CENTRALIZERS OF NILPOTENT ELEMENTS IN BASIC CLASSICAL LIE SUPERALGEBRAS IN GOOD CHARACTERISTIC LEYU HAN

2025-04-30 0 0 719.12KB 23 页 10玖币

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CENTRALIZERS OF NILPOTENT ELEMENTS IN BASIC CLASSICAL LIE

SUPERALGEBRAS IN GOOD CHARACTERISTIC

LEYU HAN

School of Mathematics, University of Birmingham,

Birmingham, B15 2TT, UK

feish.ly@gmail.com

ORCID: 0000-0002-4170-4210

Abstract. Let g=g¯

0⊕g¯

1be a basic classical Lie superalgebra over an algebraically closed ﬁeld K

whose characteristic p > 0is a good prime for g. Let G¯

0be the reductive algebraic group over Ksuch

that Lie(G¯

0) = g¯

0. Suppose e∈g¯

0is nilpotent. Write gefor the centralizer of ein gand z(ge)for the

centre of ge. We calculate a basis for geand z(ge)by using associated cocharacters τ:K×→G¯

of e. In addition, we give the classiﬁcation of ewhich are reachable, strongly reachable or satisfy the

Panyushev property for exceptional Lie superalgebras D(2,1; α),G(3) and F(4).

Keywords: basic classical Lie superalgebras, nilpotent elements, reachable elements.

Mathematics Subject Classiﬁcation 2020: 17B05, 17B20, 17B22, 17B25

1. Introduction

Let g=g¯

0⊕g¯

1be a basic classical Lie superalgebra over an algebraically closed ﬁeld Kwhose

characteristic p > 0is a good prime for g. Note that the deﬁnition for a good prime is a natural

extension of that for simple Lie algebras (see Deﬁnition 5). Let e∈g¯

0be nilpotent. We investigate

the centralizer ge={x∈g: [e, x] = 0}of ein gand the centre of centralizer z(ge) = {x∈ge:

[x, y] = 0 for all y∈ge}of ein g. A lot of research has been done on the centralizer and the centre of

centralizer of nilpotent elements in the theory of Lie algebras. Although there are similarities between

the theory of Lie superalgebras and the theory of Lie algebras, there is a lot less study in this direction

in the case of Lie superalgebras and the structural theory of nilpotent orbits in Lie superalgebras

remains to be better understood. In this paper, we calculate bases for geand z(ge)and study various

properties relating ewith geand z(ge).

Research on the centralizer of nilpotent elements and their centres in the case of Lie algebras has

been intensively developed since Springer [19] considered the centralizer Guof a unipotent element

uin a simple algebraic group G. Many mathematicians undertook further study of Gu, the reader

is referred to the introduction of [15] for an overview of research on Gu. For classical Lie algebras

over an algebraically closed ﬁeld of arbitrary characteristic, Jantzen gave an explicit account of the

structure of gein [12] and Yakimova worked out bases for z(ge)in [21]. In [15], Lawther–Testerman

dealt with the centralizer Guand its centre Z(Gu)over a ﬁeld of characteristic 0or a good prime based

on Yakimova’s results. In [6, 7], the author identiﬁed geand z(ge)for basic classical Lie superalgebras

over a ﬁeld of characteristic zero and obtained analog of results of Lawther–Testerman [15] for those

Lie superalgebras.

Deﬁne gCto be a ﬁnite-dimensional basic classical Lie superalgebra over Cand write Φfor a root

system of gC. By [11, Theorem 3.9], there exists a Chevalley basis B={eα:α∈Φ} ∪ {hi: 1 ≤i≤s}

of gCsuch that [hi, eα] = hα, αiieαand [eα, eβ] = Nα,β eα+βwhere hα, αiiis deﬁned in (2.4) and

Nα,β ∈Zcan be determined explicitly. Let gZ⊆gCbe the Chevalley Z-form of gC, i.e. gZis the

Z-span of B. Denote by hZ=hhi: 1 ≤i≤siZa Cartan subalgebra of gZ. We can view g=gK

where gK=K⊗ZgZ. It is natural to ask what is the stucture of geand z(ge)in case p > 0is a good

prime for g. For g=gl(m|n), the construction of geover a ﬁeld of prime characteristic and that of

arXiv:2210.13155v1 [math.RT] 24 Oct 2022

CENTRALIZERS OF NILPOTENT ELEMENTS IN BASIC CLASSICAL LIE SUPERALGEBRAS 2

zero characteristic are identical by [20] and [9]. However, the structure of gein case of g=sl(m|n)for

m6=n,psl(n|n),osp(m|2n)and three exceptional types have not been considered yet. In this paper,

we aim to give a description of geand further calculate the dimension of z(ge)for the above types of

Lie superalgebras.

In the remaining part of this introduction, we give a more detailed survey of our results.

Fix Kan algebraically closed ﬁeld with a good prime characteristic p > 0, see Deﬁnition 5. Let

g=gK=g¯

0⊕g¯

1be one of the Lie superalgebras in Table 2. Let G¯

0be the reductive algebraic group

over Kgiven as in Table 1 such that Lie(G¯

0) = g¯

0. Then there is a representation ρ:G¯

0→GL(g¯

such that dρ: Lie(G¯

0)→gl(g¯

1)determines the adjoint action of g¯

0on g¯

Table 1: Algebraic groups G¯

Lie superalgebras gAlgebraic groups G¯

sl(m|n), m 6=n{(A, B)∈GLm(K)×GLn(K) : det(A) = det(B)}

psl(n|n)(A, B)∈GLn(K)×GLn(K) : det(A) = det(B)}/aIn|n:a∈K×

osp(m|2n) Om(K)×Sp2n(K)

D(2,1; α) SL2(K)×SL2(K)×SL2(K)

G(3) SL2(K)×G2

F(4) SL2(K)×Spin7(K)

For each nilpotent element e∈(gC)¯

0, the Jacobson–Morozov Theorem allows one to associate an

sl2-triple {e, h, f} ⊆ (gC)¯

0to e. According to [11, Section 3], the sl2-triple can be chosen such that

{e, h, f} ⊆ (gZ)¯

0where h∈hZis of the form h=Ps

i=1 cihifor ci∈Zand e=Pα∈Φeα. Note that the

adh-grading of gZ=Lj∈ZgZ(j; adh)is given by gZ(j; adh) = {x∈gZ: [h, x] = jx}and this grading

can be extended to gC. By the representation theory of sl2(C), we can determine adh-eigenvalues of

elements of ge

C. Based on the choice of e, we also can view e∈gK. We calculate bases for geand z(ge)

for Lie superalgebras of type A(m, n),B(m, n),C(n),D(m, n)and D(2,1; α),G(3),F(4) in Sections

3–4 and 5 respectively. In particular, we have the following result.

Theorem 1. There exists a basis Be⊆gZof ge

Csuch that Beviewed in geis a basis for ge. Similarly

we can ﬁnd a basis Be

z⊆gZof z(ge

C)such that Be

zviewed in z(ge)is a basis for z(ge). Note that when

g=sl(m|n)for m6=nor psl(n|n)for n > 1, we require that char(K) = pdoes not divide mand n.

Note that in good characteristic there is a substitute for sl2-triples, so called associated cocharacters,

see Deﬁnition 6 below. Let τ:K×→G¯

0be a cocharacter associated to e. Denote by g=Lj∈Zg(j;τ)

the τ-grading on gwhere g(j;τ) = {x∈g: Ad(τ(t))(x) = tjxfor all t∈K×}. In [20, Section 3],

Wang–Zhao studied properties of the τ-grading on gwith some restrictions on p. Combining Theorem 1

with Lemma 8, we obtain the following theorem which gives a more general statement on the τ-grading

on g.

Theorem 2. Let g=g¯

0⊕g¯

1be one of the basic classical Lie superalgebras in Table 2 and e∈g¯

0be

nilpotent. Then the cocharacter τ:K×→G¯

0associated to edeﬁnes a Z-grading g=Lj∈Zg(j;τ)

such that ge=Lj≥0ge(j;τ)and dim ge(j;τ) = dim g(j;τ)−dim g((j+ 2); τ)for j≥0.

Our next result focuses on the Z-grading on z(ge). In the case of simple Lie algebras, the nilpotent

element espans the degree 2part of the adh-grading of the centre of centralizer of e. It is known as the

Brylinsky–Kostant theorem, see [1, 14]. Write z(j; adh) = z(ge

C)∩g(j; adh)and z(j;τ) = z(ge)∩g(j;τ).

Theorem 3 can be viewed as the Lie superalgebra version of the Brylinsky–Kostant theorem.

CENTRALIZERS OF NILPOTENT ELEMENTS IN BASIC CLASSICAL LIE SUPERALGEBRAS 3

Theorem 3. Let g=g¯

0⊕g¯

1be one of the basic classical Lie superalgebras in Table 2 and e∈g¯

0be

nilpotent. Then z(2; adh) = heiand z(2; τ) = hei.

The element eis called reachable if e∈[ge,ge], such element were ﬁrst considered by Elashvili and

GrÃ©laud and called compact in [3]. The element eis called strongly reachable if ge= [ge,ge]. We

say that esatisﬁes the Panyushev property [17] if in the τ-grading ge=Lj≥0g(j), the subalgebra

ge(≥1) = Lj≥1g(j)is generated by ge(1). In the case of Lie algebras, Panyushev [17] showed that

for gof type An, a nilpotent element eis reachable if and only if esatisﬁes the Panyushev property.

The result was extended to cases for gof type Bn,Cn,Dnby Yakimova [22] and for gof exceptional

types by de Graaf [2]. The author [8] gave the classiﬁcation of even elements that are reachable,

strongly reachable or satisfying the Panyushev property in basic classical Lie superalgebras over C.

Our ﬁnal result extends the results in [8] to good characteristic, which illustrates the relation between

the property of being reachable and the Panyushev property.

Theorem 4. Let g=g¯

0⊕g¯

1be one of exceptional Lie superalgebras D(2,1; α),G(3) ,F(4) and e∈g¯

be nilpotent. Then eis reachable if and only if esatisﬁes the Panyushev property except for g=G(3)

and e=x1or e=E+x1. The nilpotent orbits that are reachable, strongly reachable or satisfying the

Panyushev property are listed in Tables 14–16.

This paper is organized as follows. We ﬁrst recall some fundamental concepts of Lie superalgebras

such as basic classical Lie superalgebras, root systems and cocharacters associated to nilpotent elements

in Section 2. For each system of positive roots, we identify the highest root in it. In Sections 3–4, we

determine bases of geand z(ge)for g=sl(m|n)for m6=n,psl(n|n)and osp(m|2n). Bases of geand

z(ge)for an exceptional Lie superalgebra gare given in Section 5. Using the structure of gefor an

exceptional Lie superalgebra gin Section 5, we determine which even nilpotent elements are reachable,

strongly reachable or satisfy the Panyushev property in Section 6.

Acknowledgements. The author acknowledges ﬁnancial support from the Engineering and Physical

Sciences Research Council (EP/W522478/1). We would like to thank Simon Goodwin for very useful

discussions on the subject of this paper.

2. Preliminaries and notations

2.1. Basic classical Lie superalgebras. Finite-dimensional simple Lie superalgebras over Cwere

classiﬁed by Kac in [13]. Among those simple Lie superalgebras, we focus on basic classical Lie

superalgebras in this paper. Recall that a ﬁnite-dimensional simple Lie superalgebra g=g¯

0⊕g¯

is called a basic classical Lie superalgebra if the even part g¯

0is a reductive Lie algebra and there

exists a non-degenerate supersymmetric invariant even bilinear form (·,·)on g. Note that these Lie

superalgebras are well deﬁned over Kand remain to simple by [20, Section 2]. Below in Table 2 we

recall the list of basic classical Lie superalgebras over Kthat are not Lie algebras, they are A(m, n)for

m6=n,A(n, n)for n > 1,B(m, n),C(n),D(m, n)and three exceptional types D(2,1; α),G(3),F(4).

The explicit construction of A(m, n),A(n, n),B(m, n),C(n),D(m, n)can be found for example in [6,

Sections 3–4] and that of D(2,1; α),G(3),F(4) can be found for example in [7, Sections 4–6].

g=g¯

0⊕g¯

1g¯

A(m, n) = sl(m|n),m, n ≥1,m6=nslm(K)⊕sln(K)⊕K

A(n, n) = psl(n|n),n > 1sln(K)⊕sln(K)

B(m, n) = osp(m|2n),mis odd, m, n ≥1om(K)⊕sp2n(K)

C(n) = osp(2|2n),n≥1o2(K)⊕sp2n(K)

D(m, n) = osp(m|2n),m≥4is even, n≥1om(K)⊕sp2n(K)

D(2,1; α),α∈K\{0,−1}sl2(K)⊕sl2(K)⊕sl2(K)

G(3) sl2(K)⊕G2

F(4) sl2(K)⊕so7(K)

Table 2. Basic classical Lie superalgebras

CENTRALIZERS OF NILPOTENT ELEMENTS IN BASIC CLASSICAL LIE SUPERALGEBRAS 4

2.2. Systems of positive roots and the highest root. Let gbe one of the basic classical Lie

superalgebras in Table 2 and let Φ = Φ¯

0∪Φ¯

1be a root system for g. Let Φ+be a system of positive

roots and Π = {α1, . . . , αl}be the corresponding system of simple roots. Based on results in [6, 7],

there is in general more than one system of simple roots. For each system of positive roots, there is

an unique highest root eαin Φ+such that eα=Pl

i=1 aiαiand for any other β=Pl

i=1 biαi∈Φ+, we

have bi≤aifor all i.

In the following part of this subsection, we adopt notations in [6, Sections 3–4] and [7, Sections

4–6] for root systems of gand systems of simple roots. In particular, we use the results in [6, 7] to

determine the highest root in Φ+for each system of simple roots.

1. A(m, n)

Let V=V¯

0⊕V¯

1be a ﬁnite-dimensional vector space over Ksuch that dim V¯

0=mand dim V¯

1=n.

For a homogeneous basis {v1, . . . , vm+n}of V, we denote by ηithe parity of vi, i.e. ηi=¯

0if vi∈V¯

and ηi=¯

1if vi∈V¯

1. Let eij be the ij-matrix unit. A Cartan subalgebra hof gl(m|n)has a basis

{hi=ei,i : 1 ≤i≤m+n}and a dual basis {εηi

i∈h∗: 1 ≤i≤m+n}is deﬁned by εηi

i(hj) = δij

such that (εηi

i, εηj

j) = (−1)ηiδij . Then g=sl(V)or psl(V)has a root system Φ = Φ¯

0∪Φ¯

1such that

Φ¯

0={εηi

i−εηj

j: 1 ≤i, j ≤m+n, i 6=j, ηi=ηj}and Φ¯

1={εηi

i−εηj

j: 1 ≤i, j ≤m+n, i 6=j, ηi6=ηj}.

The system of simple roots is given by Π = {αi=εηi

i−εηi+1

i+1 : 1 ≤i<m+n}and the highest root is

eα=εη1

1−εηm+n

m+n=Pm+n−1

i=1 αi.

2. B(m, n),C(n)or D(m, n)

Let V=V¯

0⊕V¯

1be a ﬁnite-dimensional vector space over Ksuch that dim V¯

0=mand dim V¯

1= 2n

and let g=osp(V). Write l=m

2. A Cartan subalgebra hof ghas a basis {hi=ei,i −e−i,−i:

1≤i≤l+n}and a dual basis {εηi

i∈h∗: 1 ≤i≤l+n}is deﬁned by εηi

i(hj) = δij such that

(εηi

i, εηj

j)=(−1)ηiδij .

When mis odd, a root system for gis Φ¯

0={±εηi

i±εηj

j:i6=j, ηi=ηj} ∪ {±ε¯

i} ∪ {±2ε¯

i}and

Φ¯

1={±ε¯

i} ∪ {±εηi

i±εηj

j:i6=j, ηi6=ηj}. A system of simple roots is {αi=εηi

i−εηi+1

i+1 : 1 ≤i≤

l+n−1} ∪ {αl+n=εηl+n

l+n}. The associated system of positive roots is Φ+={εηi

i±εηj

j, εηk

k,2ε¯

t},

where the highest root is

eα=(εη1

1+εη2

2=α1+ 2 Pl+n

i=2 αiif η1=¯

2ε¯

1= 2 Pl+n

i=1 αiif η1=¯

When mis even, a root system for gis Φ¯

0={±εηi

i±εηj

j:i6=j, ηi=ηj} ∪ {±2ε¯

i}and Φ¯

{±εηi

i±εηj

j:i6=j, ηi6=ηj}. Note that there are two possibilities for the systems of simple roots with

the same associated system of positive roots Φ+={εηi

i±εηj

j,2ε¯

t}. There are

•Π1={αi=εηi

i−εηi+1

i+1 : 1 ≤i≤l+n−2} ∪ {αl+n−1=εηl+n−1

l+n−1−ε¯

l+n} ∪ {αl+n= 2ε¯

l+n}, and

the highest root is

eα(1) =(εη1

1+εη2

2=α1+ 2 Pl+n−1

i=2 αi+αl+nif η1=¯

2ε¯

1= 2 Pl+n−1

i=1 αi+αl+nif η1=¯

•Π2={αi=εηi

i−εηi+1

i+1 : 1 ≤i≤l+n−2}∪ {αl+n−1=εηl+n−1

l+n−1−εηl+n

l+n}∪ {αl+n=εηl+n−1

l+n−1+εηl+n

l+n},

and the highest root is

eα(2) =(εη1

1+εη2

2=α1+ 2 Pl+n−2

i=2 αi+αl+n−1+αl+nif η1=¯

2ε¯

1= 2 Pl+n−2

i=1 αi+αl+n−1+αl+nif η1=¯

3. D(2,1; α)(with α6= 0,−1)

The Lie superalgebra g=D(2,1; α)has a root system Φ=Φ¯

0∪Φ¯

1such that Φ¯

0={±2β1,±2β2,±2β3},

Φ¯

1={±β1±β2±β3}where {β1, β2, β3}is an orthogonal basis of RΦsuch that (β1, β1) = 1

(β2, β2) = −1

2α−1

2and (β3, β3) = 1

2α. According to [7, Subsection 4.2], there are four conjugacy

classes of systems of simple roots which we list in the table below.

CENTRALIZERS OF NILPOTENT ELEMENTS IN BASIC CLASSICAL LIE SUPERALGEBRAS 5

Table 3: The highest root in system of positive roots for D(2,1; α)

System of simple roots

Π = {α1, α2, α3}

Associated system of positive

roots Φ+The highest root eα

{2β1,−β1+β2−β3,2β3} {2βi,±β1+β2±β3:i= 1,2,3}2β2=α1+ 2α2+α3

{2β1,−β1−β2+β3,2β2} {2βi,±β1±β2+β3:i= 1,2,3}2β3=α1+ 2α2+α3

{2β3, β1−β2−β3,2β2} {2βi, β1±β2±β3:i= 1,2,3}2β1=α1+ 2α2+α3

{−β1+β2+β3, β1−β2+

β3, β1+β2−β3}

{2βi,±β1+β2+β3, β1−β2+

β3, β1+β2−β3:i= 1,2,3}

β1+β2+β3=

α1+α2+α3

4. G(3)

The Lie superalgebra g=G(3) has the root system Φ = Φ¯

0∪Φ¯

1such that Φ¯

0={±2δ, εi−

εj,±εi: 1 ≤i6=j≤3}and Φ¯

1={±δ±εi,±δ: 1 ≤i≤3}where {δ, ε1,ε2, ε3}are elements of

(LiCεi⊕Cδ)/C(ε1+ε2+ε3)such that (δ, δ)=2,(εi, εj)=1−3δij , and (δ, εi) = 0. According to [7,

Subsection 5.2], there are four conjugacy classes of systems of simple roots which we list in the table

below.

Table 4: The highest root in system of positive roots for G(3)

System of simple roots

Π = {α1, α2, α3}

Associated system of positive

roots Φ+The highest root eα

{δ+ε3, ε1, ε2−ε1}

{ε1, ε2,−ε3, ε2−ε1, ε1−ε3, ε2−

ε3, δ, 2δ, δ ±εi:i= 1,2,3}

2δ= 2α1+ 4α2+ 2α3

{−δ−ε3, δ−ε2, ε2−ε1}

{ε1, ε2,−ε3, ε2−ε1, ε1−ε3, ε2−

ε3, δ, 2δ, δ ±ε1, δ ±ε2,±δ−ε3}

δ−ε3= 3α1+ 4α2+ 2α3

{δ, −δ+ε1, ε2−ε1}

{ε1, ε2,−ε3, ε2−ε1, ε1−ε3, ε2−

ε3, δ, 2δ, ±δ+ε1,±δ+ε2,±δ−ε3}

ε2−ε3= 3α1+ 2α2+ 2α3

{ε1,−δ+ε2, δ −ε1}

{ε1, ε2,−ε3, ε2−ε1, ε1−ε3, ε2−

ε3, δ, 2δ, δ ±ε1,±δ+ε2,±δ−ε3}

ε2−ε3= 3α1+ 2α2+ 2α3

5. F(4)

The Lie superalgebra g=F(4) has the root system Φ=Φ¯

0∪Φ¯

1such that Φ¯

0={±δ, ±εi±εj,±εi:

i6=j, i, j = 1,2,3}and Φ¯

1={1

2(±δ±ε1±ε2±ε3)}, where {δ, ε1,ε2, ε3}is an orthogonal basis of

RΦsuch that (δ, δ) = −6and (εi, εj)=2δij . According to [7, Subsection 6.4], there are six conjugacy

classes of systems of simple roots which we list in the table below.

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CENTRALIZERSOFNILPOTENTELEMENTSINBASICCLASSICALLIESUPERALGEBRASINGOODCHARACTERISTICLEYUHANSchoolofMathematics,UniversityofBirmingham,Birmingham,B152TT,UKfeish.ly@gmail.comORCID:0000-0002-4170-4210Abstract.Letg=g0g1beabasicclassicalLiesuperalgebraoveranalgebraicallyclosedeldKwhosecharacteristicp>...

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CENTRALIZERS OF NILPOTENT ELEMENTS IN BASIC CLASSICAL LIE SUPERALGEBRAS IN GOOD CHARACTERISTIC LEYU HAN

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