Root graded Lie superalgebras which appear as the centerless cores of extended affine Lie superalgebras
MaliheYousofzadeh∗
Dedicated to Professor Jun Morita on the occasion of his 60th birthday.
Abstract. We recall the notions of extended affine Lie superalgebras and root graded Lie superalgebras. We show how a root graded Lie superalgebra can appear as the centerless core of an extended affine Lie superalgebra.
1. Extended Affine Lie superalgebras
Throughout this work,Fis a field of characteristic zero. Unless otherwise mentioned, all vector spaces are considered overF.We denote the dual space of a vector spaceV byV∗.IfV is a vector space graded by an abelian group, we denote the degree of a homogeneous elementx∈V by|x|; we also make a convention that if|x|is appeared in an expression, for an elementxofV, by default, we assume thatxis homogeneous. IfXis a subset of a groupA, the symbol⟨X⟩ means the subgroup ofA generated byX. Also we denote the cardinal number of a setS by|S|.We setδi,j := 0 ifi̸=jand δi,j := 1 ifi=j, the Kronecker delta. For a map f :A−→B and C⊆A,by f |C, we mean the restriction of f to C. Also we use⊎ to indicate the disjoint
2000Mathematics Subject Classification. 17B67.
Key words and phrases. Extended affine Lie superalgebras, Root graded Lie superalgebras.
∗This research was in part supported by a grant from IPM (No. 92170415) and partially carried out in IPM-Isfahan branch.
73
union.
In the present paper, by a symmetric formon an additive abelian group A,we mean a map (·,·) :A×A−→F satisfying
• (a, b) = (b, a) for all a, b∈A,
• (a+b, c) = (a, c)+(b, c) and (a, b+c) = (a, b)+(a, c) for alla, b, c∈A.
In this case, we set A0 := {a ∈ A | (a, A) = {0}} and call it the radical of the form (·,·). The form is called nondegenerate if A0 = {0}. If A is a vector space overF,bilinear forms are used in the usual sense.
We call a triple (L,H,(·,·)), consisting of a nonzero Lie superalgebra L=L¯0⊕L¯1,a nontrivial subalgebraHofL¯0and a nondegenerate invariant even supersymmetric bilinear form (·,·) onL,asuper-toral triple if
• L has a weight space decomposition L = ⊕α∈H∗Lα with respect to H via the adjoint representation. We note that in this case H is abelian; also as L¯0 as well as L¯1 are H-submodules of L, we have L¯0 =⊕α∈H∗Lα¯0 and L¯1 =⊕α∈H∗Lα¯1 with L¯αi :=L¯i∩ Lα, i= 0,1 [4, Pro. 2.1.1],
• the restriction of the form (·,·) to His nondegenerate.
We call R := {α ∈ H∗ | Lα ̸= {0}}, the root system of L (with respect toH). Each element of R is called a root. We refer to elements of R0 :=
{α ∈ H∗ | Lα¯0 ̸= {0}} (resp. R1 := {α ∈ H∗ | Lα¯1 ̸= {0}}) as even roots (resp. odd roots). We note that R=R0∪R1.Suppose that (L,H,(·,·)) is a super-toral triple and p : H −→ H∗ is the function mapping h ∈ H to (h,·). Since the form is nondegenerate on H, this map is one to one. So for each elementα of the imageHp of Hunderp,there is a uniquetα ∈ H representingα through the form (·,·).Now we can transfer the form onH to a form onHp,denoted again by (·,·),and defined by
(α, β) := (tα, tβ) (α, β∈ Hp).
Lemma 1.1([9, Lem. 3.1]).Suppose that(L,H,(·,·))is a super-toral triple with corresponding root systemR=R0∪R1.Then we have the following:
(i) For α, β ∈ H∗, [Lα,Lβ] ⊆ Lα+β. Also for i = 0,1 and α, β ∈ Ri, we have (L¯αi,L¯βi) = {0} unless α+β = 0; in particular, R0 = −R0 and R1 =−R1.
(ii) Suppose that α ∈ Hp and x±α ∈ L±α with [xα, x−α] ∈ H, then we have[xα, x−α] = (xα, x−α)tα.
(iii) Suppose thatα∈Ri\{0}(i∈ {0,1}), xα∈ L¯αi andx−α ∈ L¯−i α with [xα, x−α]∈ H \ {0}, then we have (xα, x−α)̸= 0 and that α ∈ Hp.
Definition 1.1. A super-toral triple (L=L¯0⊕ L¯1,H,(·,·)) (or L if there is no confusion), with root systemR=R0∪R1,is called anextended affine Lie superalgebraif
• (1) for α ∈ Ri\ {0} (i ∈ {0,1}), there are xα ∈ L¯αi and x−α ∈ L¯−i α
such that 0̸= [xα, x−α]∈ H,
• (2) for α ∈ R with (α, α) ̸= 0 and x ∈ Lα, adx :L −→ L,mapping y ∈ Lto [x, y],is a locally nilpotent linear transformation.
For an extended affine Lie superalgebra (L,H,(·,·)) with root systemR, the subsuperalgebra of L generated by Lα, for α ∈ R with (α, R) ̸= {0}, is called the core of L. (L,H,(·,·)) is called an invariant affine reflection algebra[6] ifL¯1 ={0} and it is called a locally extended affine Lie algebra [5] if L¯1 = {0} and L0 = H. Finally a locally extended affine Lie algebra (L,H,(·,·)) is called an extended affine Lie algebra[1] if L0 =H is a finite dimension subalgebra ofL.
We immediately have the following proposition:
Proposition 1.2. If (L,H,(·,·)) is an extended affine Lie superalgebra, then the triple(L¯0,H,(·,·)|L¯0×L¯0) is an invariant affine reflection algebra.
Example 1.3. Finite dimensional basic classical simple Lie superalgebras [2] and affine Lie superalgebras [7] are examples of extended affine Lie
superalgebras. ♢
One knows from [9, Cor. 3.9] that the root system of an extended affine Lie superalgebra (L,H,(·,·)) is an extended affine root supersystem in the following sense:
Definition 1.2. Suppose that A is a nontrivial additive abelian group, (·,·) :A×A−→F is a symmetric form andR is a subset of A.Set
R0:=R∩A0, R×:=R\R0,
R×re:={α∈R|(α, α)̸= 0}, Rre:=R×re∪ {0}, R×ns:={α∈R\R0 |(α, α) = 0}, Rns:=R×ns∪ {0}.
We say (A,(·,·), R) is an extended affine root supersystem if the following hold:
(S1) 0∈R, and⟨R⟩=A, (S2) R=−R,
(S3) forα∈R×re and β ∈R,2(α, β)/(α, α)∈Z,
(S4)
root string property holds in R in the sense that for α ∈Rre× and β ∈ R, there are nonnegative integers p, q with 2(β, α)/(α, α) = p−q such that
{β+kα|k∈Z} ∩R={β−pα, . . . , β+qα},
(S5) forα∈Rns andβ ∈R with (α, β)̸= 0,{β−α, β+α} ∩R̸=∅. If there is no confusion, for the sake of simplicity, we sayR is an extended affine root supersystem inA.Elements ofR0 are calledisotropic roots,ele- ments ofRreare calledreal rootsand elements ofRnsare callednonsingular roots. A subsetX of R× is called connected if each two elements α, β ∈X are connected inX in the sense that there is a chain α1, . . . , αn∈X with α1 = α, αn = β and (αi, αi+1) ̸= 0, i = 1, . . . , n−1.An extended affine root supersystemR is called irreducibleifRre̸={0}and R× is connected (equivalently, R× cannot be written as a disjoint union of two nonempty orthogonal subsets). An extended affine root supersystem (A,(·,·), R) is called a locally finite root supersystem if the form (·,·) is nondegenerate.
A locally finite root supersystem R is called a locally finite root system
if Rns = {0}; see [3]. A subset S of a locally finite root supersystem (A,(·,·), R) is called a sub-supersystemif the restriction of the form to⟨S⟩ is nondegenerate, 0 ∈ S, for α ∈ S ∩R×re, β ∈ S and γ ∈ S ∩Rns with (β, γ)̸= 0, rα(β)∈S and{γ−β, γ+β} ∩S ̸=∅.
Irreducible locally finite root systems are classified in [3] and irreducible locally finite root supersystems which are not locally finite root systems are classified in [8]. According to their classifications, we have types ˙A(I, J), C(0, J˙ ), A(ℓ, ℓ), B(I, J), BC(I, J), C(I, J), D(I, J), AB(1,3), G(1,2) and D(2,1, λ) for irreducible locally finite root supersystems which are not lo- cally finite root systems.
Definition 1.3. Suppose that (A,(·,·), R) is a locally finite root supersys- tem and Λ is an additive abelian group. A Lie superalgebraL =L¯0⊕ L¯1
is called an (R,Λ)-graded Lie superalgebra if
• the Lie superalgebraLis equipped with a⟨R⟩-gradingL=⊕α∈⟨R⟩Lα, that is
– L¯0 as well as L¯1 are ⟨R⟩-graded subspaces, – [Lα,Lβ]⊆ Lα+β for all α, β∈ ⟨R⟩,
• the support of L with respect to the⟨R⟩-grading is a subset of R,
• L0=∑
α∈R\{0}[Lα,L−α],
• the Lie superalgebra L is equipped with a Λ-grading L = ⊕λ∈ΛλL which is compatible with the ⟨R⟩-grading onL,that is
– L¯0 as well as L¯1 are Λ-graded subspaces, – Lα is a Λ-graded subspace for each α∈R, – [λL,µL]⊆λ+µL for all λ, µ∈Λ,
• there is a subsystem Φ ofRsuch thatQ⊗ZspanZR= spanQ(1⊗Φ) and that for 0̸=α∈Φ,there are 0̸=e∈0L ∩ Lα and 0̸=f ∈0L ∩ L−α withkα:= [e, f]∈ L¯0\{0}and forβ ∈Randx∈ Lβ,[kα, x] = (β, α)x (we call{kα |α∈Φ\ {0}} aset of toral elementsand refer to Φ as a grading subsystem).
Theorem 1.4([10, Thm. 2.7]). Suppose that( ˙A,(·,·)˙,R)˙ is a locally finite root supersystem and Λ is a torsion free additive abelian group. Suppose thatG=⊕λ∈ΛλG=⊕α˙∈R˙Gα˙ is a centerless( ˙R,Λ)-graded Lie superalgebra, with a grading subsystemΦ,˙ equipped with an invariant nondegenerate even supersymmetric bilinear form(·,·).Suppose that
• for λ, µ∈Λ with λ+µ̸= 0, (λG,µG) ={0},
• the form is nondegenerate on the span of a set of toral elements of G,
• for α˙ ∈R˙ \ {0}and λ∈Λ withλG¯iα˙ :=G¯i∩λG ∩ Gα˙ ̸={0} (i= 0,1), there are e∈λG¯iα˙ andf ∈−λG¯i−α˙ such thatk:= [e, f]∈ G¯0\ {0} and for β˙∈R˙ and x∈ Gβ˙, [k, x] = ( ˙β,α˙˙)x,
• for λ∈Λ\ {0} and i∈ {0,1} with λG¯i0 :=G¯i∩λG ∩ G0 ̸={0}, there are e∈λG¯i0 andf ∈−λG¯i0 such that [e, f] = 0and (e, f)̸= 0,
then G is isomorphic to the core of an extended affine Lie superalgebra modulo the center.
Suppose that V =V¯0⊕ V¯1 is a superspace and fix bases {vi|i∈I} and {vj |j ∈J} forV¯0 and V¯1 respectively, in which I and J are two disjoint nonempty index sets. Consider the Lie superalgebrapl(V) :=End(V) under the supercommutator product and forj, k∈I⊎J, define
ej,k :V −→ V; vi 7→δk,ivj (i∈I⊎J), (1) then gl(I ⊎J) := spanF{ej,k | j, k ∈ I ⊎J} is a Lie subsuperalgebra of gl(V). For A = ∑
i,j∈I⊎J
ai,jei,j ∈ gl(I⊎J), define the supertraceof A to be str(A) :=∑
i∈Iai,i−∑
j∈Jaj,j.
Example 1.5. Suppose thatI and J are two infinite index sets and ¯I and J¯ are copies of I and J respectively such that I, J,I¯and ¯J are mutually disjoint. Fori∈I (resp. j∈J), we denote by ¯i(resp. ¯j) the element of ¯I (resp. ¯J) corresponding to i(resp. j) with respect to a fixed identification.
Suppose thatV¯0 is a vector space with a basis {vi, v¯i | i∈I} and V¯1 is a vector space with a basis{vj, v¯j |j∈J}.Define the form
(·,·)0 :V¯0× V¯0 −→F
(vi, vr)7→0, (v¯i, vr¯)7→0, (v¯i, vr)7→δi,r, (vi, vr¯)7→δi,r (i, r ∈I),
and the form
(·,·)1 :V¯1× V¯1 −→F
(vj, vs)7→0, (v¯j, v¯s)7→0, (v¯j, vs)7→ −δj,s, (vj, v¯s)7→δj,s (j, s∈J).
The form (·,·) := (·,·)0 ⊕(·,·)1 is a nondegenerate supersymmetric even bilinear form on the superspace V := V¯0 ⊕ V¯1. We next consider the Lie subsuperalgebra
G:=o(I, J) :={X ∈gl((I⊎I)¯⊎(J⊎J¯))|(Xv, w) =−(−1)|X||v|(v, Xw); ∀v, w∈ V}
of gl((I⊎I)¯ ⊎(J ⊎J¯)).For i∈I and j∈J, take hi :=ei,i−e¯i,¯i and dj :=ej,j−e¯j,¯j, and set
H:= spanF{hi, dj |i∈I, j ∈J}. Define
(·,·) :G × G −→F; (x1, x2)7→str(x1x2) (x1, x2∈ G).
Also forr ∈I and s∈J, define
ϵr :H −→F; hi 7→δr,i, dj 7→0 (i∈I, j∈J) δs:H −→F; hi7→0, dj 7→δj,s (i∈I, j∈J).
We know that (G,H,(·,·)) is an extended affine Lie superalgebra withG0= Hand root system
R={±ϵi±ϵr,±δj±δs,±ϵi±δj |j, s∈J, i, r∈I, i̸=r}
which is an irreducible locally finite root supersystem of typeD(I, J).Using the same notation as in the paper, fori∈I andj ∈J, we have
tϵi = 1
2(ϵi,i−ϵ¯i,¯i) and tδj =−1
2(ϵj,j−ϵ¯j,¯j).
Moreover, for θ, θ′ ∈ {±ϵi,±δj} with θ+θ′ ∈R\ {0}, we haveGθ+θ′ = Fxθ,θ′ in which xθ,θ′ is given as in the following table:
(θ, θ′) xθ,θ′ xθ,θ′(vt) xθ,θ′(vt¯) xθ,θ′(vk) xθ,θ′(v¯k) (ϵi, ϵr) ei,¯r−er,¯i 0 δr,tvi−δi,tvr 0 0 (−ϵi,−ϵr) e¯i,r−er,i¯ δr,tv¯i−δi,tv¯r 0 0 0 (ϵi,−ϵr) ei,r−er,¯¯i δr,tvi −δi,tvr¯ 0 0
(δj, δs) ej,¯s+es,¯j 0 0 0 δs,kvj+δj,kvs (δj,−δs) ej,s−e¯s,¯j 0 0 δs,kvj −δj,kvs¯
(−δj,−δs) e¯j,s+es,j¯ 0 0 δs,kv¯j+δj,kv¯s 0 (ϵi,−δj) ei,j+e¯j,¯i 0 δi,tv¯j δj,kvi 0 (−ϵi, δj) e¯i,¯j−ej,i −δi,tvj 0 0 δj,kv¯i
(ϵi, δj) ei,¯j−ej,¯i 0 −δi,tvj 0 δj,kvi
(−ϵi,−δj) e¯i,j+e¯j,i δi,tv¯j 0 δj,kv¯i 0
for i, r, t ∈ I and j, s, k ∈ J with i ̸= r and j, k ∈ J. We also know that V = V¯0 ⊕ V¯1, under the natural action xv := x(v), for all x ∈ G and v∈ V,is a G-module. TheG-module V has a weight space decomposition V=⊕{α∈{±ϵi,±δj}|i∈I,j∈J}Vα with respect to H,where
Vϵi =Fvi, V−ϵi =Fv¯i, Vδj =Fvj, V−δj =Fv¯j; (i∈I, j∈J).
For u, v ∈ V, define γu,v : V −→ V mapping w ∈ V to (v, w)u − (−1)|u||v|(u, w)v. An easy verification shows that for u, v ∈ V¯0 ∪ V¯1, γu,v is an element of G|u|+|v| and that γu,v = −(−1)|u||v|γv,u.We also have the following table:
(u, v) γu,v(vt) γu,v(v¯t) γu,v(vk) γu,v(v¯k) (vi, vr) 0 δr,tvi−δt,ivr 0 0 (v¯i, vr) −δi,tvr δr,tv¯i 0 0 (v¯i, vr¯) δr,tv¯i−δt,iv¯r 0 0 0
(vj, vs) 0 0 0 δs,kvj+δj,kvs
(v¯j, vs) 0 0 −δj,kvs δs,kv¯j
(v¯j, vs¯) 0 0 −δs,kv¯j−δj,kvs¯ 0
(vi, vj) 0 −δi,tvj 0 δj,kvi
(v¯i, vj) −δi,tvj 0 0 δj,kv¯i
(vi, v¯j) 0 −δi,tv¯j −δj,kvi 0 (v¯i, v¯j) −δi,tv¯j 0 −δj,kv¯i 0 fori, r, t∈I andj, s, k ∈J. In particular,
γv¯i,vi =−2tϵi and γv¯j,vj = 2tδj; i∈I, j ∈J. (2)
Set
L:= (G ⊗F[t2, t−2])⊕(V ⊗Ft[t2, t−2]).
Proposition 1.6. L together with the bilinear extension of the following brackets
−(−1)|x||y|[y⊗tℓ, x⊗tk] := [x⊗tk, y⊗tℓ] := [x, y]⊗tk+ℓ,
−(−1)|x||u|[u⊗tm, x⊗tk] := [x⊗tk, u⊗tm] :=xu⊗tk+m,
−(−1)|u||v|[v⊗tn, u⊗tm] := [u⊗tm, v⊗tn] :=γu,v⊗tm+n,
for all x, y ∈ G, u, v ∈ V, k, ℓ ∈ 2Z, m, n ∈ 2Z+ 1 is a Lie superalgebra with
L¯0 = (G¯0⊗F[t2, t−2])⊕(V¯0⊗Ft[t2, t−2]) and
L¯1 = (G¯1⊗F[t2, t−2])⊕(V¯1⊗Ft[t2, t−2]).
Proof. Foru, v, w∈ V, x, y, z ∈ G and ℓ, m, n∈Z,we have [γu,v, x](w) = γu,vxw−(−1)|x|(|u|+|v|)xγu,vw
= (v, xw)u−(−1)|u||v|(u, xw)v
−(−1)|x|(|u|+|v|)((v, w)xu−(−1)|u||v|(u, w)xv)
= −(−1)|v||x|(xv, w)u+ (−1)|u||v|(−1)|u||x|(xu, w)v
−(−1)|x|(|u|+|v|)(v, w)xu+ (−1)|u||v|(−1)|x|(|u|+|v|)(u, w)xv
= −(−1)|v||x|(xv, w)u+ (−1)|u||v|(−1)|x|(|u|+|v|)(u, w)xv +(−1)|u||v|(−1)|u||x|(xu, w)v−(−1)|x|(|u|+|v|)(v, w)xu
= −(−1)|v||x|((xv, w)u)−(−1)|u||v|+|x||u|(u, w)xv)
+(−1)|u||v|(−1)|u||x|((xu, w)v−(−1)|x||v|+|u||v|(v, w)xu)
= (−(−1)|v||x|γu,xv+ (−1)|u||v|(−1)|u||x|γv,xu)(w).
So
[γu,v, x] =−(−1)|v||x|γu,xv+ (−1)|u||v|(−1)|u||x|γv,xu.
Now we have the following equalities:
[[u⊗tm, v⊗tn], x⊗tℓ] = [γu,v⊗tm+n, x⊗tℓ]
= [γu,v, x]⊗tm+n+ℓ
= −(−1)|v||x|γu,xv⊗tm+n+ℓ
+(−1)|u||v|(−1)|u||x|γv,xu⊗tm+n+ℓ
= [u⊗tm,[v⊗tn, x⊗tℓ]]
−(−1)|u||v|[v⊗tn,[u⊗tm, x⊗tℓ]]
and
[[u⊗tm, v⊗tn], w⊗tℓ] = [γu,v⊗tm+n, w⊗tℓ]
= γu,vw⊗tm+n+ℓ
= −(−1)|w||u|(−1)|w||v|γw,uv⊗tm+n+ℓ
−(−1)|w||u|(−1)|u||v|γv,wu⊗tm+n+ℓ
= −(−1)|w||u|(−1)|w||v|[γw,u⊗tm+ℓ, v⊗tn]
−(−1)|w||u|(−1)|u||v|[γv,w⊗tn+ℓ, u⊗tm]
= (−1)|w||v|[γu,w⊗tm+ℓ, v⊗tn]
−(−1)|w||u|(−1)|u||v|[γv,w⊗tn+ℓ, u⊗tm]
= −(−1)|u||v|[v⊗tn, γu,w⊗tm+ℓ] +[u⊗tm, γv,w⊗tn+ℓ]
= −(−1)|u||v|[v⊗tn,[u⊗tm, w⊗tℓ]]
+[u⊗tm,[v⊗tn, w⊗tℓ]].
Also we have
[[x⊗tm, y⊗tn], u⊗tℓ] = [[x, y]⊗tm+n, u⊗tℓ]
= [x, y]u⊗tm+n+ℓ
= (x(yu)−(−1)|x||y|y(xu))⊗tm+n+ℓ
= [x⊗tm, yu⊗tn+ℓ]−(−1)|x||y|[y⊗tn, xu⊗tm+ℓ]
= [x⊗tm,[y⊗tn, u⊗tℓ]]−(−1)|x||y|[y⊗tn,[x⊗tm, u⊗tℓ]]
and
[[x⊗tm, y⊗tn], z⊗tℓ] = [[x, y]⊗tm+n, z⊗tℓ]
= [[x, y], z]⊗tm+n+ℓ
= [x,[y, z]]⊗tm+n+ℓ−(−1)|x||y|[y,[x, z]]⊗tm+n+ℓ
= [x⊗tm,[y⊗tn, z⊗tℓ]]−(−1)|x||y|[y⊗tn,[x⊗tm, z⊗tℓ]].
