# On Chevalley restriction theorem for semi-reductive algebraic groups and its
applications

Ke Ou, Bin Shu and Yu-Feng Yao  Department of Statistics and Mathematics,
Yunnan University of Finance and Economics, Kunming, 650221, China.
<EMAIL_ADDRESS>School of Mathematical Sciences, East China Normal
University, Shanghai, 200241, China<EMAIL_ADDRESS>Department of
Mathematics, Shanghai Maritime University, Shanghai, 201306, China.
<EMAIL_ADDRESS>
###### Abstract.

An algebraic group is called semi-reductive if it is a semi-direct product of
a reductive subgroup and the unipotent radical. Such a semi-reductive
algebraic group naturally arises and also plays a key role in the study of
modular representations of non-classical finite-dimensional simple Lie
algebras in positive characteristic, and some other cases. Let $G$ ba a
connected semi-reductive algebraic group over an algebraically closed field
$\mathbb{F}$ and $\mathfrak{g}=\mathsf{Lie}(G)$. It turns out that $G$ has
many same properties as reductive groups, such as the Bruhat decomposition. In
this note, we obtain an analogue of classical Chevalley restriction theorem
for $\mathfrak{g}$, which says that the $G$-invariant ring
$\mathbb{F}[\mathfrak{g}]^{G}$ is a polynomial ring if $\mathfrak{g}$
satisfies a certain “posivity” condition suited for lots of cases we are
interested in. As applications, we further investigate the nilpotent cones and
resolutions of singularities for semi-reductive Lie algebras.

###### Key words and phrases:

Semi-reductive algebraic groups, Semi-reductive Lie algebras, Chevalley
restriction theorem, Nilpotent cone, Steinberg map, Springer resolution

###### 2010 Mathematics Subject Classification:

20G05, 20G07, 20G15, 17B10, 17B45

This work is partially supported by the National Natural Science Foundation of
China (Grant Nos. 12071136, 11671138 and 11771279), Shanghai Key Laboratory of
PMMP (No. 13dz2260400), the Fundamental Research Funds of Yunnan Province
(Grant No. 2020J0375) and the Fundamental Research Funds of YNUFE (Grant No.
80059900196).

##  1\. Introduction

Let $\mathbb{F}$ be an algebraically closed field. An algebraic group $G$ over
$\mathbb{F}$ is called semi-reductive if $G=G_{0}\ltimes U$ with $G_{0}$ being
a reductive subgroup, and $U$ the unipotent radical. Let
$\mathfrak{g}=\mathsf{Lie}(G)$, $\mathfrak{g}_{0}=\mathsf{Lie}(G_{0})$ and
$\mathfrak{u}=\mathsf{Lie}(U)$, then
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{u}$, which is called a semi-
reductive Lie algebra.

Preliminarily, the motivation of initiating the study of semi-reductive
algebraic groups and Lie algebras comes from the study of irreducible
representations of finite-dimensional non-classical simple Lie algebras in
prime characteristic (see Example 2.1). In the direction of this new study,
there have been some interesting mathematical phenomenons revealing (to see
[13], [14], [19], [22]).

In the present paper, we focus on establishing an analogue of the classical
Chevalley restriction theorem in the case of semi-reductive algebraic groups.

Recall that for a connected reductive algebraic group $L$ over $\mathbb{F}$
with $\mathfrak{l}=\mathsf{Lie}(L)$, and a maximal torus $T$ of $L$ with
$\mathfrak{h}=\mathsf{Lie}(T)$. We have the Weyl group
$\mathscr{W}=\operatorname{Nor}_{L}(T)/\operatorname{Cent}_{L}(T)$. The
classical Chevalley restriction theorem asserts that the restriction
homomorphism $\mathbb{F}[\mathfrak{l}]\rightarrow\mathbb{F}[\mathfrak{h}]$
induces an isomorphism of algebras of invariants:
$\mathbb{F}[\mathfrak{l}]^{{L}}\cong\mathbb{F}[\mathfrak{h}]^{\mathscr{W}}$
where $\mathbb{F}[\mathfrak{l}]$ (resp. $\mathbb{F}[\mathfrak{h}]$) denotes
the algebra of polynomial functions on $\mathfrak{l}$ (resp. $\mathfrak{h}$)
(see [5, Proposition 7.12] if $\operatorname{ch}(\mathbb{F})>0$ and [4, §23.1
and Appendix 23] for semisimple $\mathfrak{l}$ if
$\operatorname{ch}(\mathbb{F})=0$). The Chevalley restriction theorem plays a
key role in the study of representations of reductive Lie algebras. It has
been generalized in various ways and cases (see [8], [11] etc.). However, it
may fail if $\mathfrak{l}$ is replaced with an arbitrary algebraic Lie algebra
$\mathfrak{q}.$ In particular, one has to distinguish the adjoint and
coadjoint representations of $\mathfrak{q}.$ See §4 for a concrete example.

In this note, we will generalize the Chevalley restriction theorem to the case
of a semi-reductive algebraic group $G$ if $G$ satisfies a certain “posivity”
condition (see Convention 2.9 for the precise definition).

The generalization of the Chevalley restriction theorem seems to be quite
useful in understanding the structure of semi-reductive Lie algebras. As
applications, we investigate the nilpotent cones and resolutions of
singularities for semi-reductive Lie algebras. In particular, Propositions
5.2, 5.3 and 5.9 provide the structure of nilpotent cone $\mathcal{N}$,
Steinberg maps and resolution of singularities of $\mathcal{N}$ respectively.

Our paper is organized as follows. In §2, we recall some known results, and
investigate the structures of semi-reductive algebraic groups as well as semi-
reductive Lie algebras. In §3, the Chevalley restriction theorem for semi-
reductive Lie algebras $\mathfrak{g}$ are presented provided $\mathfrak{g}$
satisfied a certain “posivity” condition and some restrictions on the
characteristic of $\mathbb{F}.$ As the first application, we illustrate the
difference between the adjoint and coadjoint representations of $\mathfrak{g}$
in §4. As the second application, we study the nilpotent cone of
$\mathfrak{g}$ and its Springer resolution in the final section.

##  2\. Semi-redutive groups and semi-reductive Lie algebeas

###  2.1. Notions and assumptions

Throughout this paper, all vector spaces and varieties are defined over an
algebraically closed field $\mathbb{F}$.

$G=G_{0}\ltimes U$ be a semi-reductive group with $G_{0}$ being a reductive
subgroup, and $U$ the unipotent radical. Let $\mathfrak{g}=\mathsf{Lie}(G)$,
$\mathfrak{g}_{0}=\mathsf{Lie}(G_{0})$ and $\mathfrak{u}=\mathsf{Lie}(U)$,
then $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{u}$. Let
$\operatorname{pr}:\mathfrak{g}\rightarrow\mathfrak{g}_{0},\
\pi:\mathfrak{g}\rightarrow\mathfrak{u}$ be the projections. In the following,
we give some examples of semi-reductive Lie algebras (resp. semi-reductive
algebraic groups).

######  Example 2.1.

(The non-negative graded part of a restricted Lie algebra of Cartan type over
$\mathbb{F}$ of prime characteristic $p>0$) Let
$\mathscr{A}(n)=\mathbb{F}[T_{1},\ldots,T_{n}]/(T_{1}^{p},\ldots,T_{n}^{p})$,
a quotient of the polynomial ring by the ideal generated by $T_{i}^{p}$,
$1\leq i\leq n$. Set $\mathscr{L}=\text{Der}(\mathscr{A}(n))$. Then
$\mathscr{L}$ is a simple Lie algebra unless $n=1$ and $p=2$ (this simple Lie
algebra is usually called a generalized Witt algebra, denoted by $W(n)$).
Associated with the degrees of polynomial quotients, $\mathscr{L}$ becomes a
graded algebra $\mathscr{L}=\sum_{d\geq-1}\mathscr{L}_{[d]}$. Then
$\mathscr{L}$ has a filtration

$\mathscr{L}=\mathscr{L}_{-1}\supset\mathscr{L}_{0}\supset\cdots\supset\mathscr{L}_{n(p-1)-1}\supset
0.$

where

$\mathscr{L}_{i}=\bigoplus_{j\geq i}\mathscr{L}_{[j]},\,\,-1\leq i\leq
n(p-1)-1.$

Let $G=\operatorname{Aut}(\mathscr{L})$ be the automorphism group of
$\mathscr{L}$ and $\mathfrak{g}=\mathsf{Lie}(G)$. Then $G=\text{\rm
GL}(n,\mathbb{F})\ltimes U$ and $\mathfrak{g}=\mathscr{L}_{0}$ with
$\mathsf{Lie}(\text{\rm
GL}(n,\mathbb{F}))=\mathscr{L}_{[0]}(\cong\mathfrak{gl}(n,\mathbb{F}))$,
$\mathsf{Lie}(U)=\mathscr{L}_{1}=\sum_{d\geq 1}\mathscr{L}_{[d]}$ (see [9,
23]). So $G$ is a semi-reductive group and $\mathscr{L}_{0}$ is a semi-
reductive Lie algebra. According to the known results (cf. [10], [16], [17],
[20], [24] and [25], etc), the representation theory of $W(n)$ along with
other Cartan type Lie algebras are, to most extent, determined by that of its
distinguished maximal filtered subalgebra $\mathscr{L}_{0}$. So the study of
semi-reductive Lie algebra $\mathscr{L}_{0}$ becomes a crucial important
topic.

More generally, apart from $W(n)$, there are another three series of Cartan
type restricted simple Lie algebras $S(n),H(n),K(n)$ (see [15], [21], or the
forthcoming Example 2.12). Each of them is endowed with the corresponding
graded structure. Similarly, one can consider $\mathfrak{g}=\sum_{d\geq
0}X(n)_{[d]}$ for $X\in\\{S,H,K\\}$ and $G=\text{Aut}(X(n))$. The
corresponding semi-reductive groups and Lie algebras also appear as above.

######  Example 2.2.

(Parabolic subalgebras of reductive Lie algebras in any characteristic) Let
$\tilde{G}$ be a reductive algebraic group over $\mathbb{F}$ and $G$ be a
parabolic subgroup of $\tilde{G}$. Then $G$ is an semi-reductive group.

######  Example 2.3.

Let $G$ be a connected reductive algebraic group over $\mathbb{F}$ satisfying
that the characteristic of $\mathbb{F}$ is good for $G$, in the sense of [4,
§3.9]. For any given nilpotent $X\in\mathfrak{g}$, let $G_{X}$ be the
centralizer of $X$ in $G$, and $\mathfrak{g}_{X}=\mathsf{Lie}(G_{X})$. By [5,
§5.10-§5.11], $G_{X}=C_{X}\ltimes R_{X}$ is semi-reductive.

######  Example 2.4.

(Enhanced reductive algebraic groups) Let $G_{0}$ be a connected reductive
algebraic group over $\mathbb{F}$, and $(M,\rho)$ a finite-dimensional
representation of $G_{0}$ with representation space $M$ over $\mathbb{F}$.
Here and further, representations of an algebraical group always mean its
rational representations. Consider the product variety $G_{0}\times M$. Regard
$M$ as an additive algebraic group. The variety $G_{0}\times M$ is endowed
with a cross product structure denoted by $G_{0}\times_{\rho}M$, by defining
for any $(g_{1},v_{1}),(g_{2},v_{2})\in G_{0}\times M$

$\displaystyle(g_{1},v_{1})\cdot(g_{2},v_{2}):=(g_{1}g_{2},\rho(g_{1})v_{2}+v_{1}).$
(2.1)

Then by a straightforward computation it is easily known that
$\underline{G_{0}}:=G_{0}\times_{\rho}M$ becomes a group with unity $(e,0)$
for the unity $e\in G_{0}$, and $(g,v)^{-1}=(g^{-1},-\rho(g)^{-1}v)$. Then
$G_{0}\times_{\rho}M$ has a subgroup $G_{0}$ identified with $(G_{0},0)$ and a
subgroup $M$ identified with $(e,M)$. Furthermore, $\underline{G_{0}}$ is
connected since $G_{0}$ and $M$ are irreducible varieties. We call
$\underline{G_{0}}$ an enhanced reductive algebraic group associated with the
representation space $M$. What is more, $G_{0}$ and $M$ are closed subgroups
of $\underline{G_{0}}$, and $M$ is a normal closed subgroup. Actually, we have
$\underline{g}\underline{v}\underline{g}^{-1}=\rho(g)v$. Here and further
$\underline{g}$ stands for $(g,0)$ and $\underline{v}$ stands for $(e,v)$. Set
$\mathfrak{g}_{0}=\mathsf{Lie}(G_{0})$. Then $(\mathsf{d}(\rho),M)$ becomes a
representation of $\mathfrak{g}_{0}$. Naturally,
$\mathsf{Lie}(\underline{G_{0}})=\mathfrak{g}_{0}\oplus M$, with Lie bracket

$[(X_{1},v_{1}),(X_{2},v_{2})]:=([X_{1},X_{2}],\mathsf{d}(\rho)(X_{1})v_{2}-\mathsf{d}(\rho)(X_{2})v_{1}),$

which is called an enhanced reductive Lie algebra.

Clearly, $\underline{G_{0}}$ is a semi-reductive group with $M$ being the
unipotent radical.

### 2.2.

In the sequel, we always assume that $G=G_{0}\ltimes U$ is a connected semi-
reductive algebraic group over an algebraically closed field $\mathbb{F}$
where $G_{0}$ is a connected reductive subgroup and $U$ the unipotent radical.
Let $\mathfrak{g}=\mathsf{Lie}(G)$ be the Lie algebra of $G$. Recall that a
Borel subgroup (resp. Borel subalgebra) of $G$ (resp. $\mathfrak{g}$) is a
maximal solvable subgroup (resp. subalgebra) of $G$ (resp. $\mathfrak{g}$). In
the following we will illustrate the structure of Borel subgroups of $G$.

######  Lemma 2.5.

The following statements hold.

  * (1)

Suppose $B$ is a Borel subgroup of $G$. Then $B\supset U$. Furthermore,
$B_{0}:=B\cap G_{0}$ is a Borel subgroup of $G_{0}$, and $B=B_{0}\ltimes U$.

  * (2)

Any maximal torus $T$ of $G$ is conjugate to a maximal torus $T_{0}$ of
$G_{0}$.

###### Proof.

(1) As $U$ is the unipotent radical of $G$, $BU$ is still a closed subgroup
containing $B$. We further assert that $BU$ is solvable. Firstly, by a
straightforward computation we have that the $i$th derived subgroup
$\mathcal{D}^{i}(BU)$ is contained in $\mathcal{D}^{i}(B)U$. By the
solvableness of $B$, there exists some positive integer $t$ such that
$\mathcal{D}^{t}(B)$ is the identity group $\\{e\\}$. So
$\mathcal{D}^{t}(BU)\subset U$. Secondly, as $U$ is unipotent, and then
solvable. So there exists some positive integer $r$ such that
$\mathcal{D}^{r}(U)=\\{e\\}$. Hence $\mathcal{D}^{t+r}(BU)=\\{e\\}$. The
assertion is proved.

The maximality of the solvable closed subgroup $B$ implies that $BU=B$. This
is to say, $U$ is contained in the unipotent radical $B_{u}$ of $B$. Set
$B_{0}=B\cap G_{0}$ which is clearly a closed solvable subgroup of $G_{0}$. By
the same token as above, $B_{0}U$ is a closed solvable subgroup of $B$. On the
other hand, for any $b\in B$, by the definition of semi-reductive groups we
have $b=b_{0}u$ for some $b_{0}\in G_{0}$ and $u\in U$. As $U\subset B$, we
have further that $b_{0}=bu^{-1}\in B_{0}$. It follows that $B$ is contained
in $B_{0}\ltimes U$. The remaining thing is to prove that $B_{0}$ is a Borel
subgroup of $G_{0}$. It is clear that $B_{0}$ is a solvable closed subgroup of
$G_{0}$. If $B_{0}$ is not a maximal solvable closed subgroup of $G_{0}$. We
may suppose $B_{0}^{\prime}$ is a larger one properly containing $B_{0}$. Then
the solvable closed subgroup $B_{0}^{\prime}U$ of $G$ contains $B$ properly.
It contradicts with the maximality of $B$. Hence $B_{0}$ is really maximal.
Summing up, the statement in (1) is proved.

(2) Note that the maximal tori in $G$ are all conjugate (see [1, 11.3]). This
statement follows from (1). ∎

As a corollary to Lemma 2.5, we have the following fact for semi-reductive Lie
algebras.

######  Lemma 2.6.

For a semi-reductive group $G$ and $\mathfrak{g}=\mathsf{Lie}(G)$, all maximal
tori of $\mathfrak{g}$ are conjugate under adjoint action of $G$. Moreover,
$\mathfrak{g}_{\text{ss}}=\bigcup_{T}\mathsf{Lie}(T)$ where
$\mathfrak{g}_{\text{ss}}$ is the set consisting of all semisimple elements
and $T$ runs over all maximal tori of $G$.

By Lemmas 2.5 and 2.6, we can choose a maximal torus $T$ of $G$, which lies in
$G_{0}$ without loss of generality. By [1, §8.17], we have the following
decomposition of root spaces associated with
$\mathfrak{t}:=\operatorname{Lie}(T)$

$\displaystyle\mathfrak{g}=\mathfrak{t}\oplus\sum_{\alpha\in\Phi(G_{0},T)}(\mathfrak{g}_{0})_{\alpha}\oplus\sum_{\alpha\in\Phi(U,T)}\mathfrak{u}_{\alpha}$
(2.2)

where $\Phi(U,T)$ is the subset of $X(G,T)$ satisfying that for
$\mathfrak{u}:=\mathsf{Lie}(U)$,

$\mathfrak{u}=\sum_{\alpha\in\Phi(U,T)}\mathfrak{u}_{\alpha}.$

In the consequent arguments, the root system $\Phi(G_{0},T)$ will be denoted
by $\Delta$ for simplicity, which is actually independent of the choice of
$T$. We fix a positive root system $\Delta^{+}:=\Phi(G_{0},T)^{+}$. The
corresponding Borel subgroup will be denoted by $B^{+}$ which contains $T$,
and the corresponding simple system is denoted by $\Pi$.

### 2.3.

The following facts are clear.

######  Lemma 2.7.

Let $G$ be a connected semi-reductive group, and $T$ a maximal torus of
$G_{0}$.

  * (1)

Set $\mathscr{W}(G,T):=\text{\rm Nor}_{G}(T)/\text{\rm Cent}_{G}(T)$. Then
$\mathscr{W}(G,T)\cong\mathscr{W}$, where $\mathscr{W}$ is the Weyl group of
$G_{0}$. This is to say, $G$ admits the Weyl group $\mathscr{W}$ coinciding
with the one of $G_{0}$.

  * (2)

Set $\\{\dot{w}\mid w\in W\\}$ be a set of representatives in $N_{G}(T)$ of
the elements of $\mathscr{W}$. Denote by $C(w)$ the double coset
$B^{+}\dot{w}B^{+}$, and by $C_{0}(w)$ the corresponding component
$B^{+}_{0}\dot{w}B^{+}_{0}$ of the Bruhat decomposition of $G_{0}$. Then for
any $w\in\mathscr{W}$

$\displaystyle C(w)=C_{0}(w)\ltimes U.$ (2.3)

  * (3)

Let $w=s_{1}\cdots s_{h}$ be a reduced expression of $w\in\mathscr{W}$, with
$s_{i}=s_{\gamma_{i}}$ for $\gamma_{i}\in\Pi$. Then
$C(w)\cong{\mathbb{A}}^{h}\times B^{+}$.

  * (4)

For $w\in\mathscr{W}$, set $\Phi(w):=\\{\alpha\in\Delta^{+}\mid
w\cdot\alpha\in-\Delta^{+}\\}$, and $U_{w}:=\Pi_{\alpha\in\Phi(w)}U_{\alpha}$,
then $U_{w^{-1}}\times B^{+}\cong C(w)$ via sending $(u,b)$ onto $u\dot{w}b$.

  * (5)

(Bruhat Decomposition) We have
$G={\dot{\bigcup}}_{w\in\mathscr{W}}B^{+}\dot{w}B^{+}$. Therefore, for any
$g\in G$, $g$ can be written uniquely in the form $u\dot{w}b$ with
$w\in\mathscr{W}$, $u\in U_{w^{-1}}$ and $b\in B^{+}$.

###### Proof.

(1) Note that $U$ is the unipotent radical of $G$, and $G=G_{0}\ltimes U$. We
have $\text{\rm Nor}_{G}(T)=\text{\rm Nor}_{G_{0}}(T)$ and $\text{\rm
Cent}_{G}(T)=\text{\rm Cent}_{G_{0}}(T)$. The statement is proved.

(2) Note that $U\dot{w}=\dot{w}U$ and $B^{+}=B^{+}_{0}U=UB^{+}_{0}$ for
$B^{+}_{0}=B^{+}\cap G_{0}$. We have

$\displaystyle C(w)$ $\displaystyle=B^{+}_{0}U\dot{w}B^{+}_{0}U$ (2.4)
$\displaystyle=B^{+}_{0}\dot{w}UB^{+}_{0}U$ (2.5)
$\displaystyle=B^{+}_{0}\dot{w}B^{+}_{0}U$ (2.6)
$\displaystyle=C_{0}(w)\ltimes U.$ (2.7)

(3) and (4) follow from (2) along with [18, Lemma 8.3.6].

(5) follows from (2) along with the Bruhat decomposition Theorem for $G_{0}$
([18, Theorem 8.3.8]). ∎

######  Proposition 2.8.

Let $G$ be a connected semi-reductive group. Keep the notations as in Lemma
2.7. The following statements hold.

  * (1)

The group $G$ admits a unique open double coset $C(w_{0})$, where $w_{0}$ is
the longest element of $\mathscr{W}$.

  * (2)

For any given Borel subgroup $B^{+}$, let $\mathcal{B}$ denote the homogeneous
space $G/B^{+}$, and $\pi:G\rightarrow\mathcal{B}$ be the canonical morphism.
Then $\pi$ has local sections.

  * (3)

For a rational $B^{+}$-module $M$, there exists a fibre bundle over
$\mathcal{B}$ associated with $M$, denoted by $G\times^{B^{+}}M$.

###### Proof.

(1) Note that $C(w_{0})$ is a $B^{+}\times B^{+}$-orbit in $G$ under double
action, and $C(w_{0})$ is open in its closure. Thanks to Lemma 2.7(4), we have

$\dim C(w_{0})=\\#\\{\Delta^{+}\\}+\dim B^{+}=\dim G.$

On the hand, $G$ is irreducible, so $G=\overline{C(w_{0})}$. It follows that
$C(w_{0})$ is an open subset of $G$. By the uniqueness of the longest element
in $\mathscr{W}$ and Lemma 2.3(3)(5), we have that $C(w_{0})$ is a unique open
double coset.

(2) According to [18, §5.5.7], we only need to certify that (a) $\mathcal{B}$
is covered by open sets $\\{U\\}$; (b) each of such open sets has a section,
i.e. a morphism $\sigma:U\rightarrow\mathcal{B}$ satisfying
$\pi\circ\sigma=\mathsf{id}_{U}$. Thanks to [18, Theorem 5.5.5], $\pi$ is open
and separable. Let $X(w)=\pi(C(w))$. By (1), $X(w_{0})$ is an open set of
$\mathcal{B}$. By Lemma 2.7(4), $\pi$ has a section on $X(w_{0})$. Note that
$\\{gX(w_{0})\mid g\in G\\}$ constitute of open covering of $\mathcal{B}$.
Using the translation, the statement follows from the argument on $X(w_{0})$.

(3) follows from (2) and [18, Lemma 5.5.8]. ∎

###  2.4. Regular cocharacters and the positivity condition

Keep the notations and assumptions as above subsections. Let $T$ be a given
maximal torus of $G_{0}$. By [1, §12.2], there is a regular semisimple element
$t\in T$ such that $\alpha(t)\neq 1$ for any $\alpha\in\Phi(G_{0},T)$.
Furthermore, by [18, Lemma 3.2.11],
$X_{*}(T)\otimes\mathbb{F}^{\times}\rightarrow T$ with $\tau\otimes a\in
X_{*}(T)\otimes\mathbb{F}^{\times}$ mapping to $\tau(a)\in T$ gives rise to an
isomorphism of abelian groups. So there is a regular cocharacter $\tau\in
X_{*}(T)$ such that $t=\tau(a)$ for some $a\in\mathbb{F}^{\times}$. Thus, we
have

$\displaystyle\langle\alpha,\tau\rangle\neq 0\mbox{ for all
}\alpha\in\Phi(G_{0},T).$ (2.8)

By [18, §7.4.5-§7.4.6], we can further choose a regular cocharacter $\tau\in
X_{*}(T)$ for the reductive group $G_{0}$ such that

$\displaystyle\langle\alpha,\tau\rangle>0\mbox{ for all
}\alpha\in\Phi(G_{0},T)^{+}.$ (2.9)

We turn to semi-reductive case. Let $G$ be a semi-reductive algebraic group
and $\mathfrak{g}=\mathsf{Lie}(G)$.

######  Convention 2.9.

The group $G$ (or the Lie algebra $\mathfrak{g}$) is said to satisfy the
positivity condition if there exists a cocharacter $\chi\in X_{*}(T)$ such
that $\langle\alpha,\chi\rangle>0\mbox{ for\,any
}\alpha\in\Phi(G_{0},T)^{+}\cup\Phi(U,T).$

The positivity condition can be seen to valid in the following examples we are
interested  in.

######  Example 2.10.

Suppose $\mathfrak{g}$ is the Lie algebra of a semi-reductive algebraic group
$G=\text{\rm GL}(V)\times_{\nu}V$ over $\mathbb{F}$ where $\nu$ is the natural
representation of $\text{\rm GL}(V)$ (see Example 2.4). Then $\mathfrak{g}$
satisfies the positivity condition.

Actually, suppose $\operatorname{dim}V=n.$ Fix a basis $\\{v_{i}\mid 1\leq
i\leq n\\}$ of $V$ associated with the isomorphism $\text{\rm
GL}(V)\cong\text{\rm GL}(n,\mathbb{F}).$ We take the standard maximal torus
$T$ of $\text{\rm GL}(n,\mathbb{F})$ consisting of diagonal invertible
matrices. Set $B^{+}$ the Borel subgroup consisting of upper triangular
invertible matrices, which corresponds to the positive root system. And take
$\varepsilon_{i}\in X^{*}(T)$ with $\varepsilon_{i}(\text{\rm
Diag}(t_{1},\cdots,t_{n}))=t_{i}$, $i=1,\cdots,n$. Then the character group
$X^{*}(T)$ is just the free abelian group generated by $\varepsilon_{i}$,
$i=1,\cdots,n$; and the positive root system $\Phi(G_{0},T)^{+}$ consists of
all $\varepsilon_{i}-\varepsilon_{j}$ with $1\leq i<j\leq n$. Set
$\alpha_{i}=\varepsilon_{i}-\varepsilon_{i+1}$ for $i=1,\cdots,n-1$. Then
$\Pi=\\{\alpha_{1},\cdots,\alpha_{n-1}\\}$ becomes a fundamental root system
of $\text{\rm GL}(V)$, and any positive root $\varepsilon_{i}-\varepsilon_{j}$
is expressed as $\sum_{s=i}^{j-1}\alpha_{s}$ for $i<j$. Then by (2.1), we have
that $\mbox{Ad}(\textbf{t})v_{i}=t_{i}v_{i}=\varepsilon_{i}(\textbf{t})v_{i}$
for $\textbf{t}=\text{\rm Diag}(t_{1},\cdots,t_{n})\in T$. It follows that
$\Phi(V,T)=\\{\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n}\\}$ is
the set of weights of $V$ with respect to $T$.

Set $\varpi:=\sum_{i=1}^{n}(n-i+1)\varepsilon_{i}$. The corresponding
cocharacter is denoted by $\chi$, which is a regular cocharacter of $\text{\rm
GL}(V)$. Then $\langle\varepsilon_{i},\chi\rangle=(n-i+1)$ and
$\langle\alpha_{j},\chi\rangle=1$ for $1\leq i\leq n$ and $1\leq j\leq n-1$.
Hence $\text{\rm GL}(V)\times_{\nu}V$ satisfies the positivity condition 2.9.

######  Example 2.11.

Suppose $\mathfrak{g}$ is the Lie algebra of a parabolic subgroup of a
reductive algebraic group over $\mathbb{F}$ (see Example 2.2). Then the
positivity condition is assured for $\mathfrak{g}$.

We can show this below. Suppose $P$ is a parabolic subgroup of a connected
reductive algebraic group $G$. In this case, we can take a regular co-
character $\chi$ of $G$ such that $\langle\alpha,\chi\rangle>0$ for all
positive root $\alpha$ of $G$.

######  Example 2.12.

(Continued to Example 2.1) In this example, we will further make some
preliminary introduction to Cartan type Lie algebras in the classification of
finite-dimensional simple Lie algebras over an algebraically closed filed of
positive characteristic, which do not arise from simple algebraic groups (see
[7, 12, 21]). Suppose $\mathbb{F}$ is an algebraically closed field of
characteristic $p>2$ for the time being. There are four types of those Lie
algebras, and each type of them contains infinite series of ones. We will
introduce the first two types: Witt type $W(n)$ ($\geq 1$) and Special type
$S(n)$ ($n\geq 2$) (the remaining two types are Hamiltonian type $H(n)$ for
even $n\geq 2$ and contact type $K(n)$ for odd $n\geq 3$, see [15, 21] for
details).

By definition, the restricted Lie algebras of Cartan type are $p$-subalgebras
of the algebra of derivations of the truncated polynomial rings
$\mathscr{A}(n)=\mathbb{F}[T_{1},\cdots,T_{n}]/(T_{1}^{p},\ldots,T_{n}^{p})$,
whose canonical generators are denoted by $x_{i}$, $i=1,\ldots n$. Recall
$W(n)=\text{Der}(\mathscr{A}(n))$ (the $n$-th Jacobson-Witt algebra). Its
restricted mapping is the standard $p$-th power of linear operators. Denote by
$\partial_{i}$ the partial derivative with respect to the variable $x_{i}$ .
Then $\partial_{i}$, $i=1,\ldots,n$ is a basis of the $\mathscr{A}(n)$-module
$W(n)$. The Lie algebra $W(n)$ has a restricted grading
$W(n)=\bigoplus_{i=0}^{q}W(n)_{[i]}$ where
$W(n)_{[i]}=\sum_{k=1}^{n}\mathscr{A}_{i+1}\partial_{i}$ associated with the
natural grading of $\mathscr{A}$ arising from the degree of truncated
polynomials, and $q=n(p-1)-1$. Let $\Omega$ denote the exterior differential
algebra over $\mathscr{A}(n)$ in the usual sense. Then
$\Omega(n)=\sum_{i=1}^{n}\Omega(n)^{i}$ with
$\Omega(n)^{i}=\bigwedge^{i}\Omega(n)^{1}$ for
$\Omega(n)^{1}:=\text{Hom}_{\mathscr{A}(n)}(W(n),\mathscr{A}(n))$ which can be
presented as $\Omega(n)^{1}=\sum_{i=1}^{r}\mathscr{A}(n)\textsf{d}x_{i}$ where
d is an $\mathbb{F}$-linear map from $\Omega^{0}(n):=\mathscr{A}(n)$ to
$\Omega^{1}(n)$ via $\textsf{d}f:E\mapsto E(f)$ for any $f\in\mathscr{A}(n)$
and $E\in W(n)$. The exterior differential algebra $\Omega(n)$ is canonically
endowed with a $W(n)$-action (see [21, §4.2]).

Associated with the volume differential form
$\omega_{S}:=\textsf{d}x_{1}\wedge\cdots\wedge\textsf{d}x_{n}$, there is a
(Cartan-type) simple Lie algebra of Special type $S(n):=S^{\prime}(n)^{(1)}$
for $S^{\prime}(n)=\\{E\in W(n)\mid E\omega_{S}=0\\}$. Both $W(n)$ and $S(n)$
are simple Lie algebras (see [7, 12, 15, 21]), the latter of which is a graded
Lie subalgebra of the former. For $\mathfrak{g}=W(n)$ or $S(n)$, the graded
structure $\mathfrak{g}=\sum_{i\geq-1}\mathfrak{g}_{[i]}$ gives rise to the
filtration $\\{\mathfrak{g}_{i}\\}$ with $\mathfrak{g}_{i}=\sum_{j\geq
i}\mathfrak{g}_{[j]}$. Denote by $G=\text{Aut}(\mathfrak{g})$. Then $G$ is
compatible with the restricted mapping, and $G$ is connected provided that
$p\geq 5$ is additionally required for $\mathfrak{g}=W(1)$. Furthermore, the
following properties are satisfied.

  * (1)

$G=\text{\rm GL}(n)\ltimes U$ where $U$ is the unipotent radical of G
consisting of elements $\sigma\in G$:
$(\sigma-\mathsf{id}_{\mathfrak{g}})(\mathfrak{g}_{i})\subset\mathfrak{g}_{i+1}$.
So $G$ is semi-reductive.

  * (2)

$\mathsf{Lie}(G)=({C\hskip-2.0pt\mathfrak{g}})_{0}$ where

$\displaystyle{C\hskip-2.0pt\mathfrak{g}}:=\begin{cases}W(n)&\text{ if
}\mathfrak{g}=W(n);\cr\\{E\in W(n)\mid
E\omega_{S}\in\mathbb{F}\omega_{S}\\};&\text{ if
}\mathfrak{g}=S(n)\end{cases}$ (2.10)

which is a graded subalgebra of $W(n)$ (see [9, Proposition 3.2]). Actually,
${C\hskip-2.0ptS}(n)=S^{\prime}(n)+\mathbb{F}x_{1}\partial_{1}$ (see [21,
§4.2]). So
${C\hskip-2.0pt\mathfrak{g}}=({C\hskip-2.0pt\mathfrak{g}})_{[0]}\oplus({C\hskip-2.0pt\mathfrak{g}})_{1}$
with $({C\hskip-2.0pt\mathfrak{g}})_{[0]}=W_{[0]}\cong\mathfrak{gl}(n)$ and
$({C\hskip-2.0pt\mathfrak{g}})_{1}\subset W(n)_{1}$.

In particular, for $\mathfrak{g}=W(n)$ we have $G=\text{\rm GL}(n)\ltimes U$
with $\mathsf{Lie}(G)=W(n)_{0}$, and $\operatorname{Lie}(U)=W(n)_{1}$. Take a
maximal torus $T$ and a positive Borel subgroup $B^{+}$ as Example 2.10. Then
$\Phi(G_{0},T)^{+}=\\{\varepsilon_{i}-\varepsilon_{j}\mid 1\leq i<j\leq n\\}$.
For each $\textbf{t}=\text{\rm Diag}(t_{1},\cdots,t_{n})\in T,$
$\mbox{Ad}(\textbf{t})(x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}\partial_{j})=(\sum_{i=1}^{n}a_{i}\varepsilon_{i}-\varepsilon_{j})(\textbf{t})x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}\partial_{j}.$ Therefore,

$\Phi(U,T)=\left\\{\sum_{i=1}^{n}a_{i}\epsilon_{i}-\epsilon_{j}\mid 0\leq
a_{i}\leq p-1\text{ and }\sum_{i=1}^{n}a_{i}\geq 2\right\\}.$

Set $\varpi:=\sum_{i=1}^{n}(2n-i)\varepsilon_{i}$. The corresponding
cocharacter is denoted by $\chi$, which is a regular cocharacter of $\text{\rm
GL}(V)$. Then $\langle\varepsilon_{i}-\varepsilon_{j},\chi\rangle=j-i$ for
$1\leq i<j\leq n$ and

$\langle\sum_{i=1}^{n}a_{i}\varepsilon_{i}-\varepsilon_{j},\chi\rangle=\sum_{i=1}^{n}a_{i}(2n-i)-(2n-j)\geq\sum_{i=1}^{n}a_{i}n-(2n-1)\geq
2n-(2n-1)>0$

for all $\sum_{i=1}^{n}a_{i}\varepsilon_{i}-\varepsilon_{j}\in\Phi(U,T)$.
Hence associated with $W(n)$, $\mathsf{Lie}(G)$ satisfies the positivity
condition 2.9.

Associated with $S(n)$, $\mathsf{Lie}(G)$ still contains the reductive Lie
subalgebra part isomorphic to $\mathfrak{gl}(n)$ along with the unipotent part
$\mathsf{Lie}(U)\subset W(n)_{1}$. So the same arguments yield the assertion
for $\mathsf{Lie}(G)$ associated with $S(n)$.

######  Remark 2.13.

Associated with $\mathfrak{g}=H(n)$ and $K(n)$, $G=\text{Aut}(\mathfrak{g})$
still turn out to be a connected semi-reductive group. However, the above
argument is not available to the Hamiltonian algebra $H(n)$ and the contact
algebra $K(n)$. Indeed, for $H(n)$ with $n=2m,$ take a maximal torus
$\mathfrak{h}=\langle h_{i}:=x_{i}\partial_{i}-x_{i+m}\partial_{i+m}\mid 1\leq
i\leq m\rangle$, and $\alpha_{i}$ the fundamental weight with
$\alpha_{i}(h_{j})=\delta_{ij}$ for $1\leq i,j\leq m$. Note that both
$x_{1}^{2}\partial_{m+1}$ and $x_{m+1}^{2}\partial_{1}$ lie in $H(n)_{[1]},$
so that $\pm 3\alpha_{1}\in\Phi(U,T).$ Hence, the positivity condition fails
for $H(n).$ While for $K(n)$ with $n=2m+1,$ take a maximal torus
$\mathfrak{h}=\langle
h_{i}:=x_{i}\partial_{i}-x_{i+m}\partial_{i+m},h_{m+1}:=\sum_{j=1}^{2m}x_{j}\partial_{j}+2x_{2m+1}\partial_{2m+1}\mid
1\leq i\leq m\rangle$, and $\alpha_{i}$ the fundamental weight with
$\alpha_{i}(h_{j})=\delta_{ij}$ for $1\leq i,j\leq m+1$. Note that both
$3x_{1}^{2}x_{2m+1}^{\frac{p-1}{2}}\partial_{m+1}+x_{1}^{3}x_{2m+1}^{\frac{p-3}{2}}(\sum_{j=1}^{2m}x_{j}\partial_{j}-x_{2m+1}\partial_{2m+1})$
and
$3x_{m+1}^{2}x_{2m+1}^{\frac{p-1}{2}}\partial_{1}+x_{m+1}^{3}x_{2m+1}^{\frac{p-3}{2}}(\sum_{j=1}^{2m}x_{j}\partial_{j}-x_{2m+1}\partial_{2m+1})$
lie in $K(n)_{[p]},$ so that $\pm 3\alpha_{1}\in\Phi(U,T).$ Hence, the
positivity condition fails for $K(n).$

Analogue to [5, Proposition 2.11], we have the following paralleling result
for a semi-reductive algebraic group.

######  Proposition 2.14.

Suppose that $G=G_{0}\ltimes U$ is a semi-reductive algebraic group over
$\mathbb{F}$ with a maximal torus $T$ and satisfies the positivity condition
2.9. Let $\mathfrak{g}=\mathsf{Lie}(G)$ and
$\mathfrak{g}_{0}=\mathsf{Lie}(G_{0})$. Then for any $X\in\mathfrak{g}$ with
the Jordan-Chevalley decomposition $X=X_{s}+X_{n}$, we have
$X_{s}\in\overline{\mathcal{O}_{X}}$. In particular, if $X$ is nilpotent, then
$0\in\overline{\mathcal{O}_{X}}.$

###### Proof.

By Lemmas 2.5 and 2.7, any $X\in\mathfrak{g}$ under Ad$(G)$-conjugation lies
in a fixed Borel subalgebra. We first fix a Borel subgroup $B=B_{0}\ltimes U$
such that the maximal torus $T$ is contained in $B_{0}\subset G_{0}$, and
$\mathsf{Lie}(B)=\mathfrak{t}+\sum_{\alpha\in\Phi(G_{0},T)^{+}}(\mathfrak{g}_{0})_{\alpha}+\sum_{\alpha\in\Phi(U,T)}\mathfrak{u}_{\alpha}$
(keeping in mind (2.2)). Without loss of generality, we might as well suppose
$X\in\mathsf{Lie}(B)$. Consider its Jordan-Chevally decomposition
$X=X_{s}+X_{n}$. Then $X_{s},X_{n}\in\mathsf{Lie}(B)$. By assumption, we can
write

$\displaystyle
X=X_{s}+\sum_{\alpha\in\Phi(G_{0},T)^{+}}X_{{\alpha}}+\sum_{\beta\in\Phi(U,T)}X_{\beta}$
(2.11)

where $X_{s}\in\mathfrak{t},$ $X_{{\alpha}}\in(\mathfrak{g}_{0})_{\alpha}$ for
$\alpha\in\Phi(G_{0},T)^{+}$ and $X_{\beta}\in\mathfrak{u}_{\beta}$ for
$\beta\in\Phi(U,T)$.

Then $\mbox{Ad}(t)(X)=X_{s}+\sum_{\gamma\in R^{+}}{\gamma}(t)X_{\gamma}$ for
all $t\in T$, with $R^{+}:=\Phi(G_{0},T)^{+}\cup\Phi(U,T)$. Suppose $\chi$ is
a cocharacter satisfies the positivity condition 2.9. Then for all
$a\in\mathbb{F}^{\times}$, up to conjugation under $G$-action

$\mbox{Ad}(\chi(a))(X)=X_{s}+\sum_{\gamma\in
R^{+}}a^{\langle\gamma,\chi\rangle}X_{\gamma}$

with $\langle\gamma,\chi\rangle>0$ for all $\gamma\in R^{+}$, which shows that
we can extend the map $a\mapsto\mbox{Ad}(\chi(a))X$ to a morphism
$\mathbb{F}\rightarrow{\mathfrak{g}}$ by $0\mapsto X_{s}$. It follows that
$X_{s}\in\overline{G\cdot X}$ as claimed. ∎

##  3\. Chevalley Restriction Theorem

From now on, we always assume that $G=G_{0}\ltimes U$ (resp.
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{u}$) is a semi-reductive
algebraic group (resp. semi-reductive Lie algebra) over $\mathbb{F}$ of rank
$n$.

### 3.1.

Let $\mathscr{W}$ be the Weyl group of $G$ relative to $T$, i.e.,
$\mathscr{W}=N_{G}(T)/\text{C}_{G}(T)$, which can be identified with
$N_{G_{0}}(T)/\text{C}_{G_{0}}(T)$. For each $w\in\mathscr{W},$ denote by
$\dot{w}$ the presentation of $w$ in $N_{G}(T).$ Thanks to [5, §7.12] and [1,
§11.12], the following lemma holds.

######  Lemma 3.1.

For a given maximal torus $T$, let $\mathfrak{h}:=\mathsf{Lie}(T)$. Take
$h\in\mathfrak{h}$. Then

$\text{Ad}(G)(h)\cap\mathfrak{h}=\\{\dot{w}(h)|w\in\mathscr{W}\\}.$

######  Lemma 3.2.

([5, Proposition 7.12]) The map $f\longmapsto f|_{\mathfrak{h}}$ induces an
injective homomorphism
$\Phi_{0}:\,\mathbb{F}[\mathfrak{g}_{0}]^{G_{0}}\longrightarrow\mathbb{F}[\mathfrak{h}]^{\mathscr{W}}$
of $\mathbb{F}$-algebras. This homomorphism is an isomorphism if
$\text{char}(\mathbb{F})\neq 2$ or if $\text{char}(\mathbb{F})=2$ and
$\alpha\notin 2X(T)$ for all roots $\alpha$ of $G_{0}$ relative to $T$.

Consider the canonical projection
$\operatorname{pr}:\mathfrak{g}\rightarrow\mathfrak{g}_{0}$ and the imbedding
$\operatorname{i}:\mathfrak{g}_{0}\rightarrow\mathfrak{g}$ for a given semi-
reductive Lie algebra $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{u}$. Then
$\operatorname{pr}$ and $\operatorname{i}$ are both morphisms of algebraic
varieties.

######  Lemma 3.3.

The comorphism
$\operatorname{pr}^{*}:\mathbb{F}[\mathfrak{g}_{0}]\rightarrow\mathbb{F}[\mathfrak{g}]$
induces an injective homomorphism
$\operatorname{pr}^{*}:\mathbb{F}[\mathfrak{g}_{0}]^{G_{0}}\rightarrow\mathbb{F}[\mathfrak{g}]^{G},$
while
$\operatorname{i}^{*}:\mathbb{F}[\mathfrak{g}]\rightarrow\mathbb{F}[\mathfrak{g}_{0}]$
induces a surjective homomorphism
$\operatorname{i}^{*}:\mathbb{F}[\mathfrak{g}]^{G}\rightarrow\mathbb{F}[\mathfrak{g}_{0}]^{G_{0}}.$

###### Proof.

For each $f\in\mathbb{F}[\mathfrak{g}_{0}]^{G_{0}},$ denote
$F=\operatorname{pr}^{*}(f)\in\mathbb{F}[\mathfrak{g}].$ Equivalently,
$F(D+u)=f(D)$, where $D\in{\mathfrak{g}}_{0},\ u\in\mathfrak{u}$. For any
$\sigma\in G$, we have $\sigma=\sigma_{1}\cdot\sigma_{2}$ where $\sigma_{1}\in
G_{0}$ and $\sigma_{2}\in U.$ Then $\sigma\cdot(D+u)=\sigma_{1}\cdot
D+u^{\prime}$, where $u^{\prime}\in\mathfrak{u}.$ Moreover,

$(\sigma^{-1}\cdot
F)(D+u)=F(\sigma(D+u))=F(\sigma_{1}D+u^{\prime})=f(\sigma_{1}D)=f(D)=F(D+u).$

Namely, $F$ is $G$-invariant and
$\operatorname{pr}^{*}:\mathbb{F}[\mathfrak{g}_{0}]^{G_{0}}\rightarrow\mathbb{F}[\mathfrak{g}]^{G}$
is well-defined.

Since $\operatorname{i}^{*}$ is the restriction map, it is easy to verify that
$\operatorname{i}^{*}:\mathbb{F}[\mathfrak{g}]^{G}\rightarrow\mathbb{F}[\mathfrak{g}_{0}]^{G_{0}}$
is well-defined.

Note that $\operatorname{pr}\circ\operatorname{i}$ is the identity map on
$\mathfrak{g}_{0},$. Therefore,
$\operatorname{i}^{*}\circ\operatorname{pr}^{*}$ is the identity map on
$\mathbb{F}[\mathfrak{g}_{0}]^{G_{0}}.$ As a corollary,
$\operatorname{pr}^{*}$ is injective and $\operatorname{i}^{*}$ is surjective.
∎

It follows from [2] that $\mathbb{F}[\mathfrak{h}]^{\mathscr{W}}$ is generated
by $n$ algebraically independent homogeneous elements (and the unit). Then
$\mathbb{F}[\mathfrak{g}_{0}]^{G_{0}}$ is a polynomial algebra with $n$
algebraically independent homogeneous polynomials $\\{f_{i}\mid 1\leq i\leq
n\\}\subset\mathbb{F}[\mathfrak{g}_{0}]$. Denote
$F_{i}:=\operatorname{pr}^{*}(f_{i})$ for all $i=1,\cdots,n.$ Then all $F_{i}$
are homogeneous and algebraically independent, and
$\operatorname{pr}^{*}(\mathbb{F}[\mathfrak{g}_{0}]^{G_{0}})=\mathbb{F}[F_{1},\cdots,F_{n}].$

######  Theorem 3.4.

Let $G$ be a semi-reductive algebraic group satisfying the positivity
condition 2.9. Then the map $f\longmapsto f|_{\mathfrak{h}}$ induces an
injective homomorphism
$\Phi:\,\mathbb{F}[\mathfrak{g}]^{G}\longrightarrow\mathbb{F}[\mathfrak{h}]^{\mathscr{W}}$
of $\mathbb{F}$-algebras. This homomorphism is an isomorphism if
$\text{char}(\mathbb{F})\neq 2$ or if $\text{char}(\mathbb{F})=2$ and
$\alpha\notin 2X(T)$ for all roots $\alpha$ of $G_{0}$ relative to $T$.

###### Proof.

Note that $\Phi=\Phi_{0}\circ\operatorname{i}^{*}.$ For each
$X\in\mathfrak{g},$ let $X=X_{s}+X_{n}$ be the Jordan decomposition. It
follows from Proposition 2.14 that $f(X)=f(X_{s})$ for all
$f\in\mathbb{F}[\mathfrak{g}]^{G}.$ By Lemma 2.6, there is $h\in\mathfrak{h}$
such that $X_{s}\in G\cdot h.$ By Lemma 3.1, $G\cdot
h\cap\mathfrak{h}=\\{\dot{w}(h)\mid w\in\mathscr{W}\\}$ for all
$h\in\mathfrak{h}$. It follows that each $f\in\mathbb{F}[\mathfrak{g}]^{G}$ is
determined by $f|_{\mathfrak{h}}$, which is invariant under the action of
$\mathscr{W}$. This implies that $\Phi$ is injective.

On the other hand, by Lemmas 3.2 and 3.3, $\Phi$ is surjective if
$\text{char}(\mathbb{F})\neq 2$ or if $\text{char}(\mathbb{F})=2$ and
$\alpha\notin 2X(T)$ for all roots $\alpha$ of $G_{0}$ relative to $T$. ∎

######  Corollary 3.5.

Suppose $G$ is a semi-reductive algebraic group of rank $n$ satisfying the
positivity condition 2.9, and $\text{char}(\mathbb{F})\neq 2$ or if
$\text{char}(\mathbb{F})=2$ and $\alpha\notin 2X(T)$ for all roots $\alpha$ of
$G_{0}$ relative to $T$. Then $\mathbb{F}[\mathfrak{g}]^{G}$ is a polynomial
algebra generated by $n$ algebraically independent homogeneous polynomials.

###  3.2. Eamples

Theorem 3.4 entails the following good properties for those important examples
we are interested and presented before.

######  Corollary 3.6.

Let $\mathfrak{g}$ be any one from the following list:

  * (1)

the Lie algebra of an semi-reductive algebraic group $G=\text{\rm
GL}(V)\times_{\nu}V$ over $\mathbb{F}$ where $\nu$ is the natural
representation of $\text{\rm GL}(V)$ (see Example 2.10).

  * (3)

the Lie algebra of a parabolic subgroup $P$ of a reductive algebraic group
over $\mathbb{F}$ with decomposition $P=L\ltimes U$ such that the unipotent
radical $U$ contains no trivial weight space as an $L$ module (see Example
2.11).

  * (3)

$\mathsf{Lie}(G)$ associated with $W(n)$ and $S(n)$ over $\mathbb{F}$ (see
Example 2.12).

Then the invariants of $\mathbb{F}[\mathfrak{g}]$ under $G$-action must be a
polynomial ring. More precisely,
$\mathbb{F}[\mathfrak{g}]^{G}\cong\mathbb{F}[\mathfrak{h}]^{\mathscr{W}}$ as
$\mathbb{F}$-algebras, where $\mathscr{W}$ is the Weyl group of
$\mathfrak{g}$. In particular, $\mathscr{W}$ coincides with the $n$-th
symmetry group $\mathfrak{S}_{n}$ in Case (1) and Case (3).

##  4\. Application I: Comparison with the center of enveloping algebra of the
enhanced reductive Lie algebra $\underline{\mathfrak{gl}(V)}$

Suppose $\operatorname{ch}(\mathbb{F})=0$ in this section. Let $G=\text{\rm
GL}(V)\times_{\nu}V$ be the enhanced general linear group as in Corollary
3.6(1). Then $\mathfrak{g}=\mathfrak{gl}(n)\oplus V$.

### 4.1.

For each $x\in\mathfrak{gl}(n)$, we can write
$\det(t+x)=t^{n}+\sum_{i=0}^{n-1}\psi_{i}(x)t^{i}$. Then by the classical
Chevalley restriction theorem (see [5, Proposition 7.9])

$\mathbb{F}[\mathfrak{gl}(n)]^{\text{\rm
GL}(n)}=\mathbb{F}[\psi_{0},\cdots,\psi_{n-1}].$

For each $0\leq i\leq n-1,$ let
$\varphi_{i}:=\operatorname{pr}^{*}(\psi_{i})$. Namely,
$\varphi_{i}(g):=\psi_{i}(x)$ for all $g\in\mathfrak{g}$ with $g=x+v$,
$x\in\mathfrak{gl}(n)$, $v\in V$. By Theorem 3.4,

$S(\mathfrak{g}^{*})^{\mathfrak{g}}=S(\mathfrak{g}^{*})^{G}=\mathbb{F}[\mathfrak{g}]^{G}=\mathbb{F}[\varphi_{0},\cdots,\varphi_{n-1}].$

### 4.2.

Let $\mathfrak{p}$ be a parabolic subalgebra of maximal dimension in
$\mathfrak{sl}(n+1)$ consisting of $(n+1)\times(n+1)$ zero-traced matrices
with $(n+1,k)$-entries equal to zero for any $k=1,\ldots,n$.

Consider two elements $I=(\sum_{i=1}^{n}E_{ii},0)\in\mathfrak{g}$ and
$H=\frac{1}{n+1}\sum_{i=1}^{n}(E_{ii}-E_{n+1,n+1})\in\mathfrak{p}$. Then
$\mathfrak{g}=\mathbb{F}I\oplus\mathfrak{g}^{\prime}$ and
$\mathfrak{p}=\mathbb{F}H\oplus\mathfrak{p}^{\prime}$, and
$\mathfrak{g}^{\prime}\simeq\mathfrak{p}^{\prime}\simeq\mathfrak{sl}(n)\oplus
M$. Moreover, one can check that there exists an isomorphism of Lie algebras:
$\mathfrak{g}\simeq\mathfrak{p}$, sending $I$ to $H$.

By [6, Section 7], $S(\mathfrak{p})^{\mathfrak{p}^{\prime}}=\mathbb{F}[z]$ is
a polynomial ring with one indeterminate where $z$ is a harmonic element in
$\mathfrak{sl}(n+1).$ As in the proof of [6, Lemma 7.4], $[H,z]=nz$.
Therefore, we have

$\displaystyle S(\mathfrak{g})^{{\mathfrak{g}}}\simeq
S(\mathfrak{p})^{{\mathfrak{p}}}=\mathbb{F}[z]^{H}=\mathbb{F}.$ (4.1)

We finally obtain the structure of the center of $U(\mathfrak{g})$ for
$\mathfrak{g}=\mathfrak{gl}(V)\ltimes V$.

######  Proposition 4.1.

Let $\mathfrak{g}=\mathfrak{gl}(n)\oplus V$. Then the center of
$U(\mathfrak{g})$ is one-dimensional.

###### Proof.

It follows from Duflo’s theorem [3] and (4.1) that
$Z(\mathfrak{g})=U(\mathfrak{g})^{\mathfrak{g}}\simeq
S(\mathfrak{g})^{\mathfrak{g}}\simeq\mathbb{F}$. ∎

##  5\. Application II: Nilpotent cone and Steinberg map for semi-reductive
Lie algebras

Denote by $\mathcal{N}$ (resp. $\mathcal{N}_{0}$) the variety of all nilpotent
elements of $\mathfrak{g}$ (resp. $\mathfrak{g}_{0}$). Let
$\mathbb{F}[\mathfrak{g}]_{+}^{G}$ (resp.
$\mathbb{F}[\mathfrak{g}_{0}]_{+}^{G_{0}}$) be the subalgebra in
$\mathbb{F}[\mathfrak{g}]^{G}$ (resp. $\mathbb{F}[\mathfrak{g}_{0}]^{G_{0}}$)
consisting of all polynomials in $\mathbb{F}[\mathfrak{g}]^{G}$ (resp.
$\mathbb{F}[\mathfrak{g}_{0}]_{+}^{G_{0}}$) with zero constant term. It is
well-known that $\mathcal{N}_{0}$ coincides the common zeros of all
polynomials $f\in\mathbb{F}[\mathfrak{g}_{0}]_{+}^{G_{0}}$ (see [5, Lemma
6.1]).

### 5.1.

Suppose $G$ satisfies the positivity condition 2.9 and the assumption on
ch$\mathbb{F}$ in Corollary 3.5. By this corollary, there exist $n$
algebraically independent homogeneous polynomials
$F_{i}\in\mathbb{F}[\mathfrak{g}]$, $i=1,\ldots,n$ such that
$\mathbb{F}[\mathfrak{g}]^{G}=\mathbb{F}[F_{1},\cdots,F_{n}]$, of which the
precise meaning can be seen in Theorem 3.4. Furthermore, we have the following
lemma.

######  Lemma 5.1.

Let $G$ be a semi-reductive algebraic group over $\mathbb{F}$ satisfying the
positivity condition 2.9 and the assumption on ch$\mathbb{F}$ in Corollary
3.5. Then

$\mathcal{N}=\left\\{X\in\mathfrak{g}\mid f(X)=0\text{ for all
}f\in\mathbb{F}[\mathfrak{g}]_{+}^{G}\right\\}=V(F_{1},\ldots,F_{n}).$

Here and further, the notation $V(F_{1},\ldots,F_{n})$ stands for the
algebraic set of common zeros of $F_{1},\ldots,F_{n}$.

###### Proof.

Similar to [5, Section 6.1], $\mathcal{N}\supseteq\left\\{X\in\mathfrak{g}\mid
f(X)=0\text{ for all }f\in\mathbb{F}[\mathfrak{g}]_{+}^{G}\right\\}.$ Actually
Jantzen only deal with the nilpotent cone for reductive Lie algebras in [5],
but his arguments remain valid for a general algebraic Lie algebra.

If $x\in\mathfrak{g}$ is nilpotent, $0\in\overline{\mathcal{O}_{X}}$ by
Proposition 2.14. For each $F\in\mathbb{F}[\mathfrak{g}]_{+}^{G},$ since $F$
is continuous and constant in $\mathcal{O}_{X},$ $F$ is constant in
$\overline{\mathcal{O}_{X}}.$ Since $F(0)=0,$ $F(x)=0.$

Thanks to [5, Section 6.1] and the definition of $F_{i},$

$\mathcal{N}=V(F_{1},\cdots,F_{n})=V(f_{1},\cdots,f_{n})\times\mathfrak{u}=\mathcal{N}_{0}\times\mathfrak{u}.\qed$

So we have

######  Proposition 5.2.

Keep the notations and assumption as in Lemma 5.1. In particular,
$G=G_{0}\ltimes U$ denotes a connected semi-reductive algebraic group over
$\mathbb{F}$ satisfying the positivity condition 2.9, and
$\mathfrak{g}=\operatorname{Lie}(G)$. The following statements hold.

  1. (1)

$\mathcal{N}\simeq\mathcal{N}_{0}\times\mathfrak{u}\text{ as varieties.}$

  2. (2)

For any $D\in\mathfrak{g}$ with $D=x+y$, $\ x\in\mathfrak{g}_{0},\
y\in\mathfrak{u}$, we have the following decomposition of tangent spaces

$T_{D}(\mathcal{N})=T_{x}(\mathcal{N}_{0})\oplus\mathfrak{u}$

where $\mathfrak{u}$ is regarded as its tangent space.

  3. (3)

Let $\mathcal{N}_{\text{sm}}$ $\operatorname{(}resp.\
\mathcal{N}_{0,\text{sm}}\operatorname{)}$ be the smooth locus of
$\mathcal{N}$ $\operatorname{(}resp.\ \mathcal{N}_{0}\operatorname{)}$. We
have following isomorphism of varieties:

$\displaystyle\mathcal{N}_{\text{sm}}\simeq\mathcal{N}_{0,\text{sm}}\times\mathfrak{u}.$
(5.1)

  4. (4)

$\mathcal{N}$ is irreducible and normal.

###### Proof.

(1) folllows from the proof of Lemma 5.1.

(2) It is a direct consequence of (1).

(3) Thanks to (1) and (2),
$\dim(\mathcal{N})=\text{dim}(\mathcal{N}_{0})+\text{dim}\,\mathfrak{u}$ and

$\dim T_{X}(\mathcal{N})=\dim
T_{\operatorname{pr}(X)}(\mathcal{N}_{0})+\text{dim}\,\mathfrak{u},\,\,\forall\,X\in\mathcal{N}.$

Hence, $X\in\mathcal{N}$ is smooth if and only if
$\operatorname{pr}_{0}(X)\in\mathcal{N}_{\text{0}}$ is smooth. As a result,
the restriction of $\alpha$ to $\mathcal{N}_{\text{sm}}$ gives rise to the
isomorphism (5.1).

(4) Since $\mathcal{N}_{0}$ is irreducible and normal, so is $\mathcal{N}$ by
(1). ∎

###  5.2. The Steinberg map

Define canonically an adjoint quotient map

$\displaystyle\chi:\mathfrak{g}$ $\displaystyle\rightarrow\mathbb{A}^{n},$
$\displaystyle D$
$\displaystyle\mapsto(F_{1}(D),\cdots,F_{n}(D)),\quad\forall\,D\in\mathfrak{g}$

which we call the Steinberg map for $\mathfrak{g}$. Let

$\displaystyle\eta:\mathfrak{g}_{0}$ $\displaystyle\rightarrow\mathbb{A}^{n},$
$\displaystyle x$
$\displaystyle\mapsto(f_{1}(x),\cdots,f_{n}(x)),\quad\forall\,x\in\mathfrak{g}_{0}$

be the Steinberg map for $\mathfrak{g}_{0}$. Since
$F_{i}=\operatorname{pr}^{*}(f_{i})$, we have
$\chi=\eta\circ\operatorname{pr}.$

######  Proposition 5.3.

Keeping the notations and assumptions as in Corollary 5.1. In particular,
$G=G_{0}\ltimes U$ be a connected semi-reductive algebraic group with
$\mathfrak{g}=\mathsf{Lie}(G)=\mathfrak{g}_{0}\oplus\mathfrak{u}$ satisfying
the positivity condition.

  1. (1)

$\chi^{-1}(0)=\mathcal{N}_{0}+\mathfrak{u}.$ In particular,
$\chi^{-1}(0)=\mathcal{N}$.

  2. (2)

$\chi^{-1}(a)=\eta^{-1}(a)\times\mathfrak{u}$ for all
$a\in\chi(\mathfrak{g}).$ In particular, $\chi^{-1}(a)$ is irreducible of
dimension $\operatorname{dim}(\mathfrak{g})-n.$

  3. (3)

For every $a\in\chi(\mathfrak{g}),$ $\chi^{-1}(a)$ contains exactly one orbit
consisting of semisimple elements.

###### Proof.

(1) and (2) are obvious.

(3) Fix a maximal torus $\mathfrak{t}$ of $\mathfrak{g}_{0}$. Then
$\mathfrak{t}$ is a maximal torus of $\mathfrak{g}$. Thanks to Lemma 2.5, for
all semisimple elements $x_{1},x_{2}\in\chi^{-1}(a),$ there exist $g_{i}\in
G,\ i=1,2,$ such that $y_{i}:=g_{i}\cdot
x_{i}\in\mathfrak{t}\subseteq\mathfrak{g}_{0},\ i=1,2.$ Note that
$\eta(y_{i})=\chi(y_{i})=a,\ i=1,2,$ and $\mathfrak{g}_{0}$ is reductive. It
follows from [5, Proposition 7.13] that $y_{1}$ and $y_{2}$ lie in the same
$G_{0}$-orbit, namely, there is $g_{0}\in G_{0}$ such that $g_{0}\cdot
y_{1}=y_{2}.$ Moreover, $x_{2}=g_{2}^{-1}\cdot g_{0}\cdot g_{1}\cdot x_{1}.$
As a result, $x_{1}$ and $x_{2}$ are in the same $G$-orbit. Hence (3) holds. ∎

###  5.3. Springer resolution for the semi-reductive case

Let $\mathcal{B}$ (resp. $\mathcal{B}_{0}$) be the set of all Borel
subalgebras of $\mathfrak{g}$ (resp. $\mathfrak{g}_{0}$). Lemma 2.5 implies
that $\mathfrak{b}_{0}\mapsto\mathfrak{b}_{0}\oplus\mathfrak{u}$ defines an
isomorphisms between $\mathcal{B}_{0}$ and $\mathcal{B}.$

Recall that for fixed positive root system $\Delta^{+},$ denote by
$B^{+}=B_{0}^{+}\ltimes U$ (resp.
$\mathfrak{b}^{+}=\mathfrak{b}_{0}^{+}\oplus\mathfrak{u}$) the corresponding
Borel subgroup of $G$ (resp. Borel subalgebra of $\mathfrak{g}$) where
$B_{0}^{+}$ (resp. $\mathfrak{b}_{0}^{+}$) is the corresponding Borel subgroup
of $G_{0}$ (resp. Borel subalgebra of $\mathfrak{g}_{0}$) (see §2.2). Set
$\tilde{\mathcal{N}}:=\\{(x,\mathfrak{b})\in\mathcal{N}\times\mathcal{B}\mid
x\in\mathfrak{b}\\}$. Similar to the case of reductive Lie algebras, we have
the following lemma.

######  Lemma 5.4.

Keep notations as above, we have

  1. (1)

$N_{G}(\mathfrak{b}^{+})=B^{+}.$

  2. (2)

$\mathcal{B}\simeq G/B^{+},$ and $\mathcal{B}$ is a projective variety.

  3. (3)

$\tilde{\mathcal{N}}\simeq\\{(x,gB^{+})\in\mathcal{N}\times G/B^{+}\mid
g^{-1}x\in\mathfrak{b}^{+}\\}.$

###### Proof.

(1) For each $x\in\mathfrak{b}_{0},\ y\in\mathfrak{u}$, suppose $g\in
G_{0},h\in U$ such that $gh(x+y)\in\mathfrak{b}.$ Note that
$h(x)=x+x^{\prime}$, where $x^{\prime}\in\mathfrak{u}.$ Therefore,
$gh(x+y)=g(x)+g(x^{\prime})+gh(y)$. Hence, $g(x)\in\mathfrak{b}_{0}$ and
$g(x^{\prime})+gh(y)\in\mathfrak{u}.$ It follows that $g\in
N_{G_{0}}(\mathfrak{b}_{0})=B_{0}^{+}$ and $N_{G}(\mathfrak{b}^{+})\leq
B_{0}^{+}\ltimes U=B^{+}.$ Since $\operatorname{Lie}(B^{+})=\mathfrak{b}^{+},\
B^{+}\leq N_{G}(\mathfrak{b}^{+}).$ Consequently,
$N_{G}(\mathfrak{b}^{+})=B^{+}.$

(2) follows from (1) and Lemma 2.5.

(3) follows from (2). ∎

Denote by $\mathfrak{n}$ the nilpotent radical of $\mathfrak{b}^{+}$, then
$\mathfrak{n}=\mathfrak{n}_{0}\oplus\mathfrak{u}$ where $\mathfrak{n}_{0}$ is
the nilpotent radical of $\mathfrak{b}^{+}_{0}.$ One can check that $B^{+}$
can act on $G\times\mathfrak{n}$, with $x\cdot(g,n)=(x\cdot g^{-1},x\cdot n)$
for $x\in B^{+}$ and $g\in G$, $n\in\mathfrak{n}$. The arguments of [5, §6.5]
still work in our case. Therefore, the following proposition holds.

######  Proposition 5.5.

The projection $\pi:\tilde{\mathcal{N}}\rightarrow\mathcal{B}$ makes
$\tilde{\mathcal{N}}$ a $G$-equivariant vector bundle over $\mathcal{B}$ with
fiber $\mathfrak{n}.$ The assignment $(g,n)\mapsto(g\cdot n,gB^{+})$ gives a
$G$-equivariant isomorphism
$G\times^{B^{+}}\mathfrak{n}\simeq\tilde{\mathcal{N}}.$ In particular,
$\tilde{\mathcal{N}}$ is smooth.

######  Lemma 5.6.

The projection $\mu:\tilde{\mathcal{N}}\rightarrow\mathcal{N}$ is a proper
map.

###### Proof.

By definition, $\mu=\beta\circ\alpha,$ where
$\alpha:\tilde{\mathcal{N}}\hookrightarrow\mathcal{N}\times\mathcal{B}$ is a
closed immersion and
$\beta:\mathcal{N}\times\mathcal{B}\rightarrow\mathcal{N}$ is projection.
Since $\mathcal{B}$ is a projective variety, $\mu$ is a projective morphism.
Therefore $\mu$ is proper. ∎

Let $\mathcal{B}_{X}:=\\{\mathfrak{b}\in\mathcal{B}\mid X\in\mathfrak{b}\\}$
for $X\in\mathcal{N}$ and
$\mathcal{B}_{0,x}:=\\{\mathfrak{b}_{0}\in\mathcal{B}_{0}\mid
x\in\mathfrak{b}_{0}\\}$ for $x\in\mathcal{N}_{0}.$

######  Lemma 5.7.

Suppose $X=X_{0}+X_{1}\in\mathcal{N}(\mathfrak{g})$ where
$X_{0}\in\mathfrak{g}_{0},X_{1}\in\mathfrak{u}.$ Then
$\psi:\mathfrak{b}_{0}\mapsto\mathfrak{b}_{0}\oplus\mathfrak{u}$ defines a
one-to-one correspondence between $\mathcal{B}_{0,X_{0}}$ and
$\mathcal{B}_{X}.$

###### Proof.

Note that $X\in\mathfrak{b}_{0}\oplus\mathfrak{u}$ if and only if
$X_{0}\in\mathfrak{b}_{0}.$ The assertion follows immediately. ∎

######  Lemma 5.8.

If $n\in\mathcal{N}_{\text{sm}},$ then $\mu^{-1}(n)$ contains exactly one
element. Namely, there is unique Borel subalgebra of $\mathfrak{g}$ contains
$n.$

###### Proof.

It follows from Proposition 5.2 that $n=n_{0}+n_{1}$ for
$n_{0}\in\mathcal{N}(\mathfrak{g}_{0})_{\text{sm}}$ and
$n_{1}\in\mathfrak{u}.$ Since $\mathfrak{g}_{0}$ is a reductive Lie algebra,
there is exactly one Borel subalgerba $\mathfrak{b}_{0}\in\mathfrak{g}_{0}$
such that $n_{0}\in\mathfrak{b}_{0}.$

Now suppose $n\in\mathfrak{b}$ for a Borel subalgerbra $\mathfrak{b}$ of
$\mathfrak{g},$ and $\mathfrak{b}=\mathfrak{b}^{\prime}\oplus\mathfrak{u}$
where $\mathfrak{b}^{\prime}$ is a Borel subalgebra of $\mathfrak{g}_{0}.$
Since $n_{0}\in\mathfrak{b}^{\prime},$ then
$\mathfrak{b}^{\prime}=\mathfrak{b}_{0}$ and
$\mathfrak{b}=\mathfrak{b}_{0}\oplus\mathfrak{u}.$ ∎

Thanks to Proposition 5.5, Lemmas 5.6 and 5.8, $\tilde{\mathcal{N}}$ is smooth
and $\mu$ is a proper birational morphism. As a consequence, the following
result holds.

######  Proposition 5.9.

Keep notations as above, $\mu:\tilde{\mathcal{N}}\rightarrow\mathcal{N}$ is a
resolution of singularities.

######  Remark 5.10.

Let
$\tilde{\mathcal{N}_{0}}=\\{(x,\mathfrak{b})\in\mathcal{N}_{0}\times\mathcal{B}_{0}\mid
x\in\mathfrak{b}\\}$ be the Springer resolution of $\mathcal{N}_{0}.$ Assume
$G$ satisfies the positivity condition 2.9 and the assumption of
ch$\mathbb{F}$ as in Corollary 5.1, then

$\tilde{\mathcal{N}}\simeq\tilde{\mathcal{N}_{0}}\times\mathfrak{u}.$

Actually, by Proposition 5.2, $\mathcal{N}=\mathcal{N}_{0}\times\mathfrak{u}.$
Then $(x,\mathfrak{b}_{0},y)\mapsto(x+y,\mathfrak{b}_{0}+\mathfrak{u})$
defines an isomorphism between $\tilde{\mathcal{N}_{0}}\times\mathfrak{u}$ and
$\tilde{\mathcal{N}}$.

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# New family of quasi-Hermitian Hamiltonians and its application in coupled
predator-prey circles

Tengzhou Zhang  Wilczek Quantum Center and Key Laboratory of Artificial
Structures and Quantum Control, School of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai 200240, China  Zi Cai<EMAIL_ADDRESS>Wilczek
Quantum Center and Key Laboratory of Artificial Structures and Quantum
Control, School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China  Shanghai Research Center for Quantum Sciences,
Shanghai 201315, China

###### Abstract

Herein, we propose a family of non-Hermitian Hamiltonians with real spectra
other than the space-time reflection (PT) symmetric class, and discuss its
application in the predator-prey ecological processes described by a
generalized Lotka-Volterra equation. In the phase space, such a nonlinear
equation could support both chaotic and localized dynamics separated by a
dynamical critical point, which can be understood as a consequence of the
interplay between the periodicity and non-Hermiticity of its effective
Hamiltonian in the linearized equation of motion. Further, the dynamics at the
critical point (e.g. the algebraic divergence,) can be understood as an
exceptional point in the context of non-Hermitian physics. Applications to
genuine quantum systems have also been discussed.

Introduction –As a standard axiom of quantum mechanics, the Hermiticity of a
Hamiltonian guarantees the real-valuedness of an energy spectrum as well as
the probability conservation in an isolated quantum system. However, the
Hermiticity requirement can be replaced by a physically transparent condition
of space-time reflection (PT) symmetry, while the essential features of
quantum mechanics are preservedBender and Boettcher (1998); Bender (2007).
More general mathematical conditions for a non-Hermitian Hamiltonian with real
energy spectrum (dubbed quasi-Hermitian) have also been discussedMostafazadeh
(2002a, b). Physically, non-Hermitian HamiltoniansAshida et al. (2020), as a
phenomenological description of process with energy or particle flowing out of
the Hilbert space of interest, are responsible for diverse intriguing
phenomena in the contexts of classical and quantum wavesRuschhaupt et al.
(2005); Ruter et al. (2010); Peng et al. (2014); Feng et al. (2014); Bertoldi
et al. (2017); Xiao et al. (2020), topological physicsLee (2016); Yao and Wang
(2018); Yao et al. (2018); Gong et al. (2018); Liu et al. (2019); Bergholtz et
al. (2021); Wang et al. (2020), and active mattersFruchart et al. (2021).
Searching for physically transparent examples of non-Hermitian Hamiltonian
with real energy spectra is not only of fundamental interest for exploring
non-Hermitian physics in a broader context, but also of practical significance
due to its potential application in quantum sensingChen et al. (2017); Hodaei
et al. (2017) and energy transferAssawaworrarit et al. (2017); Xu et al.
(2016); Budich and Bergholtz (2020).

To this end, we propose a new class of non-Hermitian systems, whose
Hamiltonian $H$ can be expressed in terms of a product of two matrices as

$H=VD,$ (1)

where both $D$ and $V$ represent Hermitian matrices and at least one of them
is positive definite. In general, $D$ and $V$ do not commute with each other,
thus $H$ is non-Hermitian. However, the real-valuedness of the eigenvalues of
$H$ can be easily proved as follows: assuming that $\mathbf{x}$ is the
eigenstate of $H$ with eigenvalue $\lambda$:

$H\mathbf{x}=\lambda\mathbf{x}.$ (2)

According to Cholesky decomposition, a positive definite Hermitian matrix can
always be factorized into a product of a lower triangular matrix and its
conjugate transpose: $D=LL^{\dagger}$Horn and R.Johnsaon (1985); hence, we
have $H=VLL^{\dagger}$. By operating $L^{\dagger}$ on both sides of Eq.(2) one
can obtain:

$\tilde{H}\mathbf{y}=\lambda\mathbf{y}$ (3)

where $\mathbf{y}=L^{\dagger}\mathbf{x}$ and $\tilde{H}=L^{\dagger}VL$ is a
Hermitian matrix ($\tilde{H}=\tilde{H}^{\dagger}$). Eq.(2) and (3) indicate
that for a given non-Hermitian Hamiltonian $H$ satisfying Eq.(1), one can
always find a Hermitian Hamiltonian $\tilde{H}$, which shares the same set of
eigenvalues with $H$. Therefore, the eigenvalues of $H$ also have to be real.
Despite the conciseness of this mathematical proof, finding a physical system
whose Hamiltonian satisfies Eq.(1) is non-trivial, because the physical
meaning of multiplication of two operators is elusive in quantum mechanics.

In this paper, we propose a generalized Lotka-Volterra equation (GLVE) in a
one-dimensional (1D) lattice, which could exhibit chaotic or stable dynamics
in different parameter regimes. The Lotka-Volterra (LV) equation describing
the predator-prey ecological processes is a paradigmatic model in population
dynamicsA.J.Lotka (1910); Volterra (1928); Goel et al. (1971). Recently, the
GLVE has been generalized to spatially periodic systems to study the
topological phases and edge modes beyond the scope of natural scienceKnebel et
al. (2020); Yoshida et al. (2021); Tang et al. (2021). The dynamics of a
slight deviation from the stationary point of the GLVE are governed by a
linearized equation resembling the single-particle Schrodinger equation in a
lattice system. Therefore, the topological band theory can straightforwardly
be applied to such a classical systemKnebel et al. (2020); Yoshida et al.
(2021). Here, we show that if the linear expansion is performed around a
temporal periodic solution instead of the stationary point of the GLVE, the
equation of motion (EOM) of the deviation can also be described by the
Schrodinger equation, but with a time-dependent Hamiltonian which is generally
non-Hermitian but satisfies Eq.(1). The dynamical critical point in the
original nonlinear model can be understood as an exceptional point for the
effective non-Hermitian Floquet Hamiltonian.

Figure 1: (Color online)(a) Predator-prey model defined in a 1D
“diatomic”chain described by the GLVE.(4). (b)Periodic solution
$[\bar{x}(t),\bar{y}(t)]^{T}$ of the homogeneous GLVE.(5) with the conserved
quantity $V=0.131$. (c) Trajectories of $[\bar{x}(t),\bar{y}(t)]^{T}$ in the
phase space with different conserved quantities. (d) and (e) Trajectories in
the phase space of the first unit cell ($i=1$) predicted via the GLVE.(4) with
(d) $r=0.3$ and (e) $r=0.7$, $\Delta=0.05$ and $L=1024$. The initial state of
(d) and (e) is spatially inhomogeneous: $\delta_{i}(t=0)=\Delta_{i}$ with
$\Delta_{i}$ being randomly sampled from $[-\Delta,\Delta]$. The red curves
indicate the trajectory starting from the spatially homogeneous initial state
$\delta_{i}(t=0)=0$.

Model and method – We focus on the GLVE defined in a 1D “diatomic”chain (see
Fig.1 a), which reads:

$\displaystyle\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}[2-vy_{i-1}-wy_{i}]$ $\displaystyle\dot{y}_{i}$ $\displaystyle=$
$\displaystyle y_{i}[-2+vx_{i}+wx_{i+1}]$ (4)

where $i=1\cdots L$, and $L$ is the number of the unit cell, each of which
contains a prey ($x_{i}$) and predator ($y_{i}$). $v=1+r$ and $w=1-r$. $0<r<1$
is the only tunable parameter in Eq.(4) characterizing the difference between
the inter and intra unit cell coupling strengths. The linear terms in the
right side of Eq.(4) suggest an exponential growth/decay for the prey/predator
populations if there is no interspecies interaction, while the nonlinear terms
indicate the interaction between one specie and its neighbors, which suppress
the exponential growth/decay.

Starting with a simple situation where the populations of prey and predator
are site-independent $x_{i}(t)=x(t)$, $y_{i}(t)=y(t)$, Eq.(4) is reduced to a
two-species LV equation:

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle 2x-2xy$
$\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle-2y+2xy$ (5)

which was used to explain the oscillation behavior of natural populations
(e.g. the snowshoe hare and lynx) in ecological systems with predator-prey
interactions, competition and disease. Mathematically, this model is
integrable with a constant of motion $V=x+y-\ln xy-2$Goel et al. (1971).
Consequently, it supports either a steady solution $[1,1]^{T}$ (with $V=0$) or
a periodic oscillation $[\bar{x}(t),\bar{y}(t)]^{T}$ ($V>0$) (see Fig.1 b),
corresponding to a fixed point or a closed orbit around the fixed point in the
phase space respectively (see Fig.1 c).

In general, one needs to take the spatial fluctuation into account.
Considering a solution $\mathbf{v}(t)=[x_{1},y_{1},\cdots,x_{L},y_{L}]^{T}$ of
Eq.(4), one can expand it around the spatially homogeneous solutions as

$\mathbf{v}(t)=[1+\boldsymbol{\delta}(t)]\bar{\mathbf{v}}(t)$ (6)

with $\boldsymbol{\delta}(t)=[\delta_{1}^{x}(t),\delta_{1}^{y}(t),\cdots]^{T}$
($\delta_{i}^{x}(t)=\frac{x_{i}(t)-\bar{x}(t)}{\bar{x}(t)}$ and
$\delta_{i}^{y}(t)$ is likewise). A linearized equation can be derived in
terms of the dimensionless vector $\boldsymbol{\delta}(t)$. For a homogeneous
stationary solution $\bar{\mathbf{v}}_{s}(t)=[1,1,\cdots,1,1]^{T}$ , it is
shown that the linearized EOM of $\boldsymbol{\delta}(t)$ takes the identical
form of the single-particle Schrodinger equation in a 1D lattice:

$i\frac{d\boldsymbol{\delta}(t)}{dt}=H\boldsymbol{\delta}(t)$ (7)

where $H=H_{0}$ is a time-independent $2L\times 2L$ antisymmetric Hermitian
matrix:

$H_{0}=i\begin{bmatrix}0&-v&&&-w\\\ v&0&w&&\\\ &-w&0&-v&\\\ &&v&0&\ddots\\\
w&&&\ddots&\ddots\\\ \end{bmatrix}$ (8)

Unlike previous studiesKnebel et al. (2020); Yoshida et al. (2021), here we
expand the nonlinear Eq.(4) around the periodic solution
$\bar{\mathbf{v}}_{p}(t)=[\bar{x}(t),\bar{y}(t),\cdots,\bar{x}(t),\bar{y}(t)]^{T}$,
where $\bar{x}(t),\bar{y}(t)$ are the solution of Eq.(5) with a period
$T=\pi$. The linearized EOM takes the same form of Eq.(7),but with a time-
dependent “Hamiltonian”

$H(t)=H_{0}D(t),$ (9)

where $H_{0}$ has the same definition as Eq.(8), and $D(t)$ is a diagonal
matrix with dimension $2L$:

$D(t)=\begin{bmatrix}\bar{x}(t)&&&&\\\ &\bar{y}(t)&&&\\\ &&\ddots&&\\\
&&&\bar{x}(t)&\\\ &&&&\bar{y}(t)\\\ \end{bmatrix}$ (10)

Generally, $H_{0}$ and $D(t)$ are Hermitian, and $D(t)$ is positive definite
because $\bar{x}(t)$ and $\bar{y}(t)$ are positive. Therefore, although $H(t)$
is non-Hermitian, it satisfies the condition presented in Eq.(1), thus its
instantaneous eigenvalues are real.

Figure 2: (Color online) Dynamics of the average deviation $\bar{\sigma}(t)$
with different r values in a semi-log plot($r_{c}=0.64579$ is the critical
point). The inset presents the dynamics of $\bar{\sigma}(t)$ at the critical
point in the log-log plot. The initial state is chosen as
$x_{i}(t=0)=y_{i}(t=0)=1.6(1+\Delta_{i})$, where $\Delta_{i}$ is randomly
sampled from $[-\Delta,\Delta]$ and $\Delta=2\times 10^{-4}$.

Chaotic versus localized dynamics in the phase space – Before discussing the
linearized EOM, we first focus on the dynamics of the nonlinear Eq.(4), which
can be solved using the standard Runge-Kutta method. A key question is whether
the spatially homogeneous periodic solution $[\bar{x}(t),\bar{y}(t)]^{T}$ is
stable against spatial fluctuations. To address this issue, we impose a small
site-dependent perturbation on the initial state as
$\delta_{i}(t=0)=\Delta_{i}$ , where $\Delta_{i}$ is randomly sampled from a
uniform random distribution with $\Delta_{i}\in[-\Delta,\Delta]$ and
$\Delta\ll 1$ (for a spatially homogeneous solution $\delta_{i}(t=0)=0$). We
first study the dynamics in one unit cell (say, $i=1$) by plotting the
trajectories of $x_{1}(t)$ and $y_{1}(t)$ in the phase space. As shown in
Fig.1 (d), for a small $r$ (e.g. $r=0.3$), the trajectory of
$[x_{1}(t),y_{1}(t)]^{T}$ rapidly deviates from the spatially homogeneous
solution $[\bar{x}(t),\bar{y}(t)]^{T}$ after a short time, while randomly
walking in the phase space on long timescales, indicating that the solution
$[\bar{x}(t),\bar{y}(t)]^{T}$ is unstable against spatial fluctuation for
small $r$. Conversely, at a relatively large $r$ (e.g. $r=0.7$), the
trajectory of $[x_{1}(t),y_{1}(t)]$ is bounded within a finite regime around
$[\bar{x}(t),\bar{y}(t)]$ (see Fig.1 e), which resembles the Kolmogorov-
Arnold-Moser theoremV.I.Arnold (1997).

The qualitatively different dynamical behavior between the cases with small
and large values of $r$ reveal a dynamical phase transition, which can be
characterized by the average deviation:
$\bar{\delta}(t)=\sqrt{\frac{1}{L}\sum_{i}[\delta_{i}^{x}(t)]^{2}+[\delta_{i}^{y}(t)]^{2}}$.
As shown in Fig.2, $\bar{\delta}(t)$ increases exponentially (accompanied by
an oscillation) at small $r$ (a signature of chaos), while it keeps
oscillating around a finite value at a large $r$. The exponent of the
exponential divergence approaches zero at critical $r=r_{c}$, whose value
depends on the amplitude of the periodic oscillation of the spatially
homogeneous solutions. At the dynamical critical point, $\bar{\delta}(t)$
grows algebraically as $\bar{\delta}(t)\sim t^{\frac{1}{2}}$. In the
following, we will explain these observed dynamical behaviors as well as the
critical dynamics based on the properties of the non-Hermitian Hamiltonian in
Eq.(9).

Floquet dynamics with a non-Hermitian Hamiltonian – Now we focus on the
linearized EOM.(7) where the time-dependent Hamiltonian (9) is non-Hermitian
but periodic in time $H(t)=H(t+T)$. However, unlike the intensively studied
cases with periodically driven Hamiltonian, the periodic oscillation in
Hamiltonian (9) is not due to external driving, but originates from the
spontaneous oscillation in the time-independent GLVE.(4), and is self-
sustained. Thanks to the spatially translational invariance, one can perform
the Fourier transformation, after which the EOM.(7) turns into a collection of
independent $k$ modes, each of which is a two-level system governed by the
EOM:

$i\frac{d\boldsymbol{\delta}_{k}}{dt}=H_{k}(t)\boldsymbol{\delta}_{k}$ (11)

where $\boldsymbol{\delta}_{k}=[\delta_{k}^{x},\delta_{k}^{y}]^{T}$ with
$\delta_{k}^{x}=\frac{1}{\sqrt{L}}\sum_{j}e^{-ikj}\delta^{x}_{j}$ and
$\delta_{k}^{y}$ is likewise. $H_{k}$ is a $2\times 2$ matrix defined as:

$H_{k}(t)=H_{k}^{0}D(t)$ (12)

with

$H_{k}^{0}=\begin{bmatrix}0&-i(v+we^{-ik})\\\
i(v+we^{ik})&0\end{bmatrix},D(t)=\begin{bmatrix}\bar{x}(t)&\\\
&\bar{y}(t)\end{bmatrix}$ (13)

Again, $H_{k}$ is non-Hermitian if $\bar{x}(t)\neq\bar{y}(t)$, but its
instantaneous eigenvalues are real because it satisfies the condition (1).
Both $\bar{x}(t)$ and $\bar{y}(t)$ are periodic in time with a period $T$,
enabling us to employ the Floquet description of the stroboscopic dynamics of
Eq.(11) and derive a time-independent Floquet Hamiltonian $H_{k}^{F}$
satisfying:

$\mathcal{F}_{k}=e^{-iH_{k}^{F}T}=\mathcal{T}e^{-i\int_{0}^{T}dtH_{k}(t)}$
(14)

where $\mathcal{T}$ is the time-ordering operator and $\mathcal{F}_{k}$ is the
evolution operator for the k-mode within one period.

Because the explicit formalism of the periodic solution
[$\bar{x}(t),\bar{y}(t)]^{T}$ is complex, to derive an analytic formalism of
$H_{k}^{F}$, we replace the diagonal matrix in Eq.(34) by a simplified
formalism asSup :

$D(t)=\begin{cases}\mathbb{I}+\xi\hat{\sigma}^{z},\quad
nT<t<(n+\frac{1}{2})T\\\
\mathbb{I}-\xi\hat{\sigma}^{z},\quad(n+\frac{1}{2})T<t<(n+1)T\end{cases}$ (15)

where $n$ is an integer, $\mathbb{I}$ represents a $2\times 2$ unit matrix and
$\hat{\sigma}^{z}$ denotes the z-component Pauli matrix. Furthermore,
$\xi\in[0,1]$ characterizes the amplitude of the periodic oscillation, which
is determined by the initial conditions in the original LV equation. In the
following, we demonstrate that despite the simplicity of such a step-function
approximation, it can capture the essence of the non-Hermitian Floquet
physics, and explain the numerical results observed in the nonlinear Eq.(4).
By introducing $H_{k}^{\pm}=H_{k}^{0}(\mathbb{I}\pm\xi\sigma_{z})$, the
evolution operator becomes

$\mathcal{F}_{k}=e^{-i\frac{T}{2}H_{k}^{+}}e^{-i\frac{T}{2}H_{k}^{-}}=\begin{bmatrix}\frac{\cos\phi_{k}+\xi}{1+\xi}&-\frac{ie^{i\varphi_{k}}\sin\phi_{k}}{\sqrt{1-\xi^{2}}}\\\
-\frac{ie^{-i\varphi_{k}}\sin\phi_{k}}{\sqrt{1-\xi^{2}}}&\frac{\cos\phi_{k}-\xi}{1-\xi}\end{bmatrix}$
(16)

where $\phi_{k}=\frac{\Delta_{k}T}{2}\sqrt{1-\xi^{2}}$ and $\Delta_{k}$ is the
energy gap of $H_{k}^{0}$ ($\Delta_{k}=2\sqrt{(2+2\cos k)+2(1-\cos k)r^{2}}$).
$\varphi_{k}=\arg[-i(v+we^{-ik})]$. By diagonalizing the matrix presented in
Eq.(16), one can obtain the eigenvalues of $\mathcal{F}_{k}$:

$\lambda_{k}=\frac{\cos\phi_{k}-\xi^{2}\pm
2i|\sin\frac{\phi_{k}}{2}|\sqrt{\cos^{2}\frac{\phi_{k}}{2}-\xi^{2}}}{1-\xi^{2}}$
(17)

Notably, the properties of $\lambda_{k}$ considerably depend on the sign of
$\cos^{2}\frac{\phi_{k}}{2}-\xi^{2}$, resulting in qualitatively different
physical consequences. If $\cos^{2}\frac{\phi_{k}}{2}>\xi^{2}$ for all the
k-modes, it is easy to check that $|\lambda_{k}|=1$, therefore we can
introduce a real number $\theta_{k}\in[0,2\pi]$ such that $\lambda_{k}=e^{\pm
i\theta_{k}}$. Let $\epsilon_{k}$ be the quasi-energy of the Floquet
Hamiltonian $H_{k}^{F}$, since $H_{k}^{F}=\frac{i}{T}\ln\mathcal{F}_{k}$, one
can obtain $\epsilon_{k}=\frac{i}{T}\ln\lambda_{k}=\mp\frac{\theta_{k}}{T}$.
Therefore, in this case all the eigenvalues of the Floquet Hamiltonian
$H_{k}^{F}$ are real and the stroboscopic dynamics are stable. Consequently,
there is no divergence for the deviation, and the dynamics is bounded within a
finite regime around the homogeneous trajectory $[\bar{x}(t),\bar{y}(t)]$,
agreeing with our numerical observation for large $r$.

The energy gap of $H_{0}^{k}$ satisfies $\Delta_{k}\in[4r,4]$ ($0<r<1$), which
takes its minimum value $\Delta_{\mathbf{min}}=4r$ at $k=\pi$. Therefore, for
a fixed by small oscillation amplitude $\xi$, the $\pi$-mode ($k=\pi$) will
first become unstable when $r$ is below the critical value $r_{c}$ satisfying
$\cos[\pi r_{c}\sqrt{1-\xi^{2}}]=-\xi$, which indicates that
$r_{c}\rightarrow\frac{1}{2}$ in the limit $\xi\rightarrow 0$. For $r>r_{c}$,
$\lambda_{k}$ defined in Eq.(17) becomes real and $|\lambda_{k}|\neq 1$. As a
consequence, the eigenvalue of the Floquet Hamiltonian $\epsilon_{k}$ is no
longer real, but with a pair of opposite imaginary parts, which is responsible
for the exponential divergence of the deviation observed in the case with
small $r$. Obviously, such an exponential divergence predicted by the linear
analysis cannot persist forever, because the nonlinear effect will finally
take over and governs the long-time dynamics.

Critical dynamics: an algebraic divergence– Right at the critical point
$|\cos\frac{\phi_{\pi}}{2}|=\xi$, the Floquet evolution operator for the
$\pi$-mode presented in Eq.(16) become

$\mathcal{F}_{\pi}=2\xi\begin{bmatrix}1&-1\\\
1&-1\end{bmatrix}+\begin{bmatrix}-1&0\\\ 0&-1\end{bmatrix}$ (18)

Such a $2\times 2$ matrix has only one eigenvector with eigenvalue
$\lambda_{\pi}=-1$, indicating it is an exceptional point for the non-
Hermitian matrix $\mathcal{F}_{k}$. The Floquet evolution operator presented
in Eq.(24) suggests that
$\boldsymbol{\delta}_{\pi}(nT)=\mathcal{F}^{n}_{\pi}\boldsymbol{\delta}_{\pi}(0)$.
One can prove that at the critical point, the $\pi$-mode grows linearly in
timeSup . To explain the algebraic divergence of the average deviation
$\bar{\sigma}(t)\sim t^{\frac{1}{2}}$ observed at the critical point, the
dynamics of k-modes close to $k=\pi$ must be considered. Right at the critical
point, only the $\pi$-mode grows linearly, while for other k-modes close to
$\pi$-mode ($k=\pi+q$ with $|q|\ll 1$), they will first grow linearly with
time up to a saturate time scale $t_{k}^{*}$ ($\boldsymbol{\delta}_{k}(t)\sim
t$ for $t<t_{k}^{*}$), after which their dynamics start to significantly
deviate from the linear growthSup . For a k-mode with $k=\pi+q$, its saturated
time scale $t_{k}^{*}\sim 1/|q|$ ($t^{*}_{\pi}\rightarrow\infty$). As a
consequence, at a given time $t$, only those k-modes satisfying $t_{k}^{*}>t$
are still growing linearly, and their number decreases with time as
$\mathcal{N}(t)\sim 1/t$Sup . Therefore, these modes close to the $\pi$-mode
dominate the long-time dynamics and the average deviation can be expressed as:

$\bar{\sigma}(t)=\sqrt{\frac{1}{L}\sum_{k}\boldsymbol{\delta}_{k}(t)\boldsymbol{\delta}_{-k}(t)}\sim\sqrt{\mathcal{N}(t)\boldsymbol{\delta}_{\pi+q}(t)\boldsymbol{\delta}_{\pi-q}(t)}\sim
t^{\frac{1}{2}}$ (19)

which agrees with the $t^{\frac{1}{2}}$ divergence observed at the critical
point in the original GLVE.

Discussion – Finally, we discussion the relationship between our results and
the well-studied cases with the PT symmetric, or more generally, pseudo-
Hermitian Hamiltonian. In general, the condition in Eq.(1) does not have to be
related to the PT symmetry. However, in the proposed model, the original
nonlinear Eq.(4) is indeed invariant under the PT transformation:
$x_{i}\rightarrow y_{L-i}$, $y_{i}\rightarrow x_{L-i}$ and $t\rightarrow-t$.
However, once we perform the expansion around the periodic solution and derive
the linearized EOM.(7), the effective Hamiltonian $H$ becomes time-dependent
(periodic), and breaks the time-reversal symmetry ($D(t)\neq\pm D(-t)$).
Although $H$ is not PT symmetric, it satisfies the condition presented in
Eq.(1), thus its instantaneous energy spectra are real, although its Floquet
quasi-energy spectre could be complex. Mathematically, the non-Hermitian
Hamiltonian satisfying Eq.(1) can be proved to belong to a subclass of pseudo-
Hermitian HamiltonianSup , while the proposed GLVE provides a realistic
example of this type of non-Hermitian Hamiltonian and make it physically
feasible.

Conclusion and outlook – In summary, we proposed a class of non-Hermitian
Hamiltonian with real spectra other than the PT symmetric class. The Floquet
dynamics of such a non-Hermitian system was studied in the context of an GLVE,
where we found a dynamical phase transition characterized by an EP point.
Although its EOM.(7) shared the identical formalism as the Schrodinger
equation, the proposed system is classical. Therefore, a question arises
whether this type of non-Hermitian Hamiltonian satisfying Eq.(1) can be
applied to genuine quantum systems, e.g. in a quantum circuit system
simulating the imaginary time evolutionMcArdle et al. (2019); Lin et al.
(2021), where the Hermitian operators $V$ and $D$ in Eq.(1) can be understood
as the imaginary time propagators with different Hamiltonians. Another
possible application of our results is in the quadratic bosonic system, whose
Hamiltonian is Hermitian. However, owing to the commutation relation between
the bosonic operators, the corresponding Bogoliubov-de Gennes equation
resembles a Schrodinger equation but with a non-Hermitian Hamiltonian, which
falls into the category defined in Eq.(1)Sup . Such a similarity allows us to
explore the intriguing phenomena originated in non-Hermitian physics (e.g. the
skin effect) in such a Hermitian quantum systemMcDonald et al. (2018); Flynn
et al. (2020); Yokomizo and Murakami (2021). Finally, this study revealed that
the interplay between temporal periodicity and non-Hermiticity may lead to
intriguing dynamical behaviors, shedding light on investigating the
periodically-driven non-Hermitian systemsLi et al. (2019); Longhi (2017);
Koutserimpas and Fleury (2018); Höckendorf et al. (2019); Wu and An (2020);
Zhang and Gong (2020).

Acknowledgments.—This work is supported by the National Key Research and
Development Program of China (Grant No. 2020YFA0309000), NSFC of China (Grant
No.12174251), Natural Science Foundation of Shanghai (Grant No.22ZR142830),
Shanghai Municipal Science and Technology Major Project (Grant
No.2019SHZDZX01). ZC thank the sponsorship from Yangyang Development Fund.

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Supplementary material for “New family of quasi-Hermitian Hamiltonian and its
application in coupled predator-prey circles”

We first provide the details of the step-function approximation as well as the
corresponding phase diagram. The critical dynamics has been studied
analytically. We also prove the pseudo-Hermiticity of the proposed
Hamiltonian, and discussed its application in the quadratic bosonic systems.

Figure 3: (Color online) (a) Sketch of the step-function approximation where
the periodic solution $[\bar{x}(t),\bar{y}(t)]^{T}$ are replaced by the step
functions $[\tilde{x}(t),\tilde{y}(t)]^{T}$ (b) the phase diagram in the
simplified model Figure 4: (Color online) The dynamics of
$|\boldsymbol{\sigma}_{k}|$ for different k-modes that are right at or close
to $k=\pi$. $q_{0}=\frac{\pi}{64}$

##  I Step-function approximation and the corresponding phase diagram

In the main text, to derive the analytic formalism of the Floquet evolution
operator $\mathcal{F}_{k}$, we adopted the step-function approximation where
we replaced the original diagonal matrix

$D(t)=\begin{bmatrix}\bar{x}(t)&\\\ &\bar{y}(t)\end{bmatrix}$ (20)

by a step function:

$D(t)=\begin{cases}\mathbb{I}+\xi\hat{\sigma}^{z},\quad
nT<t<(n+\frac{1}{2})T\\\
\mathbb{I}-\xi\hat{\sigma}^{z},\quad(n+\frac{1}{2})T<t<(n+1)T\end{cases}$ (21)

as shown in Fig.3 (a). Taking the advantage of this approximation, we don’t
need to perform the time-order product to calculate the Floquet operator
$\mathcal{F}_{k}$, which can be directly expressed as

$\mathcal{F}_{k}=e^{-i\frac{T}{2}H_{k}^{+}}e^{-i\frac{T}{2}H_{k}^{-}}$ (22)

Notice that in such a simplified model, there are two controlling parameters
$r$ and $\xi$, while the period $T$ is fixed to be $\pi$. $\xi$ denotes the
amplitude of the periodic driving, which originates from self-sustained
oscillation in the original nonlinear model.

In spite of the simplicity of this approximation, it could indeed capture the
essence of the physics in the original model, especially its phase diagram and
the critical behavior. The phase diagram under this step-function
approximation is plotted in Fig.1 (b), where the phase boundary is determined
by the condition $\cos[\pi r_{c}\sqrt{1-\xi^{2}}]=-\xi$, at which the
$\pi$-mode start to be unstable. Notice that $r_{c}\rightarrow\frac{1}{2}$
when $\xi\rightarrow 0$, which agrees with our numerical simulation of the
nonlinear GLVE.

##  II Critical dynamics at the exceptional point

In this section, we provide more details about the explanation of the
$t^{\frac{1}{2}}$ divergence of the average deviation $\bar{\sigma}(t)$
observed at the critical point, which can be understood as a collective
behavior of the k-modes close to $k=\pi$.

$\bar{\sigma}^{2}(t)=\frac{1}{L}\sum_{i}\boldsymbol{\delta}_{i}(t)\boldsymbol{\delta}_{i}(t)=\frac{1}{L}\sum_{k}\boldsymbol{\delta}_{k}(t)\boldsymbol{\delta}_{-k}(t)$
(23)

where the momentum summation is over the k-mode in the first Brillonin Zone
$k\in[0,2\pi]$ and
$\boldsymbol{\delta}_{k}(t)=[\delta_{k}^{x}(t),\delta_{k}^{y}(t)]^{T}$.

Right at the critical point, we first focus on the $\pi$-mode, whose
stroboscopic dynamics is governed by the Floquet operator

$\mathcal{F}_{\pi}=2\xi\begin{bmatrix}1&-1\\\
1&-1\end{bmatrix}+\begin{bmatrix}-1&0\\\ 0&-1\end{bmatrix}$ (24)

Mathematically, exceptional point means that $\mathcal{F}_{\pi}$ has only one
eigenvector with an eigenvalue $-1$. Next, we will study the long-time
dynamics governed by $\mathcal{F}_{\pi}$.

The stroboscopic dynamics of $\boldsymbol{\delta}_{\pi}(t)$ with $t=nT$ (T is
the period) can be directly expressed as

$\boldsymbol{\delta}_{\pi}(nT)=\mathcal{F}^{n}_{\pi}\boldsymbol{\delta}_{\pi}(0)$
(25)

Assuming that initially $\boldsymbol{\delta}_{\pi}(0)=[a,b]^{T}$, from
Eq.(25), one can derive that

$\boldsymbol{\delta}_{\pi}(t)=(-1)^{n}\left\\{a\begin{bmatrix}1-Kt\\\
-Kt\end{bmatrix}+b\begin{bmatrix}Kt\\\ 1+Kt\end{bmatrix}\right\\}$ (26)

where $t=nT$, $K=\frac{2\xi}{T}$. In the long time limit $t\gg 1/K$, Eq.(27)
is reduced to:

$\boldsymbol{\delta}_{\pi}(t)=(b-a)Kt\begin{bmatrix}1\\\ 1\end{bmatrix}$ (27)

which indicates a linear divergence of $|\boldsymbol{\delta}_{\pi}(t)|$ at the
critical point. This agrees very well with the numerical results as shown in
Fig.4, where the envelope of $|\boldsymbol{\delta}_{\pi}(t)|$ growth linearly
in time.

According to Eq.(23), all the k-modes contribute to $\bar{\sigma}(t)$, while
at the critical point, only the $\pi$-mode and those k-mode close to it
dominate the long-time dynamics of $\bar{\sigma}(t)$. Now we focus on those
k-modes close to $\pi$-mode with $k=\pi+q$ and $q\ll 1$. As shown in Fig.4,
for a k-mode that slight deviates from $k=\pi$, the envelope of
$|\boldsymbol{\delta}_{\pi+q}(t)|$ behavior resembles a sine function:
initially, it grows linearly in time, while after a characteristic time scale
$t^{*}_{q}$, it will significantly deviate from the linear function. Such a
characteristic time scale is roughly a quarter of the period of the sine
function, which in turns, is proportional to $1/|q|$, as shown in Fig.4.

As a consequence, the closer a k-mode is to $k=\pi$, the longer it can
contribute to the linear growth dynamics. In another word, at a fixed time t,
only those k-modes satisfying $t^{*}_{q}>t$ can still contribute to the linear
growth. Considering the fact that the k-modes are equally spaced in the
momentum space, we can find that the number of those k-mode contributing to
the linear growth decay in time as $\mathcal{N}(t)\sim 1/t$. In summary, at
long time, the summation in Eq.(23) can be approximately replaced by the
summation over those k-modes that still contribute to the linear growth

$\bar{\sigma}^{2}(t)=\frac{1}{L}\sum_{k}|\boldsymbol{\delta}_{k}(t)|^{2}\sim\sum^{\prime}_{q}|\boldsymbol{\delta}_{\pi+q}(t)|^{2}$
(28)

where the summation of is over those $\mathcal{N}(t)$ k-modes, each of which
is still growing linearly at time t $|\boldsymbol{\delta}_{\pi+q}(t)|\sim t$.
Therefore, one can obtain $\bar{\sigma}(t)\sim t^{\frac{1}{2}}$, which agrees
with our numerical results in the nonlinear GLVE.

##  III Pseudo-Hermiticity of the effective Hamiltonian

A non-Hermitian Hamiltonian with real spectre is known as pseudo-Hermitian.
Here, we will prove that the Hamiltonian proposed in the maintext belongs to a
subclass of pseudo-Hermitian Hamiltonian. Mathematically, a Hamiltonian H is
pseudo-Hermitian if it satisfies:

$H^{\dagger}=\eta^{-1}H\eta$ (29)

where $\eta$ is a Hermitian matrix. The Hamiltonian in the present study takes
the form as:

$H=VD$ (30)

where both $V$ and $D$ are Hermitian. Thus we have

$H^{\dagger}=DV=DVDD^{-1}=DHD^{-1}$ (31)

As a consequence, $H$ satisfies the pseudo-Hermitian condition (29) with
$\eta=D^{-1}$.

##  IV An application in quantum system: the Bdg equation for quadratic
bosonic system

In this section, we discuss a possible application of our results to a genuine
quantum system. Even though the Hamiltonian of this model is Hermitian, its
equation of motion takes the same form of a Schrodinger Equation, but with a
non-Hermitian Hamiltonian, which is due to the commutation relation between
the bosonic operators.

Considering a quadratic bosonic system with the Hamiltonian:

$H=\omega a^{\dagger}a+\Delta(a^{\dagger}a^{\dagger}+aa)$ (32)

The BdG equation of motion can be expressed as:

$i\frac{d}{dt}\left(\begin{array}[]{c}a\\\ a^{\dagger}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\omega&\Delta\\\ -\Delta&\
-\omega\end{array}\right)\left(\begin{array}[]{c}a\\\ a^{\dagger}\\\
\end{array}\right),$ (33)

Formally, Eq.(33) resemble the Schrodinger Equation
$i\frac{\partial\psi}{\partial t}=\tilde{H}\psi$, where $\tilde{H}$ is a non-
Hermitian matrix that can be expressed as: $\tilde{H}=DH_{0}$ with

$H_{0}=\begin{bmatrix}\omega&\Delta\\\
\Delta&\omega\end{bmatrix},D=\begin{bmatrix}1&\\\ &-1\end{bmatrix}$ (34)

where both $H_{0}$ and $D$ are Hermitian. If $\omega>|\Delta|$, $H_{0}$ is
positive definite, thus $\tilde{H}$ satisfies the condition proposed in the
maintext, thus its spectre are real, which indicates that the dynamics of the
BdG Eq.(33) are stable. On the contrary, for $\omega<|\Delta|$, one of the
eigenvalue of $H_{0}$ become negative, thus the neither $H_{0}$ nor $D$ are
positive definite, and $\tilde{H}$ has a pair of complex eigenvalues with the
opposite imaginary part. Physically, this corresponds to the parametric
instability of the quadratic bosonic systems.

More generally, it is easy to generalize the quadratic Hamiltonian (32) into a
lattice system, where similar argument can be used to analyze the stability of
these systems. The similarity between the BdG equations and the Schrodinger
Equation with non-Hermitian Hamiltonian provides a new perspective to analyze
these kinds of bosonic systems, where intriguing phenomena originated in non-
Hermitian physics (e.g. the skin effect) might be explored in such a Hermitian
quantum system.