then

$\mathrm{dim}\mathrm{H}^{2}_{\mathrm{diff}}(\mathfrak{sl}(2),\mathcal{D}_{\bar{\lambda},\mu})=2\left\\{\begin{array}[]{lllll}\Gamma_{n-1}^{k}&\text{if}\quad\sigma_{n}<k-1,\\\\[6.0pt]
\Gamma_{n-1}^{k}+3&\text{if}\quad\sigma_{n}=k-1,\\\\[6.0pt]
\Gamma_{n-1}^{k}+{3\over 2}(s-1)&\text{if}\quad\sigma_{n}=k,\\\\[6.0pt]
\Gamma_{n-1}^{k}+{3\over
2}(s+r)(s+r-1)-3r&\text{if}\quad\sigma_{n}=k+1,\,\text{ and }\,\max{t_{i}}\geq
2\\\\[6.0pt] \Gamma_{n-1}^{k}&\text{if}\quad\sigma_{n}=k+1,\,\text{ and
}\,\max{t_{i}}=1\\\\[6.0pt] \Gamma_{n-1}^{k}+{3\over
2}s(s-1)&\text{if}\quad\sigma_{n}=k+m,\quad m\geq 2,\\\\[6.0pt]
\end{array}\right..$

where $s$ is the number of $t_{i}>\sigma_{n}-k$, $r$ is the number of
$t_{i}=1$ and $\Gamma_{n-1}^{k}=\begin{pmatrix}n+k-2\\\ k\end{pmatrix}$.

## References

  * [1] Basdouri Okba and Nasri Elamine, $\mathfrak{sl}(2)$ -invariant 3-ary differential operators cohomology of $\mathfrak{sl}(2)$ acting on 3-ary differential operators, Int. J. Geom. Methods Mod. Phys. 14, 1750015 (2017) [17 pages], https://doi.org/10.1142/S0219887817500153
  * [2] Basdouri Imed, Hammami Sarra and Messaoud Olfa, Second cohomology space of $\mathfrak{sl}(2)$ acting on the space of bilinear bidifferential operators
  * [3] Bouarroudj S, Cohomology of the vector fields Lie algebras on $\mathbb{R}\mathbb{P}^{1}$ acting on bilinear differential operators, International Journal of Geometric Methods in Modern Physics (2005), 2; N 1, 23-40.
  * [4] Bouarroudj S, Ovsienko V, Three cocycles on Diff($S^{1}$) generalizing the Schwarzian derivative, Internat. Math. Res. Notices 1 (1998), 25–39.
  * [5] Feigin B L, Fuchs D B, Homology of the Lie algebras of vector fields on the line, Func. Anal. Appl., 14 (1980) 201–212.
  * [6] Fuchs D B, Cohomology of infinite-dimensional Lie algebras, Plenum Publ. New York, 1986.
  * [7] Gargoubi H, Sur la géométrie de l’espace des opérateurs différentiels linéaires sur $\mathbb{R}$, Bull. Soc. Roy. Sci. Liège. Vol. 69, 1, 2000, 21–47.
  * [8] Lecomte B P A, On the cohomology of $\mathfrak{sl}(n+1;\mathbb{R})$ acting on differential operators and $\mathfrak{sl}(n+1;\mathbb{R})$-equivariant symbols, Indag. Math. NS. 11 (1), (2000), 95 114.
  * [9] A. Nijenhuis, R.W. Richardson, Cohomology and deformations in graded Lie algebras. Bull. Amer. Math. Soc., 72 (1966), 1-29.


# Romanian Multiword Expression Detection Using Multilingual Adversarial
Training and Lateral Inhibition

Andrei-Marius Avram1, Verginica Barbu Mititelu2, Dumitru-Clementin Cercel1
1University Politehnica of Bucharest, Faculty of Automatic Control and
Computers
2Romanian Academy Research Institute for Artificial Intelligence
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Multiword expressions are a key ingredient for developing large-scale and
linguistically sound natural language processing technology. This paper
describes our improvements in automatically identifying Romanian multiword
expressions on the corpus released for the PARSEME v1.2 shared task. Our
approach assumes a multilingual perspective based on the recently introduced
lateral inhibition layer and adversarial training to boost the performance of
the employed multilingual language models. With the help of these two methods,
we improve the F1-score of XLM-RoBERTa by approximately 2.7% on unseen
multiword expressions, the main task of the PARSEME 1.2 edition. In addition,
our results can be considered SOTA performance, as they outperform the
previous results on Romanian obtained by the participants in this competition.

##  1 Introduction

The correct identification and handling of multiword expressions (MWEs) are
important for various natural language processing (NLP) applications, such as
machine translation, text classification, or information retrieval. For
example, in machine translation, if an MWE is not recognized as such and is
literally translated rather than as an expression, the resulting translation
either is confusing or has the wrong meaning Zaninello and Birch (2020). In
text classification, MWEs recognition can provide important information about
the topic or sentiment of a text Catone et al. (2019), while in information
retrieval, MWEs can clarify the meaning of a query and improve the accuracy of
search results Englmeier and Contreras (2021).

The PARSEME COST Action111https://typo.uni-konstanz.de/parseme/. organized
three editions Savary et al. (2017); Ramisch et al. (2018, 2020) of a shared
task that aimed at improving the identification of verbal MWEs (VMWEs) in
text. This work improves the results obtained in PARSEME 1.2 Ramisch et al.
(2020) for the Romanian language. We investigate the advantages of using
Romanian monolingual Transformer-based Vaswani et al. (2017) language models
together with merging all the datasets for each language presented at the
competition in a single corpus and then fine-tuning several multilingual
language models on it. Additionally, for the latter, we aim to enhance the
overall system’s performance by generating language-independent features, with
the help of two techniques, namely the lateral inhibition layer Păis (2022) on
top of the language models and adversarial training Lowd and Meek (2005)
between languages.

Our experiments show that by employing these two algorithms, the results of
the cross-lingual robustly optimized BERT approach (XLM-RoBERTa) Conneau et
al. (2020) improve by 2.7% on unseen MWEs when trained on the combined
dataset. Additionally, we report state-of-the-art (SOTA) results with the
monolingual training of Romanian Bidirectional Encoder Representations from
Transformer (RoBERT) Dumitrescu et al. (2020) in comparison with the results
obtained at the PARSEME 1.2 edition, achieving an F1-score of 60.46%, an
improvement of over 20%.

##  2 Dataset

The PARSEME multilingual corpus was annotated with several types of VMWEs, to
serve as training and testing material for the shared task. The quality of the
manual annotation was further enhanced by a semi-automatic way of ensuring
annotation consistency. For edition 1.2, the corpus contained 14 languages:
Basque, Chinese, French, German, Hebrew, Hindi, Irish, Italian, Modern Greek,
Polish, Portuguese, Romanian, Swedish, and Turkish.

The types of VMWEs (i.e., universal, quasi-universal, and language-specific
types) annotated therein are described in the annotation
guidelines222https://parsemefr.lis-lab.fr/parseme-st-guidelines/1.2/.. The
types of VMWEs annotated for Romanian are as follows: VID (verbal idiom) like
"fura somnul" (eng., "steal sleep-the", "fall asleep”), LVC.full (light verb
construction with a semantically bleached verb) like "da citire" (eng., "give
reading", "read”), LVC.cause (light verb construction in which the verb has a
causative meaning) like "da foc" (eng., "give fire", "put on fire”), and IRV
(inherently reflexive verb) like "se gândi" (eng., "Refl.Cl. think", "think").

The whole corpus version 1.2 contains 5.5 million tokens with 68k VMWEs
annotations, split into train, dev, and test sets, on the one hand for
controlling the distribution of unseen VMWEs both in dev with respect to test
and in test with respect to train+dev, and on the other hand in ensuring a
sufficient number of unseen VMWEs in the test set for each language. The
Romanian training corpus contains 195k tokens in which 1,218 VMWEs are
annotated. The Romanian dev set contains 134,340 tokens and 818 annotated
VMWEs; the Romanian test set includes 685,566 tokens and 4,135 annotated
VMWEs. The frequency of occurrence of VMWEs in Romanian ranges from 8% (for
LVC.full) to 22% (for LVC.cause), with an average of 12%, thus being quite
redundant Barbu Mititelu et al. (2019).

##  3 System Description

###  3.1 Monolingual Training

We experiment with four BERT-based models (first two monolingual and last two
multilingual) for MWE identification using only the Romanian part of the
PARSEME 1.2 corpus, namely the RoBERT, the Distilled Romanian BERT (Distil-
RoBERT) Avram et al. (2022a), the multilingual BERT (M-BERT) Kenton and
Toutanova (2019), and the XLM-RoBERTa Conneau et al. (2020). We follow the
standard sequence tagging procedure described in the original BERT model and
fine-tune the embeddings produced by the last layer for the input tokens to
predict the corresponding MWE labels using a feed-forward layer.

###  3.2 Multilingual Training

Our second and principal line of work here combines all the training sets of
the corpora. Therefore, we train the two multilingual language models on the
resulting dataset and then evaluate the models on the Romanian test set of the
PARSEME 1.2 shared task. In addition, we improve the performance of the system
by forcing the embeddings of the respective language models to depend less on
their source language and more on the semantic specificities of an MWE using a
lateral inhibition layer and adversarial training.

The general architecture of our multilingual training methodology is depicted
in Figure 1. It is divided into three major components: a multilingual BERT
model that acts as a feature extractor $F$ and produces the embeddings of the
tokens, a classifier $C$ whose role is to identify the MWEs in the given
texts, and a language discriminator $LG$ whose role is to recognize the
language of the input. We employ the lateral inhibition layer before feeding
the embeddings to $C$ and adversarially train $LG$ by reversing its gradient
before backpropagating through $F$. Further details on these two methods are
given below.

Figure 1: The multilingual training architecture. We use a multilingual BERT-
based model to extract the embeddings from the input tokens (green). All these
embeddings are fed into a classifier with a lateral inhibition layer to
predict the MWE labels (blue) and into an adversarially trained language
discriminator (orange). The block arrow depicts the forward pass, and the
dotted arrow the backward pass.

###  3.3 Lateral Inhibition

The neural inhibitory layer, modelled after the biological process of lateral
inhibition in the brain, has been successfully used for the named entity
recognition (NER) task in the past Păis (2022); Avram et al. (2022b); Mitrofan
and Păis (2022). We envisage that since the terms recognised by NER are just a
subset of the MWEs identification, both being grounded in sequence tagging,
introducing this layer into our model would also bring improvements in the
final performance of our system. However, in the previous work, the neural
inhibitory layer was mainly used to enhance the quality of the extracted named
entities. In contrast, in this work, we employ it to achieve language-
independent embeddings out of the multilingual transformer models.

The main idea behind the lateral inhibitory layer is quite simple. Given the
embeddings $X$ produced by a language model and a weight matrix $W$ with a
bias $b$, the output $Y$ of this layer is described in the following formula:

$Y=X*Diag(H(X*ZeroDiag(W^{T})+b))$ (1)

where $Diag$ is a function that creates a matrix whose main diagonal is the
vector given as input, $ZeroDiag$ is a function that sets a given matrix with
the zero value on the main diagonal, and $H$ is the Heaviside step function.

Equation 1 works well for the forward pass. However, since the Heaviside step
function is not differentiable, the lateral inhibition layer approximates the
respective gradients with the gradients of the parameterized Sigmoid function
Wunderlich and Pehle (2021), a technique known as surrogate gradient learning
Neftci et al. (2019).

###  3.4 Adversarial Training

Adversarial training of neural networks has been a highly influential area of
research in recent years, particularly in fields such as computer vision with
generative unsupervised models Gui et al. (2021). Adversarial training has
also been used to train predictive models Zhao et al. (2022), and in recent
research, both multilingual and cross-lingual adversarial neural networks were
introduced Hu et al. (2019); Guzman-Nateras et al. (2022). These networks are
designed to learn discriminative representations that are invariant to
language. In this study, we utilize the same methodology to learn task-
specific representations in a multilingual setting, trying to improve the
predictive capabilities of the employed multilingual transformer models.

Our methodology closely follows the domain adversarial neural network
algorithm (DANN) Ganin et al. (2016), the difference here being that instead
of reversing the gradient to create domain-independent features, we reverse it
to generate language-independent embeddings out of the multilingual
transformer models. As is the case for our system, DANN has in its composition
a feature extractor $F$, a label classifier $C$, and a domain classifier $D$
that is replaced in our work with a language classifier $LG$. Thus, the
gradient computation of each component can be formalized in the following
equations:

$\begin{array}[]{c}\theta_{C}=\theta_{C}-\alpha\frac{\partial
L_{y}}{\partial\theta_{C}}\\\ \theta_{LG}=\theta_{LG}-\alpha\frac{\partial
L_{lg}}{\partial\theta_{LG}}\\\ \theta_{F}=\theta_{F}-\alpha(\frac{\partial
L_{y}}{\partial\theta_{F}}-\lambda\frac{\partial
L_{lg}}{\partial\theta_{F}})\end{array}$ (2)

where $\theta_{C}$ are the parameters of the label classifier, $L_{y}$ is the
loss obtained by the label classifier when predicting the class labels $y$,
$\theta_{LG}$ are the parameters of the language classifier, $L_{lg}$ is the
loss obtained by the language classifier when predicting the language labels
$d$, $\theta_{F}$ are the parameters of the feature extractor, $\lambda$ is
the hyperparameter used to reverse the gradients, and $\alpha$ is the learning
rate.

##  4 Results

###  4.1 Monolingual Training

Model | Global MWE | Unseen MWE
---|---|---
P | R |  F1 | P | R | F1
MTLB-STRUCT | 89.88 | 91.05 | 90.46 | 28.84 | 41.47 | 34.02
TRAVIS-mono | 90.80 | 91.39 | 91.09 | 33.05 | 51.51 | 40.26
RoBERT | 90.73 | 93.74 | 92.21 | 52.97 | 70.69 | 60.56
Distil-RoBERT | 87.56 | 90.40 | 88.96 | 41.06 | 62.77 | 49.65
M-BERT | 90.39 | 90.11 | 90.25 | 46.82 | 51.09 | 48.86
XLM-RoBERTa | 90.72 | 91.46 | 91.09 | 51.54 | 62.77 | 56.61
Table 1: The results of the models trained on the monolingual Romanian set. Model | Global MWE | Unseen MWE
---|---|---
P | R |  F1 | P | R | F1
M-BERT | 91.34 | 88.46 | 89.88 | 49.90 | 48.12 | 48.99
M-BERT + LI | 90.78 | 88.85 | 89.81 | 45.06 | 45.15 | 45.10
M-BERT + Adv | 89.14 | 90.13 | 89.63 | 46.27 | 56.44 | 50.85
M-BERT + LI + Adv | 89.95 | 88.78 | 89.36 | 45.44 | 50.30 | 47.74
XLM-RoBERTa | 91.23 | 92.53 | 91.87 | 52.92 | 64.55 | 58.16
XLM-RoBERTa + LI | 91.12 | 92.02 | 91.02 | 52.11 | 61.19 | 56.28
XLM-RoBERTa + Adv | 89.45 | 92.87 | 91.12 | 54.91 | 63.96 | 59.09
XLM-RoBERTa + Adv + LI | 90.49 | 92.61 | 91.53 | 55.01 | 64.47 | 59.36
Table 2: The results of the multilingual models trained on the multilingual
combined dataset and evaluated on the Romanian set. LI means lateral
inhibition, and Adv means multilingual adversarial training.

Table 1 shows the results of our monolingual training. We report both the
overall scores (called global MWE) and the scores of the identified MWEs that
do not appear in the training set (called unseen MWE), as well as the results
of the best overall system (MTLB-STRUCT) Taslimipoor et al. (2020) and the
results of the best system on Romanian (TRAVIS-mono) Kurfalı (2020). All our
monolingual models outperform the MTLB-STRUCT and TRAVIS-mono systems by more
than 8% on unseen MWE, with RoBERT achieving an improvement of more than 20%.
We believe that this is due to the more intensive hyperparameter search that
we performed and the text preprocessing which consisted of things like
replacing the letters with diacritics in Romanian to the standard used in pre-
training or making sure that the tokenizer produces cased subtokens333These
text preprocessing techniques are suggested at
https://github.com/dumitrescustefan/Romanian-Transformers..

Both the highest global MWE and unseen MWE performance were achieved by the
monolingual RoBERT model, with F1-scores of 92.21% and 60.56%, respectively.
The second highest performance was obtained by the XLM-RoBERTa model, although
it is a multilingual model. Thus, XLM-RoBERTa outperformed the other
monolingual model, Distil-RoBERT, by 2.1% on global MWE and 7% on unseen MWE.
This phenomenon has also been noticed by Conneau et al. (2020), showing the
raw power of multilingual models pre-trained on a large amount of textual
data.

###  4.2 Multilingual Training

Table 2 shows the results for the multilingual training of both M-BERT and
XLM-RoBERTa. As in the monolingual training case, XLM-RoBERTa achieves better
performance, coming out on top with an F1-score of 58.16% in comparison with
the 48.99% F1-score obtained by M-BERT. We also notice that the simple
multilingual training (i.e., without lateral inhibition and adversarial
training) improves the results of the two models when trained on the
monolingual Romanian set.

The adversarial training improves the performance of both M-BERT and XLM-
RoBERTa in multilingual training. At the same time, the lateral inhibition
layer brought improvements only to the later when it was combined with
adversarial training. Thus, by merging the two methodologies, we outperform
the XLM-RoBERTa’s results trained on monolingual data (i.e., around 2.7% on
unseen MWEs), which was the main target of the competition, being behind
RoBERT with only 1.2%.

##  5 Conclusions

The detection and processing of MWEs play an important role in various areas
of NLP. This paper made notable improvements in unseen Romanian MWE
identification by employing a lateral inhibition layer and adversarial
training to multilingual large language models like XLM-RoBERTa. This way, we
were able to improve the results of XLM-RoBERTa. In addition, we achieved SOTA
results on this task with a simple fine-tuning of RoBERT that involved a
better hyperparameter search and text preprocessing pipeline, respectively.

Future work considers an analysis of the language-independent embeddings
produced in the multilingual training, together with more experiments on other
languages, to validate the generalization of this approach. In addition, we
intend to add these results in LiRo - the public benchmark for Romanian NLP
models Dumitrescu et al. (2021).

## References

  * Avram et al. (2022a) Andrei-Marius Avram, Darius Catrina, Dumitru-Clementin Cercel, Mihai Dascalu, Traian Rebedea, Vasile Pais, and Dan Tufis. 2022a.  Distilling the knowledge of Romanian BERTs using multiple teachers.  In _Proceedings of the Thirteenth Language Resources and Evaluation Conference_ , pages 374–384, Marseille, France. European Language Resources Association.
  * Avram et al. (2022b) Andrei-Marius Avram, Vasile Păis, and Maria Mitrofan. 2022b.  Racai@ smm4h’22: Tweets disease mention detection using a neural lateral inhibitory mechanism.  In _Proceedings of The Seventh Workshop on Social Media Mining for Health Applications, Workshop & Shared Task_, pages 1–3.
  * Barbu Mititelu et al. (2019) Verginica Barbu Mititelu, Mihaela Cristescu, and Mihaela Onofrei. 2019.  The Romanian corpus annotated with verbal multiword expressions.  In _Proceedings of the Joint Workshop on Multiword Expressions and WordNet (MWE-WN 2019)_ , pages 13–21, Florence, Italy. Association for Computational Linguistics.
  * Catone et al. (2019) Maria Carmela Catone, Mariacristina Falco, Alessandro Maisto, Serena Pelosi, and Alfonso Siano. 2019.  Automatic text classification through point of cultural interest digital identifiers.  In _International Conference on P2P, Parallel, Grid, Cloud and Internet Computing_ , pages 211–220. Springer.
  * Conneau et al. (2020) Alexis Conneau, Kartikay Khandelwal, Naman Goyal, Vishrav Chaudhary, Guillaume Wenzek, Francisco Guzmán, Édouard Grave, Myle Ott, Luke Zettlemoyer, and Veselin Stoyanov. 2020.  Unsupervised cross-lingual representation learning at scale.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 8440–8451.
  * Dumitrescu et al. (2020) Stefan Dumitrescu, Andrei-Marius Avram, and Sampo Pyysalo. 2020.  The birth of romanian bert.  In _Findings of the Association for Computational Linguistics: EMNLP 2020_ , pages 4324–4328.
  * Dumitrescu et al. (2021) Stefan Daniel Dumitrescu, Petru Rebeja, Beata Lorincz, Mihaela Gaman, Andrei Avram, Mihai Ilie, Andrei Pruteanu, Adriana Stan, Lorena Rosia, Cristina Iacobescu, et al. 2021.  Liro: Benchmark and leaderboard for romanian language tasks.  In _Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 1)_.
  * Englmeier and Contreras (2021) Kurt Englmeier and Pedro Contreras. 2021.  Aspect fusion as design paradigm for legal information retrieval.  In _International Conference on Intelligent Human Systems Integration_ , pages 547–553. Springer.
  * Ganin et al. (2016) Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. 2016\.  Domain-adversarial training of neural networks.  _The journal of machine learning research_ , 17(1):2096–2030.
  * Gui et al. (2021) Jie Gui, Zhenan Sun, Yonggang Wen, Dacheng Tao, and Jieping Ye. 2021.  A review on generative adversarial networks: Algorithms, theory, and applications.  _IEEE Transactions on Knowledge and Data Engineering_.
  * Guzman-Nateras et al. (2022) Luis Guzman-Nateras, Minh Van Nguyen, and Thien Nguyen. 2022.  Cross-lingual event detection via optimized adversarial training.  In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 5588–5599.
  * Hu et al. (2019) Ke Hu, Hasim Sak, and Hank Liao. 2019.  Adversarial training for multilingual acoustic modeling.  _arXiv preprint arXiv:1906.07093_.
  * Kenton and Toutanova (2019) Jacob Devlin Ming-Wei Chang Kenton and Lee Kristina Toutanova. 2019.  Bert: Pre-training of deep bidirectional transformers for language understanding.  In _Proceedings of NAACL-HLT_ , pages 4171–4186.
  * Kurfalı (2020) Murathan Kurfalı. 2020.  Travis at parseme shared task 2020: How good is (m) bert at seeing the unseen?  In _Proceedings of the Joint Workshop on Multiword Expressions and Electronic Lexicons_ , pages 136–141.
  * Lowd and Meek (2005) Daniel Lowd and Christopher Meek. 2005.  Adversarial learning.  In _Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining_ , pages 641–647.
  * Mitrofan and Păis (2022) Maria Mitrofan and Vasile Păis. 2022.  Improving romanian bioner using a biologically inspired system.  In _Proceedings of the 21st Workshop on Biomedical Language Processing_ , pages 316–322.
  * Neftci et al. (2019) Emre O Neftci, Hesham Mostafa, and Friedemann Zenke. 2019.  Surrogate gradient learning in spiking neural networks: Bringing the power of gradient-based optimization to spiking neural networks.  _IEEE Signal Processing Magazine_ , 36(6):51–63.
  * Păis (2022) Vasile Păis. 2022.  Racai at semeval-2022 task 11: Complex named entity recognition using a lateral inhibition mechanism.  In _Proceedings of the 16th International Workshop on Semantic Evaluation (SemEval-2022)_ , pages 1562–1569.
  * Ramisch et al. (2018) Carlos Ramisch, Silvio Ricardo Cordeiro, Agata Savary, Veronika Vincze, Verginica Barbu Mititelu, Archna Bhatia, Maja Buljan, Marie Candito, Polona Gantar, Voula Giouli, Tunga Güngör, Abdelati Hawwari, Uxoa Iñurrieta, Jolanta Kovalevskaitė, Simon Krek, Timm Lichte, Chaya Liebeskind, Johanna Monti, Carla Parra Escartín, Behrang QasemiZadeh, Renata Ramisch, Nathan Schneider, Ivelina Stoyanova, Ashwini Vaidya, Abigail Walsh, Cristina Aceta, Itziar Aduriz, Jean-Yves Antoine, Špela Arhar Holdt, Gözde Berk, Agnė Bielinskienė, Goranka Blagus, Loic Boizou, Claire Bonial, Valeria Caruso, Jaka Čibej, Matthieu Constant, Paul Cook, Mona Diab, Tsvetana Dimitrova, Rafael Ehren, Mohamed Elbadrashiny, Hevi Elyovich, Berna Erden, Ainara Estarrona, Aggeliki Fotopoulou, Vassiliki Foufi, Kristina Geeraert, Maarten van Gompel, Itziar Gonzalez, Antton Gurrutxaga, Yaakov Ha-Cohen Kerner, Rehab Ibrahim, Mihaela Ionescu, Kanishka Jain, Ivo-Pavao Jazbec, Teja Kavčič, Natalia Klyueva, Kristina Kocijan, Viktória Kovács, Taja Kuzman, Svetlozara Leseva, Nikola Ljubešić, Ruth Malka, Stella Markantonatou, Héctor Martínez Alonso, Ivana Matas, John McCrae, Helena de Medeiros Caseli, Mihaela Onofrei, Emilia Palka-Binkiewicz, Stella Papadelli, Yannick Parmentier, Antonio Pascucci, Caroline Pasquer, Maria Pia di Buono, Vandana Puri, Annalisa Raffone, Shraddha Ratori, Anna Riccio, Federico Sangati, Vishakha Shukla, Katalin Simkó, Jan Šnajder, Clarissa Somers, Shubham Srivastava, Valentina Stefanova, Shiva Taslimipoor, Natasa Theoxari, Maria Todorova, Ruben Urizar, Aline Villavicencio, and Leonardo Zilio. 2018.  Annotated corpora and tools of the PARSEME shared task on automatic identification of verbal multiword expressions (edition 1.1).  LINDAT/CLARIAH-CZ digital library at the Institute of Formal and Applied Linguistics (ÚFAL), Faculty of Mathematics and Physics, Charles University.
  * Ramisch et al. (2020) Carlos Ramisch, Bruno Guillaume, Agata Savary, Jakub Waszczuk, Marie Candito, Ashwini Vaidya, Verginica Barbu Mititelu, Archna Bhatia, Uxoa Iñurrieta, Voula Giouli, Tunga Güngör, Menghan Jiang, Timm Lichte, Chaya Liebeskind, Johanna Monti, Renata Ramisch, Sara Stymme, Abigail Walsh, Hongzhi Xu, Emilia Palka-Binkiewicz, Rafael Ehren, Sara Stymne, Matthieu Constant, Caroline Pasquer, Yannick Parmentier, Jean-Yves Antoine, Carola Carlino, Valeria Caruso, Maria Pia Di Buono, Antonio Pascucci, Annalisa Raffone, Anna Riccio, Federico Sangati, Giulia Speranza, Renata Ramisch, Silvio Ricardo Cordeiro, Helena de Medeiros Caseli, Isaac Miranda, Alexandre Rademaker, Oto Vale, Aline Villavicencio, Gabriela Wick Pedro, Rodrigo Wilkens, Leonardo Zilio, Monica-Mihaela Rizea, Mihaela Ionescu, Mihaela Onofrei, Jia Chen, Xiaomin Ge, Fangyuan Hu, Sha Hu, Minli Li, Siyuan Liu, Zhenzhen Qin, Ruilong Sun, Chenweng Wang, Huangyang Xiao, Peiyi Yan, Tsy Yih, Ke Yu, Songping Yu, Si Zeng, Yongchen Zhang, Yun Zhao, Vassiliki Foufi, Aggeliki Fotopoulou, Stella Markantonatou, Stella Papadelli, Sevasti Louizou, Itziar Aduriz, Ainara Estarrona, Itziar Gonzalez, Antton Gurrutxaga, Larraitz Uria, Ruben Urizar, Jennifer Foster, Teresa Lynn, Hevi Elyovitch, Yaakov Ha-Cohen Kerner, Ruth Malka, Kanishka Jain, Vandana Puri, Shraddha Ratori, Vishakha Shukla, Shubham Srivastava, Gozde Berk, Berna Erden, and Zeynep Yirmibeşoğlu. 2020.  Annotated corpora and tools of the PARSEME shared task on semi-supervised identification of verbal multiword expressions (edition 1.2).  LINDAT/CLARIAH-CZ digital library at the Institute of Formal and Applied Linguistics (ÚFAL), Faculty of Mathematics and Physics, Charles University.
  * Savary et al. (2017) Agata Savary, Carlos Ramisch, Silvio Ricardo Cordeiro, Federico Sangati, Veronika Vincze, Behrang QasemiZadeh, Marie Candito, Fabienne Cap, Voula Giouli, Ivelina Stoyanova, Antoine Doucet, Kübra Adalı, Verginica Barbu Mititelu, Eduard Bejček, Ismail El Maarouf, Gülşen Eryiğit, Luke Galea, Yaakov Ha-Cohen Kerner, Chaya Liebeskind, Johanna Monti, Carla Parra Escartín, Jolanta Kovalevskaitė, Simon Krek, Lonneke van der Plas, Cristina Aceta, Itziar Aduriz, Jean-Yves Antoine, Greta Attard, Kirsty Azzopardi, Loic Boizou, Janice Bonnici, Mert Boz, Ieva Bumbulienė, Jael Busuttil, Valeria Caruso, Manuela Cherchi, Matthieu Constant, Monika Czerepowicka, Anna De Santis, Tsvetana Dimitrova, Tutkum Dinç, Hevi Elyovich, Ray Fabri, Alison Farrugia, Jamie Findlay, Aggeliki Fotopoulou, Vassiliki Foufi, Sara Anne Galea, Polona Gantar, Albert Gatt, Anabelle Gatt, Carlos Herrero, Uxoa Iñurrieta, Glorianna Jagfeld, Milena Hnátková, Mihaela Ionescu, Natalia Klyueva, Svetla Koeva, Viktória Kovács, Taja Kuzman, Svetlozara Leseva, Sevi Louisou, Teresa Lynn, Ruth Malka, Héctor Martínez Alonso, John McCrae, Helena de Medeiros Caseli, Ayşenur Miral, Amanda Muscat, Joakim Nivre, Michael Oakes, Mihaela Onofrei, Yannick Parmentier, Caroline Pasquer, Maria Pia di Buono, Belem Priego Sanchez, Annalisa Raffone, Renata Ramisch, Erika Rimkutė, Monica-Mihaela Rizea, Katalin Simkó, Michael Spagnol, Valentina Stefanova, Sara Stymne, Umut Sulubacak, Nicole Tabone, Marc Tanti, Maria Todorova, Zdenka Urešová, Aline Villavicencio, and Leonardo Zilio. 2017.  Annotated corpora and tools of the PARSEME shared task on automatic identification of verbal multiword expressions (edition 1.0).  LINDAT/CLARIAH-CZ digital library at the Institute of Formal and Applied Linguistics (ÚFAL), Faculty of Mathematics and Physics, Charles University.
  * Taslimipoor et al. (2020) Shiva Taslimipoor, Sara Bahaadini, and Ekaterina Kochmar. 2020.  Mtlb-struct@ parseme 2020: Capturing unseen multiword expressions using multi-task learning and pre-trained masked language models.  In _Proceedings of the Joint Workshop on Multiword Expressions and Electronic Lexicons_ , pages 142–148.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  _Advances in neural information processing systems_ , 30.
  * Wunderlich and Pehle (2021) Timo C Wunderlich and Christian Pehle. 2021.  Event-based backpropagation can compute exact gradients for spiking neural networks.  _Scientific Reports_ , 11(1):12829.
  * Zaninello and Birch (2020) Andrea Zaninello and Alexandra Birch. 2020.  Multiword expression aware neural machine translation.  In _Proceedings of the 12th Language Resources and Evaluation Conference_ , pages 3816–3825.
  * Zhao et al. (2022) Weimin Zhao, Sanaa Alwidian, and Qusay H Mahmoud. 2022.  Adversarial training methods for deep learning: A systematic review.  _Algorithms_ , 15(8):283.