LIE SUPERALGEBRAS GRADED BY
THE ROOT SYSTEM A(m,n)
Georgia Benkart1
Alberto Elduque2
February 25, 2002
We determine the Lie superalgebras that are graded by the root systems
of the basic classical simple Lie superalgebras of type A(m, n).
ż1. Introduction
Our investigation is a natural extension of work on the problem of classifying
Lie algebras graded by finite root systems. Many important classes of Lie algebras
such as the affine and toroidal Lie algebras and various generalizations of them,
such as the intersection matrix Lie algebras of Slodowy [S], which arise in the study
of singularities, or the extended affine Lie algebras of [AABGP], exhibit a grading
by a finite (possibly nonreduced) root system ". The formal definition, as first
given by Berman and Moody in [BM], depends on a finite-dimensional split simple
Lie algebra g over a field F of characteristic zero having a root space decomposition
g = h •" gµ relative to a split Cartan subalgebra h. Such a Lie algebra g is
µ""
an analogue over F of a finite-dimensional complex simple Lie algebra.
Definition 1.1. A Lie algebra L over F is graded by the (reduced) root
system " or is "-graded if
("G1) L contains as a subalgebra a finite-dimensional split simple Lie algebra g =
h •" gµ whose root system is " relative to a split Cartan subalgebra
µ""
h = g0;
("G2) L = Lµ, where Lµ = {x " L | [h, x] = µ(h)x for all h " h} for
µ""*"{0}
µ " " *" {0}; and
("G3) L0 = [Lµ, L-µ].
µ""
2000 Mathematics Subject Classification. Primary 17A70.
1
Support from National Science Foundation Grants #DMS 9810361 (at the Mathematical
Sciences Research Institute, Berkeley) and #DMS 9970119 is gratefully acknowledged.
2
Supported by the Spanish DGI (BFM2001-3239-C03-03).
Typeset by AMS-TEX
1
2 GEORGIA BENKART, ALBERTO ELDUQUE
The second condition in Definition 1.1 describes the grading, while the first
provides a uniformity to the structure and enables the representation theory of g
to be used to study L. Condition ("G3) insures that the space L0 is connected to
the root spaces. Without it, any central ideal could be added to L. If only ("G1)
and ("G2) are assumed, then the subalgebra L = [Lµ, L-µ] •" Lµ
µ"" µ""
is "-graded.
There is a parallel notion of a Lie algebra graded by the nonreduced root system
BCr introduced and studied in [ABG2] (see also [BS] for the BC1-case). The Lie
algebras graded by finite root systems (both reduced and nonreduced) decompose
relative to the adjoint action of g into a direct sum of finite-dimensional irreducible
g-modules. There is one possible isotypic component corresponding to each root
length and one corresponding to 0 (the sum of the trivial g-modules). Thus, for
the simply-laced root systems only adjoint modules and trivial modules occur. For
the doubly-laced root systems, copies of the module having the highest short root
as its highest weight also can occur. For type BCr, there are up to four isotypic
components, except when the grading subalgebra g has type D2 <" A1× A1, where
=
there are five possible isotypic components. The complexity increases with the
number of isotypic components. The representation theory of g is an essential
ingredient in the classification of the Lie algebras graded by finite root systems,
which has been accomplished in the papers [BM], [BZ], [N], [ABG1], [ABG2], [BS].
Our focus here and in [BE1], [BE2] is on Lie superalgebras graded by the root
systems of the finite-dimensional basic classical simple Lie superalgebras A(m, n),
B(m, n), C(n), D(m, n), D(2, 1; Ä…) (Ä… " F\{0, -1}), F(4), and G(3). (A standard
reference for results on simple Lie superalgebras is Kac s seminal paper [K1].)
Let g be a finite-dimensional split simple basic classical Lie superalgebra over
a field F of characteristic zero with root space decomposition g = h •" gµ
µ""
relative to a split Cartan subalgebra h. Thus, g is an analogue over F of a complex
simple Lie superalgebra whose root system " is of type A(m, n) (m e" n e" 0,
m + n e" 1), B(m, n) (m e" 0, n e" 1), C(n) (n e" 3), D(m, n) (m e" 2, n e" 1),
D(2, 1; Ä…) (Ä… " F \ {0, -1}), F(4), and G(3). These Lie superalgebras can be
characterized by the properties of being simple, having reductive even part, and
having a nondegenerate even supersymmetric bilinear form. Imitating Definition
1.1, we say
Definition 1.2. ([BE1, Defn. 2.1]) A Lie superalgebra L over F is graded
by the root system " or is "-graded if
(i) L contains as a subsuperalgebra a finite-dimensional split simple basic clas-
sical Lie superalgebra g = h •" gµ whose root system is " relative to
µ""
a split Cartan subalgebra h = g0;
(ii) ("G2) and ("G3) of Definition 1.1 hold for L relative to the root system
".
The B(m, n)-graded Lie superalgebras were determined in [BE1]. These Lie
superalgebras differ from others because of their complicated structure and most
closely resemble the Lie algebras graded by the nonreduced root systems BCr. The
A(m, n)-GRADED LIE SUPERALGEBRAS 3
"-graded Lie superalgebras for " = C(n), D(m, n), D(2, 1; Ä…) (Ä… " F \ {0, -1}),
F(4), and G(3) were fully described in [BE2]. Therefore, the only remaining case
is that of " = A(m, n) (m e" n e" 0, m + n e" 1). Complete results (Theorem 3.10
and Corollary 3.12 below) will be established here for the case m = n. The Lie
superalgebras graded by the root system A(n, n) are truly exceptional for several
reasons, and we determine only those A(n, n)-graded Lie superalgebras that are
completely reducible g-modules (in Theorem 4.2 and Corollary 4.4).
Our modus operandi for investigating the A(m, n)-graded Lie superalgebras
will mimic that used in our previous papers, whose main steps can be summarized
as follows:
Procedure 1.3.
1. The determination of the finite-dimensional irreducible g-modules whose
nonzero weights relative to the Cartan subalgebra h are roots.
2. The proof of the complete reducibility of any "-graded Lie superalgebra L
as a module for g.
3. The computation of Homg(V " W, X) for any g-modules V, W, X in Step 1.
4. The determination of the multiplication in any "-graded L by combining
the previous information.
For A(m, n)-graded Lie superalgebras, Step 1 is achieved by arguments similar
to those used in the C(n)-case of [BE2]. The complete reducibility in Step 2 can be
proved along the same lines as in [BE2] provided m = n. However, for m = n, it is
no longer true that an A(n, n)-graded Lie superalgebra is completely reducible as
a module for its grading subsuperalgebra g, and this is where many complications
arise. This case will be considered in the last section of the paper.
The distinctive feature of the A(m, n)-graded Lie superalgebras, m = n, is that
Homg(g " g, g) is two-dimensional here, while in all the other cases it is spanned
by the Lie bracket.
Steps 1 and 2 above will give that any A(m, n)-graded Lie superalgebra L for
m = n, when viewed as a module for the grading subsuperalgebra g, is a direct
sum of two types of irreducible g-modules adjoint and trivial ones. By collecting
isomorphic summands, we may assume that there are F-vector superspaces A and
D so that
(1.4) L = (g " A) •" D,
where D is the sum of all the trivial g-modules. The multiplication on L makes A
a superalgebra, while D is a Lie subsuperalgebra acting as superderivations of A.
The problem of classifying the A(m, n)-graded Lie superalgebras reduces to one of
determining the possibilities for A and D and of finding the multiplication. Step 3
will show that Homg(g " g, g) is two-dimensional, while Homg(g " g, F) is spanned
by the supertrace. As a result, the product in L must be that given in (3.1) below.
The Jacobi identity will impose several restrictions on A and D, which eventually
will lead to the classification of such Lie superalgebras L (Step 4).
Throughout the paper, F will denote a fixed but arbitrary field of characteristic
zero. Unadorned tensor products will be assumed to be over F.
4 GEORGIA BENKART, ALBERTO ELDUQUE
ż2. The g-module structure of
A(m, n)-graded Lie superalgebras (m > n)
The following result, proved in [BE1, Lemma 2.2], plays a key role in examining
"-graded Lie superalgebras.
Lemma 2.1. Let L be a "-graded Lie superalgebra, and let g be its grading sub-
superalgebra. Then L is locally finite as a module for g.
As a consequence, each element of a "-graded Lie superalgebra L, in particu-
lar each weight vector of L relative to the Cartan subalgebra h of g, generates a
finite-dimensional g-module. Such a finite-dimensional module has a g-composition
series whose irreducible factors have weights which are roots of g or 0. Next we
determine (Step 1 in Procedure 1.3) which finite-dimensional irreducible g-modules
have nonzero weights that are roots of g.
Throughout we will identify the split simple Lie superalgebra g of type A(m, n),
m > n e" 0, with the special linear Lie superalgebra slm+1,n+1. For simplicity of
notation, set p = m + 1 and q = n + 1, so that g = slp,q, p > q e" 1.
The diagonal matrices in g form a Cartan subalgebra h, and the corresponding
even and odd roots and a system of simple roots of g are given by [K1, Sec. 2]:
"0 = {µi - µj | 1 d" i < j d" p} *" {´r - ´s | 1 d" r < s d" q},
Å»
"1 = {Ä…(µi - ´r) | 1 d" i d" p, 1 d" r d" q},
(2.2) Å»
= {µ1 - µ2, . . . , µp-1 - µp, µp - ´1, ´1 - ´2, . . . , ´q-1 - ´q},
where for h = diag(a1, . . . , ap, b1, . . . , bq) " h, µi(h) = ai and ´r(h) = br for any
i, r. The corresponding Cartan matrix is
ëÅ‚ öÅ‚
0
.
.
ìÅ‚ ÷Å‚
.
ìÅ‚ ÷Å‚
Am 0
ìÅ‚ ÷Å‚
0
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
-1
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
(2.3) 0 . . . 0 -1 0 1 0 . . . 0
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
-1
ìÅ‚ ÷Å‚
ìÅ‚ ÷Å‚
0
ìÅ‚
0 . An ÷Å‚
íÅ‚ Å‚Å‚
.
.
0
(if n = 0 it is just the (m + 1) × (m + 1) upper left corner above), where
ëÅ‚ öÅ‚
2 -1
ìÅ‚ -1 ÷Å‚
ìÅ‚ ÷Å‚
.
ìÅ‚ ÷Å‚
.
.
ìÅ‚ ÷Å‚
Ar =
ìÅ‚ ÷Å‚
-1
ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚
-1 2 -1
-1 2
A(m, n)-GRADED LIE SUPERALGEBRAS 5
is the r × r Cartan matrix correponding to slr+1. In terms of the standard matrix
units {Ei,j | 1 d" i, j d" p + q}, the coroots h1, . . . , hm+n+1 have the following
expressions:
hi = Ei,i - Ei+1,i+1 (1 d" i d" m + n + 1, i = m + 1 = p)
hp = Ep,p + Ep+1,p+1.
All this is valid too in case m = n e" 1, although there the hi s are the classes of
the elements above modulo the center (and hence they are linearly dependent).
According to [K1, Prop. 2.3]), É " h" = HomF(h, F) is a dominant weight (hence
the highest weight of some finite-dimensional irreducible g-module) if and only if
É(hi) " Ze"0 for all i = p, and É(hp) " Z. Therefore, the roots that are dominant
weights are µ1-µp, ´1-´q (this one does not appear if q = 1, that is, if n = 0), ´1-µp
and µ1 - ´q (the highest root of g). Consequently, these are the candidate highest
weights for irreducible g-modules occurring in A(m, n)-graded Lie superalgebras.
Now the Lie superalgebra g has a Z-gradation, g = g-1 •" g0 •" g1 with g0 = g0
Å»
and g1 = g-1 •" g1, which corresponds to the partition of the (p + q) × (p + q)-
Å»
matrices into blocks of size p × p, p × q, q × p and q × q (if p = q one has to factor
out the one-dimensional center). The Lie algebra g0 consists of the block diagonal
matrices, and g1 (respectively g-1) consists of the block strictly upper (resp. lower)
triangular matrices. Kac [K2, Sec. 2] has shown for a finite-dimensional irreducible
g-module V = V (›) that V = {x " V | g1.x = 0} is an irreducible g0-submodule
of highest weight ›, and V is a quotient of the induced module U(g) "U(g •"g1) V ,
0
which as a vector space is isomorphic to U(g-1) " V (where U( ) denotes the
universal enveloping algebra). Thus, the weights of V are of the form É + ½, where
É is a weight of the g0-module V and ½ is a weight of U(g-1). Hence ½ is either 0
or a sum of roots of the form ´r - µi.
Assume that › is the root ´1 - µp. Then no weight É + ½ as above is a root or
0 unless ½ = 0. Thus V = V and g-1.V = 0. This is a contradiction since g is
simple and V must be a faithful g-module. Now assume that › is either µ1 - µp or
´1 -´q. The same argument gives that the possible ½ s involved are either 0 or roots
of the form ´r - µi, so that V = V + g-1.V . If V = V , we reach a contradiction
as before, but if V = V , then weights of the form ´r - µi appear in V . Reasoning
with the lowest weight instead of the highest one, we see that necessarily the lowest
weight of V has to be ´q - µ1, which corresponds to the adjoint module, whose
highest weight is µ1 - ´q, a contradiction. Therefore, the only possibility left is that
› = µ1 - ´q, and V is the adjoint module.
Let us summarize the assertions above in the following:
Theorem 2.4. Let g be a split simple Lie superalgebra of type A(m, n), with
m e" n e" 0, m + n e" 1. The only finite-dimensional irreducible g-modules whose
weights relative to the Cartan subalgebra of diagonal matrices (modulo the center
if necessary) are either roots or 0 are exactly the adjoint and the trivial modules
(possibly with the parity changed).
For showing complete reducibility, we can adapt the proof of complete reducibil-
ity in [BE2, Prop. 3.1] to our setting. But from now on, the condition m > n is
necessary.
6 GEORGIA BENKART, ALBERTO ELDUQUE
Proposition 2.5. Let g be a split simple Lie superalgebra of type A(m, n), m > n e"
0, with split Cartan subalgebra h. Assume V is a locally finite g-module satisfying
(i) h acts semisimply on V ;
(ii) any composition factor of any finite-dimensional submodule of V is isomor-
phic to the adjoint module g or to a trivial module (possibly with the parity
changed).
Then V is a completely reducible g-module.
Proof. With the same reductions as in [BE2, Prop. 3.1], it suffices to show that if
X is a submodule of V and Y is a submodule of X such that either Y is an adjoint
module and X/Y is a trivial one-dimensional module, or if Y is trivial and X/Y is
<"
adjoint, then X Y •" X/Y . Since m > n, the superform given by the supertrace:
=
(x, y) str(xy) is a nondegenerate supersymmetric bilinear form on g = slm+1,n+1.
Let C be the Casimir element corresponding to g (see [K1, comments before Prop.
5.2.6]). Then one checks easily that C acts as the nonzero scalar m - n on the
adjoint module (and it acts trivially on the trivial module). Hence X is the direct
sum of the kernel of the action of C and of its image, and thus it is completely
reducible.
Proposition 2.5 is no longer true if m = n, and this is the source of many
of the difficulties encountered in that case. To illustrate this, consider the Lie
superalgebras sln+1,n+1 and pgln+1,n+1 as modules over the simple Lie superalgebra
psln+1,n+1 of type A(n, n). In both instances, there is a composition series with an
adjoint and a trivial module, but there is no complete reducibility.
The final preparatory step (number 3 in Procedure 1.3) needed to study A(m, n)-
graded Lie superalgebras is the computation of the spaces of module homomor-
phisms Homg(g " g, F) and Homg(g " g, g). It is clear that Homg(g " g, F) is
spanned by the even supersymmetric bilinear form induced by the supertrace:
(x, y) str(xy). As for the latter we have
Proposition 2.6. Let g be a split simple classical Lie superalgebra of type A(m, n)
(m > n e" 0). Then Homg(g " g, g) is two-dimensional and spanned by the Lie
2
bracket and by the map given by (x, y) x " y = xy + yx - str(xy)I, for any
m - n
x, y " g = slm+1,n+1.
Proof. The Lie bracket and the symmetrized product x " y are module homomor-
phisms and linearly independent. Hence it is enough to check that the dimension
of Homg(g " g, g) d" 2.
Let g = g-1 •" g0 •" g1 be the Z-gradation considered above. Here g0 = g0 =
Å»
slp •" slq •" Fc (p = m + 1 and q = n + 1 as before), where c is the block diagonal
q p
matrix c = diag ( Ip, Iq), which acts as the identity on g1 and as minus the
q - p q - p
identity on g-1. Moreover, g1 and g-1 are contragredient irreducible g0-modules.
Then the matrix v = E1,p+q is a highest weight vector of g1 relative to the Cartan
subalgebra h of g0 with respect to the set of simple (even) roots. Also w = Ep+q,1
is a lowest weight vector for g-1. As in [BE2, proof of Lemma 5.1], v " w generates
A(m, n)-GRADED LIE SUPERALGEBRAS 7
g " g as a g-module, so any Õ " Homg(g " g, g) is determined by Õ(v " w), which
belongs to h because v " w has weight 0. In particular, Õ restricts to a nonzero
g0-module homomorphism g1 " g-1 g0. But as a module for [g0, g0] = slp •" slq,
" "
g1 = V " W and g-1 = V " W , where V is the natural p-dimensional module
"
for slp and W the natural q-dimensional module for slq. Since Homsl (V " V , slp)
p
"
<"
is one-dimensional (as V " V glp = FIp •" slp), and the same is true for q if
=
q e" 1, it follows that the dimension of Hom[g ,g0](g1 " g-1, [g0, g0]) is two if q e" 1
0
and one if q = 1 (n = 0). As a consequence, dim Homg(g " g, g) d" 3 if q e" 1, and
it is d" 2 if q = 1. Thus we are done when n = 0 or when n e" 1, unless there is a
Õ " Homg(g " g, g) such that Õ(g1 " g-1) = Fc. If such a Õ exists, then for any
x " g1 and yÄ…1 " gÄ…1,
Õ(y1 " [x, y-1]) = Õ([x, y1] " y-1) - [x, Õ(y1 " y-1)] " F[c, x] = Fx.
In particular, for 1 d" i = j d" p, and r e" p + 1,
Õ(g1 " Ei,j) = Õ(g1 " [Ei,r, Er,j]) Ä…" FEi,r,
so Õ(g1 " Ei,j) = 0 must hold since q e" 2. In a similar vein, Õ(g1 " Er,s) = 0
for any p + 1 d" r = s d" p + q. As the elements Ei,j and Er,s generate [g0, g0]
as a g0-module, it follows that Õ(g1 " [g0, g0]) = 0. Also, for any 1 d" i d" p,
Õ(g1 " [Ei,p+q, Ep+q,i]) Ä…" FEi,p+q. But
[E2,p+q, Ep+q,2] = E2,2 + Ep+q,p+q = E2,2 - E1,1 + E1,1 + Ep+q,p+q,
so
Õ(g1 " [E2,p+q, Ep+q,2]) Ä…" Õ(g1 " (E2,2 - E1,1)) + Õ(g1 " [E1,p+q, Ep+q,1])
Ä…" 0 + FE1,p+q = FE1,p+q
too. Hence Õ(g1 " (E2,2 + Ep+q,p+q)) = 0 and
Õ(g1 " g0) = Õ(g1 " [g0, g0]) + Õ(g1 " (E2,2 + Ep+q,p+q)) = 0.
In the same way one proves that Õ(g0 " g-1) = 0. Because g = g-1 •" g0 •" g1 is the
eigenspace decomposition for ad c, it follows that Õ(g1 " g1) = 0 = Õ(g-1 " g-1)
also.
Finally,
Õ(g0 " g1) = Õ([g-1, g1] " g1) Ä…" [g-1, Õ(g1 " g1)] + Õ(g1 " g0) = 0
and also Õ(g-1 " g0) = 0. Hence
Õ(g " g)1 = Õ((g1 + g-1) " g0) + Õ(g0 " (g1 + g-1)) = 0.
Å»
But Õ(g " g) is a nonzero ideal of g, which is simple. This contradiction shows that
no Õ with Õ(g1 " g-1) = Fc exists to complete the proof.
8 GEORGIA BENKART, ALBERTO ELDUQUE
ż3. The structure of the A(m, n)-graded Lie superalgebras (m > n)
The results of Section 2 show that any A(m, n)-graded Lie superalgebra L with
m > n e" 0 is the direct sum of adjoint and trivial modules (possibly with a change
of parity) for the grading subalgebra g. After collecting isomorphic summands, we
may suppose that there are superspaces A = A0 •" A1 and D = D0 •" D1 so that
Å» Å» Å» Å»
L = (g " A) •" D, and a distinguished element 1 " A0 which allows us to identify
Å»
the grading subalgebra g with g " 1, Observe first that D is a subsuperalgebra of
L, since it is the (super)centralizer of g.
For determining the multiplication on L, we may apply the same type of argu-
ments as in [BZ] and [BE1,2]. Indeed, fixing homogeneous basis elements {ai}i"I of
A and choosing ai, aj, ak with i, j, k " I, we see that the projection of the product
[g"ai, g"aj] onto g"ak determines an element of Homg(g"g, g), which is spanned
by the supercommutator and the symmetrized product. Thus, there exist scalars
k
¾i,j and Ńk so that
i,j
Å» k
i
[x " ai, y " aj] = (-1) y ¾i,j[x, y] " ak + Ńk x " y " ak
i,j
g"A
k"I k"I
Å» k
i
= (-1) y [x, y] " ¾i,jak + x " y " Ńk ak .
i,j
k"I k"I
Å» Å»
(Our convention is that y = e whenever y " ge for e = 0, 1, etc.) Defining
Å»
k
ć% : A × A A by ai ć% aj = 2 ¾i,jak, and [ , ] : A × A A by [ai, aj] =
k"I
2 Ńk ak and extending them bilinearly, we obtain two products ć% and [ , ]
k"I i,j
on A. (The factors of 2 are simply for convenience.) Necessarily the first is super-
commutative and the second superanticommutative, because the products on g and
L are superanticommutative. Using similar reasoning and taking into account that
Homg(g " g, F) is spanned by the supertrace, we see that there exist a (super) skew
symmetric form | : A×A D and an even bilinear map D×A A: (d, a) da
with d1 = 0 such that the multiplication in L is given by:
1 1
[f " a, g " a ] = (-1)! [f, g] " a ć% a + f " g " [a, a ] + str(fg) a | a
2 2
ŻŻ
(3.1)
[d, f " a] = (-1)df f " da,
[d, d ] (the product in D)
for homogeneous elements f, g " g, a, a " A, d, d " D. Additionally, 1 ć% a = 2a for
any a " A and [1, A] = 0.
There is a unique unital multiplication aa on A such that ać%a = aa +(-1) a a
and [a, a ] = aa - (-1) a a for any homogeneous elements a, a " A. When we
refer to the algebra A, it is this unital multiplication that will be tacitly assumed.
Å» Å»
1
Now the Jacobi superidentity J(z1, z2, z3) = (-1)z z3[[z1, z2], z3] = 0 (cyclic
permutation of the homogeneous elements z1, z2, z3), when specialized with homo-
geneous elements d1, d2 " D and f " a " g " A shows that Ć : D EndF(A):
A(m, n)-GRADED LIE SUPERALGEBRAS 9
Ć(d)(a) = da, is a representation of the Lie superalgebra D. When it is specialized
with homogeneous elements d " D and f " a, g " a " g " A, we obtain
Å» Å»
[d, [f " a, g " a ]] = [[d, f " a], g " a ] + (-1)d(f%2Å‚)[f " a, [d, g " a ]],
and using (3.1), we see that this is the same as:
(3.2)
1 1
Å» Å»
(-1)d(f+! )(-1)! [f, g] " d(a ć% a ) + f " g " d([a, a ]) + str(fg)[d, a | a ]
2 2
1 1
ŻŻ Ż
= (-1)df (-1)(d%2ł)! [f, g] " (da) ć% a + f " g " [(da), a ] + str(fg) da | a
2 2
1 1
Å» Å»
+ (-1)d(f%2ł+! )(-1)! [f, g] " a ć% (da ) + f " g " [a, (da )] + str(fg) a | da .
2 2
When f = E1,2 and g = E2,1, the elements [f, g] and f"g are linearly independent
and str(fg) = 1. Hence (3.2) is equivalent to:
Å»
(i) d(a ć% a ) = (da) ć% a + (-1)da ć% (da ),
(ii) d([a, a ]) = [(da), a ] + [a, (da )],
Å»
(iii) [d, a | a ] = da | a + (-1)d a | da ,
for any homogeneous d " D and a, a " A. Items (i) and (ii) can be combined to
give
(3.3) Ć is a representation as superderivations: Ć : D DerF(A),
while (iii) says that
(3.4) | is invariant under the action of D.
For z1 " a1, z2 " a2, z3 " a3 " g " A, the Jacobi superidentity is equivalent to
the two relations
Å» Å»
1
0 =str(z1z2z3) (-1)a a3 a1 | a2a3
(3.5)
Å» Å» Å» Å» Å»
2 1
- (-1)z z3str(z1z3z2) (-1)(a +a2)a3 a1 | a3a2
10 GEORGIA BENKART, ALBERTO ELDUQUE
(3.6)
Å» Å» Å» Å»
1
0 = - (-1)z z3+a1a3z1z2z3 " (a1, a2, a3)
Å» Å» Å» Å» Å» Å»
1
+ (-1)(z +z2)z3+(a1+a2)a3z1z3z2 " (a1, a3, a2)
1
Å» Å» Å» Å»
1
- (-1)z z3+a1a3str(z1z2)z3 " a1 | a2 a3 - [[a1, a2], a3]
m - n
str(z1z2z3)
Å» Å» Å» Å»
1 1
- (-1)z z3 I " (-1)a a3[a1, a2a3]
m - n
str(z1z3z2)
Å» Å» Å» Å» Å» Å»
1 1
+ (-1)(z +z2)z3 I " (-1)(a +a2)a3[a1, a3a2] ,
m - n
where (a1, a2, a3) = (a1a2)a3 - a1(a2a3), (the associator). The first corresponds
to the D-component and the second to the (g " A)-component. Here (3.6) makes
sense inside glp,q " A ‡" slp,q " A = g " A, and I is the identity matrix.
Now suppose z1 = E1,2, z2 = E2,3 and z3 = E3,1 in g. They are all even if m e" 2,
while z1 is even and z2 and z3 are odd if m = 1 (so n = 0). Then zizi+1zi+2 = Ei,i,
zizi+2zi+1 = 0, str(zizi+1) = 0, str(zizi+1zi+2) = Ä…1 (indices modulo 3). As a
result, (3.5) gives:
Å» Å»
1
(3.7) (-1)a a3 a1 | a2a3 = 0.
If m e" 2 , E1,1, E2,2, E3,3, and I are linearly independent, so Equation 3.6 gives
(a1, a2, a3) = 0, that is
(3.8) A is associative.
Then (3.6) becomes simply
1
Å» Å» Å» Å»
1
0 = (-1)z z3+a1a3str(z1z2)z3 " a1 | a2 a3 - [[a1, a2], a3] ,
m - n
and since str(z1z2)z3 is not identically 0 on g,
1
(3.9) a1 | a2 a3 = [[a1, a2], a3],
m - n
for any a1, a2, a3 " A.
Now, if m = 1, the expression in (3.6) with the zi s chosen as above is a linear
combination of E1,1, E2,2 and E3,3 with coefficients in A. The coefficient of E1,1 is
A(m, n)-GRADED LIE SUPERALGEBRAS 11
Å» Å» Å» Å»
1 1
-(-1)a a3(a1, a2, a3) - (-1)a a3[[a1, a2], a3] = 0,
while the coefficient of E3,3 is
Å» Å» Å» Å»
3 1
(-1)a a2(a3, a1, a2) - (-1)a a3[[a1, a2], a3] = 0.
Hence
Å» Å» Å»
1
(a3, a2, a1) = -(-1)(a +a2)a3(a3, a1, a2)
for any homogeneous a1, a2, a3 " A. Permuting cyclically twice, we determine that
(a1, a2, a3) = -(a1, a2, a3),
so that A is associative in this case also, and then (3.9) is satisfied too.
In this way, we have arrived at our main theorem. The last sentence in it is a
consequence of condition ("G3) in Definition 1.2.
Theorem 3.10. Assume L = (g " A) •" D is a superalgebra over a field F of
characteristic zero where g = slm+1,n+1, m > n e" 0, A is unital F-superalgebra,
and D is Lie superalgebra, and with multiplication as in (3.1). Then L is a Lie
superalgebra if and only if
" A is a unital associative superalgebra,
" D is a Lie subsuperalgebra of L and Ć : D DerF(A) (Ć(d)(a) = da) is a
representation of D as superderivations on the algebra A,
Å»
1
" [d, a1 | a2 ] = da1 | a2 + (-1)d a1 | da2 ,
Å» Å»
1
" (-1)a a3 a1 | a2a3 = 0,
1
" a1 | a2 a3 = [[a1, a2], a3],
m-n
for any homogeneous elements d " D and a1, a2, a3 " A.
Moreover, the A(m, n)-graded Lie superalgebras (for m > n e" 0) are exactly
these superalgebras with the added constraint that
D = A | A .
Remark 3.11. Let A be any unital associative superalgebra. Then ad[A,A] is a
subsuperalgebra of DerF(A). Consider the Lie superalgebra
def
L(A) = (g " A) •" ad[A,A],
with g = slm+1,n+1 (m > n e" 0), with multiplication given by (3.1), with ad[A,A]
1
in place of D, and with a | a = ad[a,a for any a, a " A. Then Theorem
]
m-n
3.10 shows that L(A) is an A(m, n)-graded Lie superalgebra.
12 GEORGIA BENKART, ALBERTO ELDUQUE
Moreover, for any A(m, n)-graded Lie superalgebra L with coordinate superal-
<"
gebra A, Theorem 3.10 implies that L/Z(L) L(A). Thus L is a cover of L(A) (a
=
central extension of L(A) which is perfect, L = [L, L]).
Any perfect Lie superalgebra L has a unique (up to isomorphism) universal
central extension, which is also perfect, called the universal covering superalgebra
of L. Two perfect Lie superalgebras L1 and L2 are said to be centrally isogenous if
<"
L1/Z(L1) L2/Z(L2).
=
To see how these concepts apply in our case, consider the Lie superalgebra
glm+1,n+1(A). This is the Lie superalgebra Mm+1,n+1(F) " A of the associative su-
peralgebra of block partitioned matrices of size (m+1)+(n+1) × (m+1)+(n+1)
tensored (as superalgebras over F) with the associative superalgebra A. Its com-
mutator subalgebra slm+1,n+1(A) = [glm+1,n+1(A), glm+1,n+1(A)] is an A(m, n)-
graded Lie superalgebra (since A is unital) with A as a coordinate superalgebra.
Thus, Theorem 3.10 and Remark 3.11 give:
Corollary 3.12. The A(m, n)-graded Lie superalgebras with m > n e" 0 are pre-
cisely the Lie superalgebras which are centrally isogeneous to the Lie superalgebras
slm+1,n+1(A) for A a unital associative superalgebra A.
The universal central extension of the Lie superalgebra slm+1,n+1(A) with m = n
and m + n e" 3 has been shown to be the Steinberg Lie superalgebra stm+1,n+1(A)
in [MP].
ż4. A(n, n)-graded Lie superalgebras
The situation when m = n is much more involved than the previous one, due to
the fact that the complete reducibility result in Proposition 2.5 is no longer valid
in this case, as already noted above. However, when L is an A(n, n)-graded Lie
superalgebra with grading subsuperalgebra g, in one respect the situation is even
simpler than for m = n, because of the next result:
Proposition 4.1. Let g be a split simple classical Lie superalgebra of type A(n, n)
(n > 0). Then Homg(g " g, g) is spanned by the Lie bracket.
Proof. Write x for the class of a matrix x " slp,p modulo the center F I2p (p =
Å»
n + 1). As in the proof of Proposition 2.6, let v = 1,2p and w = 2p,1. Then
any Õ " Homg(g " g, g) is determined by Õ(v " w), which belongs to the Cartan
subalgebra h, and it is annihilated by i,i+1 for 2 d" i d" p-1 and p+1 d" i d" 2p-2.
Therefore, Õ(v " w) is a linear combination of (p - 1)1,1 - (2,2 + · · · + p,p) and
(p - 1)2p,2p - (p+1,p+1 + · · · + 2p-1,2p-1), so that dim Homg(g " g, g) d" 2. If it
were 2, there would exist a Õ " Homg(g " g, g) such that Õ(v " w) = (p - 1)1,1 -
(2,2 + · · · + p,p), but since [Ep,p+1, E1,2p] = 0 = [Ep,p+1, E2p,1], it follows that
0 = [p,p+1, Õ(v " w)] = [p,p+1, (p - 1)1,1 - (2,2 + · · · + p,p)] = p,p+1, a
contradiction. Consequently, dim Homg(g " g, g) = 1, and hence it is spanned by
the Lie bracket.
A(m, n)-GRADED LIE SUPERALGEBRAS 13
Therefore, any A(n, n)-graded Lie superalgebra with grading subsuperalgebra
g acting completely reducibly on L satisfies the hypotheses of [BE2, Lemma 4.1],
where º is the supersymmetric form on g = pslp,p induced by the supertrace on
slp,p (we will denote it by str too), and as an immediate consequence we obtain the
following:
Theorem 4.2. Let L be a "-graded Lie superalgebra over F with grading subsu-
peralgebra g of type A(n, n) (n > 0), and assume that L is a completely reducible
module over g. Then there is a unital (super)commutative associative superalgebra
A and a superspace D such that L = (g " A) •" D, with multiplication given by
[f " a, g " a ] = (-1)! [f, g] " aa + str(fg) a | a
(4.3)
[d, L] = 0
for homogeneous elements f, g " g, a, a " A, d, d " D; where | : A × A D
is a super skew symmetric bilinear even form with A | A = D and satisfying the
Å» Å»
1
two-cocycle condition, (-1)a a3 a1a2, a3 = 0.
Corollary 4.4. A "-graded Lie superalgebra L with grading subalgebra g corre-
sponding to a root system " of type A(n, n) and acting completely reducibly on L
is a covering of a Lie superalgebra g " A, where A is a unital supercommutative
associative superalgebra.
A precise description of the structure of an arbitrary A(n, n)-graded Lie su-
peralgebra is yet to be fully resolved. We can prove that, even though complete
reducibility fails, as a module over the grading superalgebra g = psln+1,n+1, such a
Lie superalgebra must be a direct sum of copies of gln+1,n+1, sln+1,n+1, pgln+1,n+1,
psln+1,n+1, and trivial modules. However, the large number of possibilities for ele-
ments in the various spaces Homg(V "W, X), where V, W, X are among the modules
above, makes accomplishing Step 4 in Procedure 1.3 a daunting task in this case.
References
[AABGP] B.N. Allison, S. Azam, S. Berman, Y. Gao, A. Pianzola, Extended Affine Lie Algebras
and Their Root Systems, Memoirs Amer. Math. Soc. 126, vol. 603, 1997.
[ABG1] B.N. Allison, G. Benkart, Y. Gao, Central extensions of Lie algebras graded by finite
root systems, Math. Ann. 316 (2000), 499-527.
[ABG2] B.N. Allison, G. Benkart, Y. Gao, Lie Algebras Graded by the Root Systems BCr,
r e" 2, Memoirs Amer. Math. Soc., Providence, R.I., 2001 (to appear).
[BE1] G. Benkart and A. Elduque, Lie superalgebras graded by the root system B(m, n), sub-
mitted; Jordan preprint archive: http://mathematik.uibk.ac.at/jordan/ (paper 108).
[BE2] G. Benkart and A. Elduque, Lie superalgebras graded by the root systems C(n), D(m, n),
D(2,1;Ä…), F(4), and G(3), submitted; Jordan preprint archive:
http://mathematik.uibk.ac.at/jordan/ (paper 112).
[BS] G. Benkart and O. Smirnov, Lie algebras graded by the root system BC1, J. Lie Theory
(to appear).
14 GEORGIA BENKART, ALBERTO ELDUQUE
[BZ] G. Benkart and E. Zelmanov, Lie algebras graded by finite root systems and intersection
matrix algebras, Invent. Math. 126 (1996), 1 45.
[BM] S. Berman and R.V. Moody, Lie algebras graded by finite root systems and the inter-
section matrix algebras of Slodowy, Invent. Math. 108 (1992), 323 347.
[K1] V.G. Kac, Lie superalgebras, Advances in Math. 26 (1977), 8 96.
[K2] V.G. Kac, Representations of classical superalgebras; Differential and Geometrical
Methods in Mathematical Physics II, Lect. Notes in Math., vol. 676, Springer-Verlag,
Berlin, Heidelberg, New York, 1978, pp. 599 626.
[MP] A.V. Mikhalev and I.A. Pinchuk, Universal central extensions of the matrix Lie su-
peralgebras sl(m, n, A); Combinatorial and computational algebra (Hong Kong, 1999),
Contemp. Math., vol. 264, Amer. Math. Soc., Providence, R.I., 2000, pp. 111 125.
[N] E. Neher, Lie algebras graded by 3-graded root systems, Amer. J. Math. 118 (1996),
439 491.
[S] P. Slodowy, Beyond Kac-Moody algebras and inside; Lie Algebras and Related Topics,
Canad. Math. Soc. Conf. Proc. 5, Britten, Lemire, Moody eds., 1986, pp. 361-371.
E-mail address: benkart@math.wisc.edu
E-mail address: elduque@posta.unizar.es
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