Homology, Homotopy and Applications, vol.4(2), 2002, pp.259–275

DEFINING RELATIONS FOR CLASSICAL LIE SUPERALGEBRAS WITHOUT CARTAN MATRICES

P. GROZMAN, D. LEITES and E. POLETAEVA

(communicated by Clas Löfwall)

Abstract The analogs of Chevalley generators are offered for simple (and close to them) ℤ-graded complex Lie algebras and Lie superalgebras of polynomial growth without Cartan matrix. We show how to derive the defining relations between these generators and explicitly write them for a “most natural” (“distinguished” in terms of Penkov and Serganova) system of simple roots. The results are given mainly for Lie superalgebras whose component of degree zero is a Lie algebra (other cases being left to the reader). Observe presentations presentations of exceptional Lie superalgebras and Lie superalgebras of hamiltonian vector fields.

To Jan–Erik Roos on his sixty–fifth birthday

§1. Preliminaries

After Berezin formulated to one of us (DL) the problem which in modern terms would sound “define supermanifolds and differential supergeometry” it was natural to look after examples of Lie superalgebras that naturally appear in mathematics. Homotopy rings with respect to Whitehead’s product are examples of such Lie rings, but nobody, so far, described these natural rings in any case. By description we mean identification of the semisimple part and radical. A reason for such careless attention to these rings becomes clear soon after one tackles the problem: they are nilpotent, hence, not so interesting in a sense (simple Lie (super)algebras have a richer structure and interesting representation theory).

A paper by C. Löfvall and J.-E. Roos [LR] is a break-through: in a similar problem they made a very interesting observation: they not only found traces left by simple Lie superalgebras where nothing indicated them, they also identified these superalgebras as a “positive part” of certain twisted loops algebras with values in simple Lie superalgebras. The paper [LR] is, clearly, the first in a series to appear,

We were financially supported: P.G. by the Swedish Institute in 1993–95; D.L. by I. Bendixson grant, an NSF grant via IAS, Princeton, in 1989 and by NFR during 1986–1999; D.L. and E.P. by SFB-170 in June–July of 1990. We are particularly thankful to Yu. Kochetkov, actually a co-author, for calculating the relations for 𝔰𝔥(0|n). Received February 27, 2001, revised July 3, 2001; published on July 12, 2002. 2000 Mathematics Subject Classification: 17A70, 17B01, 17B70. Key words and phrases: Lie superalgebras, defining relations. \begin{matrix}c \\ ◯ \end{matrix} 2002, P. Grozman, D. Leites and E. Poletaeva. Permission to copy for private use granted.

where, among other things, Löfvall and Roos will need presentations (in other words, generators and defining relations) of the positive part 𝔤_{>} = ∑_{i>0} 𝔤_{i} of certain (twisted)

loop superalgebra 𝔤 = ∑_{i>0} 𝔤_{i}, associated with the simple (or a close to the simple)

finite dimensional Lie superalgebras.

The results obtained here (an extension of [GLP]), and those of [GL1], [GL2], [Y], [LSe], as well as those which, though listed as open problems, are supplied with an instruction how to solve them, describe how to shorthand the presentation needed, both generators and relations, compare with appalling presentation of [T] or implicit presentations of the positive parts of vectorial algebras in [FF].

We hope that our results contribute to the fest on the occasion of Jan-Erik’s birthday and make his calculations, if not life, easier.

Namely, we observe that even in the absence of 𝔤_{0} there is a concise way to encode the presentation. Indeed, in the majority of cases 𝔤_{>} is the direct sum of irreducible 𝔤_{0}-modules 𝔤_{i}, and, as Lie superalgebra, 𝔤_{>} is generated by 𝔤_{1}. If 𝔤_{1} is irreducible, as in cases of Löfvall and Roos, then, instead of dim 𝔤_{1} generators, it suffices to take just one, any one vector, say the lowest weight vector.

The space of relations (same as the space of generators in the general case) must not split into the direct sum of irreducible 𝔤_{0}-modules, but, nevertheless, one can list only the vacuum vectors, i.e., the lowest AND highest weight vectors (since some modules can be glued in indecomposable conglomerates, we need both). Mathematica-based package SuperLie ([G]) helps to find these vacuum vectors.

To reduce volume of the paper, we did not reproduce the standard homological interpretation of relations and spectral sequence leading to the answer; for Lie algebras it is expressed in [LP] and its superization is straightforward via the Sign Rule. Observe only that since relations represent homology class, they can be “pure” or “dirty” if defined modulo boundaries.

History and an overview. The traditional way to determine classical simple finite dimensional Lie algebra (over ℂ) is via Chevalley generators, though other generators are possible. For discussion of other possibilities with examples see [GL1]. Recently a presentation of simple Lie superalgebras of the four Cartan series of vector fields was given [LP] and, together with Serre relations for affine Kac–Moody algebras [K1], this completed description of presentations of simple ℤ-graded Lie algebras of polynomial growth (presentations with respect to other choices of generators are certainly possible).

Here we consider simple ℤ-graded Lie superalgebras of polynomial growth (and close to them “classical” Lie superalgebras, such as central extensions of the simple ones, their algebras of differentiations, etc.). Their list is conjecturally ([LS1]) completed and consists of

• the finite dimensional ones (classification results by Kaplansky and Nahm-Rittenberg-Scheunert [FK], [NRS] were skillfully rounded up by Kac [K2]),

• the vectorial algebras, i.e., algebras of vector fields, (classification announced [LS1] and partly proved [LS2] by Leites and Schepochkina; the proof was again quickly rounded up by Kac and Cheng [K3], bar some gaps, see [Sh5]),

• the twisted loop algebras (with symmetrizable Cartan matrix [vdL]), or obtained as twisted loops [FLS], and

• the stringy (i.e., vectorial algebras pertaining to string theories) algebras (for their intrinsic definition and list see [GLS1]).

In terms of presentations, another subdivision is more natural:

(a) the algebras of the form 𝔤(A) with Cartan matrix A (subdivided into subclasses (a_{s}) with symmetrizable Cartan matrix and (a_{n}) with non-symmetrizable Cartan matrix,

(q) the series 𝔭𝔰𝔮 and its relatives (central extensions, exterior differentiations, etc.) and

(v) the vectorial algebras and their relatives, the members of the subclass are easily recognized by lack of the property “if α is a root, then so is −α”.

For 𝔤(A) (both subcases) a very redundant presentation is given in [LSS] and, the minimal one, in [GL2]; their q-quantization (for symmetrizable A) is described in [Y]. The redundant presentation [LSS], though very long, has an advantage: it only involves Serre relations. Regrettably, it is so redundant that practically it is useless.

For 𝔮(n) and 𝔮(n)^{(1)} see [LSe]; presentation of twisted loops is an open problem. For presentations of vectorial Lie algebras see [LP]; the series with 𝔤_{0} a Lie alge-

bra see [GLP]; here we also consider several cases left in [GLP] as open problems and the Lie superalgebras of Hamiltonian vector fields and its central extension: Poisson superalgebra. The last two cases are of interest in relation with spinoroscillator representations and its q-quantization, see [Kl], [LSh].

Though several results describing presentations of simple and close to them vectorial superalgebras were obtained a while ago ([Ko], [T]) they are given in the form too bulky to grasp or implicit ([U]; [FF]). A simplification of presentations is desirable: for 𝔤_{0} = 𝔰𝔥(0|n) the dimension of the space of relations computed in [T] for n = 5 is equal to 420 and grows with n, whereas the total number of vacum vectors in the space of relations is < 10 and does not grow with n, cf. Tables 2.1.2.

Problem formulation. Consider ℤ-graded Lie superalgebras 𝔤 = ⊕_{i∈ℤ} 𝔤_{i} of the

following two types:

(1) vectorial Lie superalgebras, i.e., of finite depth d (in the above sum i ⩾ d), cf. [LP];

(2) of infinite depth, but not of the form 𝔤(A) (the algebras 𝔤(A) being already considered in complementing each other papers [GL2] and [Y]) or of type 𝔮 (for whose presentation see [LSe]).

Among these algebras we will first consider the ones for which 𝔤_{0} is a Lie algebra. The general case is an open problem, which can be solved any time for any given 𝔤 via the lines indicated here and with the help of Grozman’s SuperLie package.

Observe that in [GL2] and [Y] all bases (systems of simple roots or, rather, corresponding generators) are considered. For the vectorial algebras and superalgebras and for loop algebras with values in vectorial superalgebras we have considered below just one of the possible bases. It can well happen that presentations corresponding to some other base is nicer in some sense: e.g., for a rank n simple Lie algebra, Serre relations corresponding to 3n Chevalley generators though numerous (∼ n^{2}) are very simple and easy to compute, unlike a handful of independent on n but more intricate relations between the pair of “Jacobson generators” considered in [GL1]. Nevertheless, both presentations are needed. For a method of passage from base to base (an analog of the Weyl group) see [PS].

Let 𝔤_{+} = ⊕_{i>0} 𝔤_{i} and 𝔤_{−} = ⊕_{i<0} 𝔤_{i}; let 𝔫_{±} be a maximal nilpotent subalgebra of 𝔤_{0}

described in textbooks (e.g., [OV]) if 𝔤_{0} is a Lie algebra or in [PS] (see also refs. therein) if 𝔤_{0} is a Lie superalgebra. We decompose 𝔤 into the sum 𝔑_{−}⊕𝔥⊕𝔑_{+}, where 𝔑_{±} = 𝔫_{±} ⊕ 𝔤_{±}, and for the cases when 𝔤_{0} is a Lie algebra (purely even) describe the defining relations. The relations obtained for vectorial Lie superalgebras are not very simple-looking, cf. [LP].

Notice that, unlike the case of finite dimensional simple Lie algebras, the bases, i.e., systems of simple roots, correspond not to maximal solvable Lie superalgebras (described in [Shc]) but to what is called Borel subalgebras in [PS].

Open problems are listed in §4.

§1. Generators in some vectorial Lie superalgebras and associated loops

1.1. Generators of 𝔳𝔢𝔠𝔱(0|n)

Set ∂_{i} = \frac{∂}{∂x_{i}}. Some of the generators of 𝔳𝔢𝔠𝔱(0|n) generate its subalgebras as indicated (i.e., the X_{i}^{+} and Y generate 𝔑_{+}; the X_{i}^{±} generate 𝔰𝔩(1|n)):

𝔰𝔩(1|n)
𝔑_{+}  x_{1}∂_{2}, …, x_{n−1}∂_{n}, x_{n} ∑ x_{i}∂_{i} x_{n}x_{n−1}∂_{1}
𝔑_{−}  x_{2}∂_{1}, …, x_{n}∂_{n−1}, ∂_{n}
notations  X_{1}^{±}, …, X_{n−1}^{±}, X_{n}^{±}  Y

The generators of 𝔰𝔳𝔢𝔠𝔱(0|n) are the same as of 𝔳𝔢𝔠𝔱(0|n) but without the boldfaced element X_{n}^{+} = x_{n} ∑ x_{i}∂_{i}. The loop algebras have two more generators: for 𝔳𝔢𝔠𝔱(0|n) set

X_{0}^{−} = x_{1} … x_{n}∂_{n} · t^{−1} and X_{0}^{+} = ∂_{1} · t.

For 𝔰𝔳𝔢𝔠𝔱(0|n) set

X_{0}^{−} = x_{1} … x_{n−1}∂_{n} · t^{−1} and X_{0}^{+} = ∂_{1} · t.

It is not clear that this choice of generators (the highest weight vector of 𝔤_{−1} and the lowest weight vector of 𝔤_{1}) which gives nice-looking relations for Lie algebras (and even Lie superalgebra with Cartan matrix) is the best when Lie superalgebras are very non-symmetric.

1.2. Generators of 𝔨(1|n)

In what follows we will by abuse of language write just f instead of either H_{f}, the Hamiltonian vector field generated by f or K_{f}, the contact vector field generated by f; in so doing we must remember that in either case (H_{f} or K_{f}) the degree of the vector field generated by a monomial f of degree k is equal to k − 2.

Some of the generators of 𝔨(1|n) generate the following subalgebras (for n = 2k > 6 and n = 2k + 1 > 5, respectively):

𝔬𝔰𝔭(2k|2)
𝔑_{+}  tη_{1} ξ_{1}η_{2} … η_{n}ξ_{n−1} ξ_{n}ξ_{n−1} η_{1}η_{2}η_{3}
𝔑_{−}  ξ_{1} η_{1}ξ_{2} … η_{n−1}ξ_{n} η_{n}η_{n−1}
notations  X_{0}^{±} X_{1}^{±} … X_{n−1}^{±} X_{n}^{±}  Y
𝔬𝔰𝔭(2k + 1|2)
𝔑_{+}  tη_{1} ξ_{1}η_{2} … η_{n}ξ_{n−1} ξ_{n}θ  η_{1}η_{2}η_{3}
𝔑_{−}  ξ_{1} η_{1}ξ_{2} … η_{n−1}ξ_{n} η_{n}θ
notations  X_{0}^{±} X_{1}^{±} … X_{n−1}^{±} X_{n}^{±}  Y

The generators of 𝔥(0|n) and 𝔭𝔬(0|n) are those above without the boldfaced element X_{0}^{+} = tη_{1}.

The loop algebras 𝔥(0|n)^{(1)} and 𝔭𝔬(0|n)^{(1)} have two more generators:

Y_{0}^{−} = ξ_{1} … ξ_{n}η_{n}η_{n−1} … η_{2} · t^{−1} and Y_{0}^{+} = η_{1} · t.

It is not clear that this choice of generators is the best and it is desirable to experiment with other choices.

In small dimensions (n < 7) relations look differently and are to be computed separately. Besides, the generators look different. Though presentations of some of these algebras were considered, it is advisabable to revise it (the results of A. Nilsson are unpublished and those of [FNZ], as well of [T], should be presented in a more user-friendly form).

§2. Relations

2.1. Relations for 𝔑_{−} of 𝔨(1|n), 𝔭𝔬(0|n) and 𝔥(0|n)

Clearly, 𝔑_{−} for 𝔳𝔢𝔠𝔱(0|n) and 𝔰𝔳𝔢𝔠𝔱(0|n) coincides with 𝔫_{−} for 𝔰𝔩(1|n) while 𝔑_{−} for 𝔨(1|n) and 𝔭𝔬(0|n) coincides with 𝔫_{−} for 𝔬𝔰𝔭(n|2), which are known [LSe], [GL2].

2.1.1. Relations for 𝔑_{−} of 𝔥(0|n)

The Lie algebra 𝔥(0|n) is generated by the same elements as 𝔨(1|n) and 𝔭𝔬(0|n) but the relations are different: for 𝔥(0|n) there is an additional relation of weight (0, …, 0) with respect to 𝔬(n) because (for n > 1)

H_{2}(𝔤_{−1}) = S^{2}(𝔤_{−1}) = R(2π) ⊕ R(0).

The corresponding cycle of weight 0 is

{ξ_{1}, η_{1}} + ⋯ + {ξ_{n}, η_{n}} (+{θ, θ} if n is odd ).  (∗)

The relation expressed in terms of generators looks awful. It can be beautified as follows. In the space of relations corresponding to the other irreducible component Homology, Homotopy and Applications, vol. 4(2), 2002 264

the subspace of relations of weight 0 is of dimension n−1. Therefore, n−1 summands in (∗) vanish; for role of survivor select the simplest one of them, say, the following one: {ξ_{1}, η_{1}} = 0.

2.1.2. Relations for 𝔑_{+} of 𝔥(0|n), n > 4

H_{2}(𝔤_{+}) is the direct sum of irreducible 𝔤_{0}-modules with the following lowest weights with respect to 𝔬(n), for notations see Tables in [OV]:

For n = 2l, l ⩾ 5:

N  the lowest weight  the corresponding cycle
1  −2(ε_{1} + ε_{2} + ε_{3})  η_{1}η_{2}η_{3} ∧ η_{1}η_{2}η_{3}
2  −2(ε_{1} + ε_{2})  ∑_{i} η_{1}η_{2}η_{i} ∧ η_{1}η_{2}ξ_{i}
3  −2ε_{1}  ∑_{i,j} η_{1}η_{i}ξ_{j} ∧ η_{1}η_{i}ξ_{j} − 2 ∑_{i<j} η_{1}η_{i}η_{j} ∧ η_{1}ξ_{i}ξ_{j}
4  0  ∑_{i,j} ∑_{k⩽l}(η_{i}η_{j}ξ_{k} ∧ ξ_{i}ξ_{j}η_{k} + η_{i}η_{j}η_{k} ∧ ξ_{i}ξ_{j}ξ_{k})
5 −(2ε_{1} + ε_{2} + ε_{3} + ε_{4} + ε_{5})  η_{1}η_{2}η_{3} ∧ η_{4}η_{5}η_{1} − η_{1}η_{2}η_{4} ∧ η_{3}η_{5}η_{1}+
+η_{1}η_{2}η_{5} ∧ η_{3}η_{4}η_{1}
6  −ε_{1}  η_{1} ∑_{i} η_{i}η_{i+1}η_{i+2} ∧ ξ_{i}ξ_{i+1}ξ_{i+2}

For small l the relations look differently; the form of relations 1) – 3) is the same as in the general case, the new in form relations are (here ∑_{cycl} means the cyclic permutation of η_{1}η_{2}η_{3}):

l N the lowest weight  the corresponding cycle
3 5  −2ε_{1}  η_{1}η_{2}ξ_{2} ∧ η_{1}η_{3}ξ_{3} − η_{1}η_{2}ξ_{3} ∧ η_{1}η_{3}ξ_{2} − η_{1}η_{2}η_{3} ∧ η_{1}ξ_{2}ξ_{3}
3 6 −2(ε_{3} − ε_{1} − ε_{2})  η_{1}η_{2}ξ_{3} ∧ η_{1}η_{2}ξ_{3}
4 5 ε_{4} − ε_{1} + ε_{2} + ε_{3} ∑_{cycl} η_{1}η_{2}ξ_{4} ∧ η_{3}η_{4}ξ_{4} − ∑_{i} ∑_{cycl}(η_{1}η_{2}ξ_{4} ∧ η_{3}η_{i}ξ_{i}+
η_{1}η_{2}η_{3} ∧ ∑_{i} η_{i}ξ_{i}ξ_{4})
4 6 −(2ε_{1} + ε_{2} + ε_{3}) η_{1}η_{2}η_{3} ∧ η_{1}η_{4}ξ_{4} − η_{1}η_{2}η_{4} ∧ η_{1}η_{3}ξ_{4} + η_{1}η_{3}η_{4} ∧ η_{1}η_{2}ξ_{4}
5 6 −(2ε_{1} − ε_{2} − ε_{3}− η_{1}η_{2}η_{3} ∧ η_{4}η_{5}η_{1} − η_{1}η_{2}η_{4} ∧ η_{3}η_{5}η_{1} + η_{1}η_{2}η_{5} ∧ η_{3}η_{4}η_{1}
−ε_{4} + ε_{5})
6  −ε_{1}  η_{1} ∑_{i}(η_{i}η_{i+1}η_{i+2} ∧ ξ_{i}ξ_{i+1}ξ_{i+2})

For n = 2l + 1, l ⩾ 5:

N the lowest weight  the corresponding cycle
1 −2(ε_{1} + ε_{2} + ε_{3})  η_{1}η_{2}η_{3} ∧ η_{1}η_{2}η_{3}
2  −2(ε_{1} + ε_{2})  ∑_{i} η_{1}η_{2}η_{i} ∧ η_{1}η_{2}ξ_{i}
3  −2ε_{1}  ∑_{i,j} η_{1}η_{i}ξ_{j} ∧ η_{1}η_{i}ξ_{j} − 2 ∑_{i<j} η_{1}η_{i}η_{j} ∧ η_{1}ξ_{i}ξ_{j}
4  0  ∑_{i,j} ∑_{k⩽l}(η_{i}η_{j}ξ_{k} ∧ ξ_{i}ξ_{j}η_{k} + η_{i}η_{j}η_{k} ∧ ξ_{i}ξ_{j}ξ_{k})
5 −(2ε_{1} + ε_{2} + ε_{3} η_{1}η_{2}η_{3} ∧ η_{4}η_{5}η_{1} − η_{1}η_{2}η_{4} ∧ η_{3}η_{5}η_{1} + η_{1}η_{2}η_{5} ∧ η_{3}η_{4}η_{1}
+ε_{4} + ε_{5})
6  −ε_{1}  η_{1} ∑_{i} η_{i}η_{i+1}η_{i+2} ∧ ξ_{i}ξ_{i+1}ξ_{i+2}

The space H_{1}(𝔫_{+}; 𝔤_{1}) is responsible for the following relations (element indicated should vanish for i > 3):

{ξ_{1}η_{2}, η_{1}η_{2}η_{3}}, {ξ_{2}η_{3}, η_{1}η_{2}η_{3}}, (ad ξ_{3}η_{4})^{2}(η_{1}η_{2}η_{3}),
{ξ_{i}η_{i+1}, η_{1}η_{2}η_{3}}, {ξ_{n−1}ξ_{n}, η_{1}η_{2}η_{3}}.

2.1.3. Relations for 𝔑_{+} of 𝔨(1|n), n > 4 For the X_{i}^{+} the relations are the same as for 𝔫_{+} of 𝔬𝔰𝔭(n|2), cf. [GL2]. The relations between the X_{i}^{+}, 1 ⩽ i ⩽ n and Y are the same as for 𝔥(0|n). New

relations involving X_{0}^{+} and Y are:

N the lowest weight  the corresponding cycle
1  −4ε_{1}  ∑_{i}(−q_{1}^{2}q_{i} ∧ q_{1}^{2}p_{i}) + (n + 2)tq_{1} ∧ q_{1}^{3}
2  −3ε_{1} − ε_{2}  q_{1}^{3} ∧ tq_{2} + q_{1}^{2}q_{2} ∧ tq_{1}

2.1.4. Relations between 𝔑_{+} and 𝔑_{−} for 𝔥(0|n) and 𝔨(1|n), n > 4

These relations are as for 𝔬𝔰𝔭(2|2n) unless they involve Y; and the new extra ones are:

[Y, X_{0}^{−}] = η_{2}η_{3} [Y, X_{i}^{−}] = 0 for i > 0.

2.2. Relations for 𝔳𝔢𝔠𝔱(0|n) and 𝔰𝔳𝔢𝔠𝔱(0|m), m > 2 2.2.1. Relations for 𝔑_{+} of 𝔳𝔢𝔠𝔱(0|n), n > 2

The space H_{1}(𝔫_{+}; 𝔤_{1}) is spanned by (adp_{1}q_{2})^{3}q_{1}^{3} ∧ p_{1}q_{2}, q_{1}^{3} ∧ p_{2}q_{3}, …, q_{1}^{3} ∧ p_{n−1}q_{n}, and q_{1}^{3} ∧ p_{n}^{2}.

H_{2}(𝔤_{+}) is the direct sum of irreducible 𝔤_{0}-modules with the following lowest weights:

N  the lowest weight  the corresponding cycle
1  2(ε_{n} + ε_{n−1} − ε_{1})  ξ_{n}ξ_{n−1}∂_{1} ∧ ξ_{n}ξ_{n−1}∂_{1}
2 2ε_{n} + ε_{n−1} + ε_{n−2} − ε_{1} − ε_{2}  ξ_{n}ξ_{n−1}∂_{2} ∧ ξ_{n}ξ_{n−2}∂_{1} − ξ_{n}ξ_{n−1}∂_{1} ∧ ξ_{n}ξ_{n−2}∂_{2}
3  2ε_{n} + ε_{n−1} − ε_{1}  ∑_{i} ξ_{n}ξ_{i}∂_{1} ∧ ξ_{n}ξ_{n−1}∂_{i}
4  ε_{n} + ε_{n−1} + ε_{n−2}+  ξ_{n}ξ_{n−1}∂_{1} ∧ ξ_{n−2}ξ_{n−3}∂_{1} − ξ_{n}ξ_{n−2}∂_{1} ∧ ξ_{n−1}ξ_{n−3}∂_{1}+
+ε_{n−3} − 2ε_{1}  ξ_{n}ξ_{n−3}∂_{1} ∧ ξ_{n−1}ξ_{n−2}∂_{1}
5  2ε_{n}  ∑_{i,j} ξ_{n}ξ_{i}∂_{i} ∧ ξ_{n}ξ_{j}∂_{j}
6  ε_{n} + ε_{n−1} + ε_{n−2} − ε_{1}  ∑_{i}(ξ_{n}ξ_{n−1}∂_{1} ∧ ξ_{n−2}ξ_{i}∂_{i} + ξ_{n−2}ξ_{n−3}∂_{1} ∧ ξ_{n}ξ_{i}∂_{i}+
ξ_{n−2}ξ_{n}∂_{1} ∧ ξ_{n−1}ξ_{i}∂_{i})
7  2ε_{n} + ε_{n−1} − ε_{1}  ∑_{i} ξ_{n}ξ_{i}∂_{i} ∧ ξ_{n}ξ_{n−1}∂_{1}
8  2ε_{n}  (∑ ξ_{n}ξ_{i}∂_{i}) ∧ (∑ ξ_{n}ξ_{j}∂_{j})

The corresponding relations for 𝔰𝔳𝔢𝔠𝔱 are the relations 1) – 4).

The relations for 𝔳𝔢𝔠𝔱(0|4) are

N the lowest weight  the corresponding cycle
1  2(ε_{4} + ε_{3} − ε_{1})  ξ_{4}ξ_{3}∂_{1} ∧ ξ_{4}ξ_{3}∂_{1}
2  2ε_{4} + ε_{3} − ε_{1}  ∑ ξ_{4}ξ_{i}∂_{1} ∧ ξ_{4}ξ_{3}∂_{i}
3  2ε_{4}  ∑_{i,j} ξ_{4}ξ_{i}∂_{j} ∧ ξ_{4}ξ_{j}∂_{i}
4 ε_{4} + ε_{3} + ε_{2} − ε_{1} ∑(ξ_{4}ξ_{3}∂_{1} ∧ ξ_{2}ξ_{i}∂_{i} + ξ_{4}ξ_{2}∂_{1} ∧ ξ_{3}ξ_{i}∂_{i} + ξ_{3}ξ_{2}∂_{1} ∧ ξ_{4}ξ_{i}∂_{i})
5  2ε_{4}  (∑ ξ_{4}ξ_{i}∂_{i}) ∧ (∑ ξ_{4}ξ_{i}∂_{i})
6  2ε_{4} + ε_{3} − ε_{1}  (∑ ξ_{4}ξ_{i}∂_{i}) ∧ ξ_{4}ξ_{3}∂_{1}

The corresponding relations for 𝔰𝔳𝔢𝔠𝔱(0|4) are the relations 1) – 3). The relations for 𝔳𝔢𝔠𝔱(0|3) are

N the lowest weight  the corresponding cycle
1  2(ε_{3} + ε_{2} − ε_{1})  ξ_{3}ξ_{2}∂_{1} ∧ ξ_{3}ξ_{2}∂_{1}
2  2ε_{3}  ξ_{3}ξ_{2}∂_{1} ∧ ξ_{3}ξ_{1}∂_{2} + ξ_{3}ξ_{1}∂_{2} ∧ ξ_{3}ξ_{2}∂_{1}
3  2ε_{3}  (∑ ξ_{3}ξ_{i}∂_{i}) ∧ (∑ ξ_{3}ξ_{i}∂_{i})
4  2ε_{3} + ε_{2} − ε_{1}  (∑ ξ_{3}ξ_{i}∂_{i}) ∧ ξ_{3}ξ_{2}∂_{1}

The corresponding relations for 𝔰𝔳𝔢𝔠𝔱(0|3) ≃ 𝔰𝔭𝔢(3) are the relations 1) – 2).

2.2.2. Relations between 𝔑_{+} and 𝔑_{−} for 𝔳𝔢𝔠𝔱(0|n), n > 3, and 𝔰𝔳𝔢𝔠𝔱(0|m), m > 2

These relations are as for 𝔰𝔩(1|n) unless they involve Y; the extra relations are:

[Y, X_{n}^{−}] = (x_{n−1}∂_{1}) = [X_{n−1}^{−}, […, [X_{2}^{−}, X_{1}^{−}]] …]; [Y, X_{i}^{−}] = 0 for i > 0.

2.2.3. Relations for 𝔑_{±} of 𝔳𝔢𝔠𝔱(0|n)^{(1)}, n > 3

The new relations that involve X_{0}^{+} are (we only indicate the terms to be equated to zero):

\underline{For n = 3, 𝔑_{+}}: [X_{0}^{+}, X_{0}^{+}], [X_{0}^{+}, X_{2}^{+}], [X_{0}^{+}, Y], [X_{1}^{+}, [X_{0}^{+}, X_{1}^{+}]], [Y, [X_{0}^{+}, X_{1}^{+}]]−

[X_{0}^{+}, X_{3}^{+}], (adX_{1}^{+})^{2} [X_{1}^{+}, Y], [Y, (adX_{1}^{+})^{2} Y], [[X_{0}^{+}, X_{1}^{+}], [X_{0}^{+}, X_{3}^{+}]], [[X_{0}^{+}, X_{3}^{+}], [X_{2}^{+}, X_{3}^{+}]], [[X_{3}^{+}, [X_{0}^{+}, X_{1}^{+}]], [Y, [X_{1}^{+}, X_{2}^{+}]]] + \frac{1}{2} [(adX_{1}^{+})^{2} Y, [X_{2}^{+}, [X_{0}^{+}, X_{3}^{+}]]].

\underline{For n = 3, 𝔑_{−}}: [X_{0}^{−}, X_{1}^{−}], (adX_{0}^{−})^{2} X_{2}^{−}, (adX_{0}^{−})^{2} X_{3}^{−}, (adX_{2}^{−})^{2} X_{0}^{−}, [[X_{0}^{−}, X_{2}^{−}], [X_{0}^{−}, X_{3}^{−}]], [[X_{2}^{−}, [X_{0}^{−}, X_{3}^{−}]], [X_{3}^{−}, [X_{0}^{−}, X_{2}^{−}]]], [[[X_{0}^{−}, X_{3}^{−}], [X_{1}^{−}, X_{2}^{−}]], [[X_{0}^{−}, X_{3}^{−}], [X_{2}^{−}, X_{3}^{−}]]] + 2 [[X_{3}^{−}, [X_{1}^{−}, [X_{0}^{−}, X_{2}^{−}]]], [[X_{0}^{−}, X_{3}^{−}], [X_{2}^{−}, X_{3}^{−}]]]

\underline{For n = 4, 𝔑_{+}}: [X_{0}^{+}, X_{0}^{+}], [X_{0}^{+}, X_{2}^{+}], [X_{0}^{+}, X_{3}^{+}], [X_{0}^{+}, Y], (adX_{1}^{+})^{2} X_{0}^{+}, [Y, [X_{0}^{+}, X_{1}^{+}]], [[X_{0}^{+}, X_{1}^{+}], [X_{0}^{+}, X_{4}^{+}]], [[X_{0}^{+}, X_{1}^{+}], [X_{2}^{+}, Y]]−[X_{0}^{+}, X_{4}^{+}], [[X_{0}^{+}, X_{4}^{+}], [X_{3}^{+}, X_{4}^{+}]], [[X_{2}^{+}, Y], [X_{3}^{+}, [X_{0}^{+}, X_{4}^{+}]]], [[X_{4}^{+}, [X_{0}^{+}, X_{1}^{+}]], [Y, [X_{2}^{+}, X_{3}^{+}]]] − \frac{1}{2} [[X_{3}^{+}, [X_{0}^{+}, X_{4}^{+}]], [Y, [X_{1}^{+}, X_{2}^{+}]]], [[[X_{0}^{+}, X_{4}^{+}], [X_{1}^{+}, X_{2}^{+}]], [[X_{1}^{+}, Y], [X_{2}^{+}, X_{3}^{+}]]] + [[[X_{0}^{+}, X_{1}^{+}], [X_{3}^{+}, X_{4}^{+}]], [[X_{1}^{+}, X_{2}^{+}], [X_{2}^{+}, Y]]]

\underline{For n = 4, 𝔑_{−}}: [X_{0}^{−}, X_{0}^{−}], [X_{0}^{−}, X_{1}^{−}], [X_{0}^{−}, X_{2}^{−}], (adX_{3}^{−})^{2} X_{0}^{−}, [[X_{0}^{−}, X_{3}^{−}], [X_{0}^{−}, X_{4}^{−}]], [[X_{0}^{−}, X_{4}^{−}], [X_{3}^{−}, [X_{0}^{−}, X_{4}^{−}]]],

[[X_{2}^{−}, [X_{0}^{−}, X_{3}^{−}]], [X_{4}^{−}, [X_{0}^{−}, X_{3}^{−}]]],
[[X_{4}^{−}, [X_{0}^{−}, X_{3}^{−}]], [[X_{0}^{−}, X_{4}^{−}], [X_{2}^{−}, X_{3}^{−}]]],
[[X_{0}^{−}, X_{4}^{−}], [X_{3}^{−}, X_{4}^{−}]], [[X_{2}^{−}, X_{3}^{−}], [X_{4}^{−}, [X_{0}^{−}, X_{3}^{−}]]]],

\begin{matrix}[[[X_{1}^{−}, X_{2}^{−}], [[X_{0}^{−}, X_{4}^{−}], [X_{3}^{−}, X_{4}^{−}]]], [[X_{4}^{−}, [X_{0}^{−}, X_{3}^{−}]], [X_{4}^{−}, [X_{2}^{−}, X_{3}^{−}]]]]+ \\ \frac{1}{2} [[[X_{1}^{−}, X_{2}^{−}], [[X_{0}^{−}, X_{4}^{−}], [X_{3}^{−}, X_{4}^{−}]]], [[X_{3}^{−}, [X_{0}^{−}, X_{4}^{−}]], [X_{4}^{−}, [X_{2}^{−}, X_{3}^{−}]]]] \end{matrix}

\underline{For n = 5, 𝔑_{+}}: [X_{0}^{+}, X_{0}^{+}], [X_{0}^{+}, X_{2}^{+}], [X_{0}^{+}, X_{3}^{+}], [X_{0}^{+}, X_{4}^{+}], [X_{0}^{+}, Y],

(adX_{1}^{+})^{2} X_{0}^{+}, [Y, [X_{0}^{+}, X_{1}^{+}]], [[X_{0}^{+}, X_{1}^{+}], [X_{0}^{+}, X_{5}^{+}]], [[X_{0}^{+}, X_{5}^{+}], [X_{4}^{+}, X_{5}^{+}]], [[X_{3}^{+}, Y], [X_{2}^{+}, [X_{0}^{+}, X_{1}^{+}]]]+[X_{0}^{+}, X_{5}^{+}], [[X_{3}^{+}, Y], [X_{4}^{+}, [X_{0}^{+}, X_{5}^{+}]]] = 0, [[X_{5}^{+}, [X_{0}^{+}, X_{1}^{+}]], [Y, [X_{3}^{+}, X_{4}^{+}]]], [[Y, [X_{3}^{+}, X_{4}^{+}]], [[X_{0}^{+}, X_{5}^{+}], [X_{1}^{+}, X_{2}^{+}]]] + [[Y, [X_{1}^{+}, X_{2}^{+}]], [[X_{0}^{+}, X_{5}^{+}], [X_{3}^{+}, X_{4}^{+}]]]

\begin{matrix}[[[X_{3}^{+}, Y], [X_{3}^{+}, [X_{1}^{+}, X_{2}^{+}]]], [[X_{4}^{+}, X_{5}^{+}], [X_{2}^{+}, [X_{0}^{+}, X_{1}^{+}]]]]− \\ [[[X_{2}^{+}, X_{3}^{+}], [X_{5}^{+}, [X_{0}^{+}, X_{1}^{+}]]], [[X_{3}^{+}, X_{4}^{+}], [Y, [X_{1}^{+}, X_{2}^{+}]]]], \end{matrix}

[[[X_{4}^{+}, X_{5}^{+}], [[X_{1}^{+}, Y], [X_{2}^{+}, X_{3}^{+}]]], [[X_{3}^{+}, [X_{1}^{+}, X_{2}^{+}]], [Y, [X_{3}^{+}, X_{4}^{+}]]]] =

\frac{1}{2} [[[X_{2}^{+}, X_{3}^{+}], [X_{4}^{+}, X_{5}^{+}]], [[X_{4}^{+}, X_{5}^{+}], [X_{3}^{+}, [X_{1}^{+}, Y]]]]+

\frac{1}{2} [[[X_{3}^{+}, Y], [X_{4}^{+}, X_{5}^{+}]], [[X_{4}^{+}, X_{5}^{+}], [X_{3}^{+}, [X_{1}^{+}, X_{2}^{+}]]]]

\underline{For n = 5, 𝔑_{−}}: [X_{0}^{−}, X_{1}^{−}], [X_{0}^{−}, X_{2}^{−}], [X_{0}^{−}, X_{3}^{−}], (adX_{0}^{−})^{2} X_{4}^{−}, (adX_{4}^{−})^{2} X_{0}^{−}, [[X_{0}^{−}, X_{4}^{−}], [X_{0}^{−}, X_{5}^{−}]], [[X_{4}^{−}, [X_{0}^{−}, X_{5}^{−}]], [X_{5}^{−}, [X_{0}^{−}, X_{4}^{−}]]], [[X_{5}^{−}, [X_{0}^{−}, X_{4}^{−}]], [[X_{0}^{−}, X_{5}^{−}], [X_{3}^{−}, X_{4}^{−}]]], [[[X_{0}^{−}, X_{5}^{−}], [X_{3}^{−}, X_{4}^{−}]], [[X_{0}^{−}, X_{5}^{−}], [X_{4}^{−}, X_{5}^{−}]]], [[[X_{2}^{−}, X_{3}^{−}], [X_{5}^{−}, [X_{0}^{−}, X_{4}^{−}]]], [[X_{4}^{−}, X_{5}^{−}], [X_{3}^{−}, [X_{0}^{−}, X_{4}^{−}]]]], [[[X_{4}^{−}, X_{5}^{−}], [X_{3}^{−}, [X_{0}^{−}, X_{4}^{−}]]], [[X_{2}^{−}, X_{3}^{−}], [[X_{0}^{−}, X_{5}^{−}], [X_{4}^{−}, X_{5}^{−}]]]]

2.2.4. The periplectic series

Recall that the compatible (with parity) ℤ-gradings of 𝔰𝔭𝔢(n) are of the form 𝔰𝔭𝔢(n) = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} and there are two such cases both with 𝔤_{0} = 𝔰𝔩(n): (here id is the standard 𝔰𝔩(n)-module):

a) 𝔤_{1} = S^{2}(id), 𝔤_{−1} = E^{2}(id^{∗});

b) 𝔤_{1} = E^{2}(id), 𝔤_{−1} = S^{2}(id^{∗}) .

Let 𝔫^{±} be the maximal nilpotent subalgebras of 𝔤_{0}. Set

𝔪^{+} = 𝔫^{+} ⊕ 𝔤_{1}; 𝔪^{−} = 𝔫^{−} ⊕ 𝔤_{−1}

Denote by X^{+} (resp. X^{−}) a vector of lowest (highest) weight in the 𝔤_{0}-module 𝔤_{1} (resp. 𝔤_{−1}). The first term ⊕_{p+q=2} E_{1}^{p,q} of the spectral sequence converging to H_{2}(𝔪^{±})

consists of

E_{1}^{2,0} = H_{2}(𝔫^{±}), E_{1}^{1,1} = H_{1}(𝔫^{±}; 𝔤_{±1}), E_{1}^{0,2} = H_{0}(𝔫^{±}; E^{2}(𝔤_{±1})).

Since we already know H_{2}(𝔫^{±}), we are only interested in the other two summands. In case (a), (resp. (b)), H_{1}(𝔫^{±}; 𝔤_{±1}) is the same as for 𝔪^{+} of 𝔰𝔭(2n) and for 𝔪^{−} of 𝔬(2n) (resp. for 𝔪^{−} of 𝔬(2n) and 𝔰𝔭(2n) of 𝔪^{+}), we explicitly have:

(adX_{n}^{+})^{3}(X^{+}) = 0, (adX_{n}^{−})^{2}(X^{−}) = 0  (a)
(adX_{n}^{+})^{2}(X^{+}) = 0, (adX_{n}^{−})^{3}(X^{−}) = 0  (b)

with [X_{i}^{+}, X^{+}] = [X_{i}^{−}, X^{−}] = 0 for i < n in both cases (a) and (b).

Let φ_{i} be the ith fundamental weight of 𝔤_{0}, R(χ) the (space of the) irreducible representation with highest weight χ. Now, for the 𝔰𝔩(n)-modules 𝔤_{1} = R(2φ_{1}) in case (a) and 𝔤_{1} = R(φ_{2}) in case (b), we have:

\begin{matrix}S^{2}(R(2φ_{1})) = R(4φ_{1}) ⊕ R(2φ_{2}) \\ S^{2}(R(φ_{2})) = \begin{cases}R(2φ_{2}) ⊕ R(φ_{4}) & if n > 3 \\ R(2φ_{2}) & if n = 3. \end{cases} \end{matrix}

Therefore, we have the relations

[X^{±}, X^{±}] = 0  for both cases a) and b)

and the relations

\begin{matrix}(a) & [X^{+}, [X_{n}^{+}, [X_{n}^{+}, X^{+}]]] = 0, & [X^{−}, [X_{n}^{−}, [X_{n−1}^{−}, X^{−}]]] = 0; \\ (b) & [X^{+}, [X_{n}^{+}, [X_{n}^{+}, X^{+}]]] = 0, & [X^{−}, [X_{n}^{−}, [X_{n−1}^{−}, X^{−}]]] = 0. \end{matrix}  (P eR_{±})

of which the first in case (a) and the second in case (b) are only defined if n > 3.

2.3.1 Exceptional loop algebras: 𝔡(ε)^{(3)}

Let ε be a primitive cubic root of 1 and 𝔡(ε) the deform of the Lie superalgebra 𝔬𝔰𝔭(4|2) corresponding to the value of parameter equal to ε, i.e., 𝔡(ε) = 𝔤(A) for any of the following Cartan matrices (cf. [GL2]):

(\begin{matrix}0 & −1 & ε^{2} \\ 1 & 0 & ε \\ ε^{2} & −ε & 0 \end{matrix}) , or (\begin{matrix}2 & −1 & 0 \\ ε & 0 & ε^{2} \\ 0 & −1 & 2 \end{matrix}) , or (\begin{matrix}2 & −1 & 0 \\ 1 & 0 & ε \\ 0 & −1 & 2 \end{matrix}) , or (\begin{matrix}2 & −1 & 0 \\ 1 & 0 & ε^{2} \\ 0 & −1 & 2 \end{matrix}) .

The algebra 𝔡(ε) has an outer automorphism of order 3; select the generators of the maximal nilpotent subalgebras of 𝔡(ε)^{(3)} as follows. Let X_{1}^{±}, X_{2}^{±}, X_{3}^{±} be the Chevalley generators of 𝔡(ε). Set

\begin{matrix}Y_{1}^{+} = εX_{1}^{+} + ε^{2}X_{2}^{+} + X_{3}^{+}, & Y_{2}^{+} = [X_{3}^{−}, [X_{1}^{−}, X_{2}^{−}]]; \\ Y_{1}^{−} = εX_{1}^{−} + ε^{2}X_{2}^{−} + X_{3}^{−}, & Y_{2}^{+} = [X_{1}^{+}, X_{2}^{+}] − [X_{1}^{+}, X_{3}^{+}] + [X_{2}^{+}, X_{3}^{+}]. \end{matrix}

The relations between these generators are:

[Y_{2}^{+}, Y_{2}^{+}] = 0,
[[Y_{1}^{+}, Y_{2}^{+}], [Y_{2}^{+}, [Y_{1}^{+}, Y_{1}^{+}]]] = 0,
(ad[Y_{1}^{+}, Y_{1}^{+}])^{3}([Y_{1}^{+}, Y_{2}^{+}]) = 0,

[[Y_{1}^{+}, Y_{1}^{+}], [[Y_{2}^{+}, [Y_{1}^{+}, Y_{1}^{+}]], [[Y_{2}^{+}, [Y_{1}^{+}, Y_{1}^{+}]], [[Y_{1}^{+}, Y_{1}^{+}], [Y_{2}^{+}, [Y_{1}^{+}, Y_{1}^{+}]]]]]] = 64Y_{1}^{+},

[[Y_{1}^{+}, Y_{2}^{+}], [[Y_{2}^{+}, [Y_{1}^{+}, Y_{1}^{+}]], [[Y_{2}^{+}, [Y_{1}^{+}, Y_{1}^{+}]], [[Y_{1}^{+}, Y_{1}^{+}], [Y_{2}^{+}, [Y_{1}^{+}, Y_{1}^{+}]]]]]] = −96Y_{2}^{+}.

(adY_{2}^{−})^{3}Y_{1}^{−} = 0,
[[Y_{1}^{−}, Y_{2}^{−}], [Y_{1}^{−}, Y_{2}^{−}]] − 2[Y_{2}^{−}, [Y_{2}^{−}, [Y_{1}^{−}, [Y_{1}^{−}]] = 0,
(ad[Y_{1}^{−}, Y_{1}^{−}])^{2}[Y_{1}^{−}, Y_{2}^{−}] = 0,

[[Y_{1}^{−}, Y_{1}^{−}], [[Y_{2}^{−}, [Y_{1}^{−}, Y_{1}^{−}]], [[Y_{1}^{−}, Y_{2}^{−}], [Y_{2}^{−}, [Y_{1}^{−}, Y_{1}^{−}]]]]] = −64Y_{1}^{−},

[[Y_{1}^{−}, Y_{2}^{−}], [[Y_{2}^{−}, [Y_{1}^{−}, Y_{1}^{−}]], [[Y_{1}^{−}, Y_{2}^{−}], [Y_{2}^{−}, [Y_{1}^{−}, Y_{1}^{−}]]]]] = −64Y_{2}^{−}.

2.3.2. Stringy superalgebras

For the exceptional stringy superalgebra 𝔨𝔞𝔰^{L} introduced in [GLS1] (and in [CK]) the relations are computed in [GLS1] and, in another form, in [CK]. Observe that the relations of 𝔨𝔞𝔰 are not simply the relations for 𝔨𝔞𝔰^{L} that do not involve the extra generators.

Among the stringy superalgebras 𝔨^{L}(1|6) is one of the most interesting: it possesses a nondegenerate invariant symmetric bilinear form and, therefore, can be q-quantized, cf. [LSp].

The basis of 𝔤_{+} for the standard ℤ-grading of 𝔨^{L}(1|6):

\begin{matrix}X_{1}^{+} = ξ_{1}η_{2}, & X_{2}^{+} = ξ_{2}η_{3}, & X_{3}^{+} = ξ_{2}ξ_{3}, & X_{0}^{+} = tη_{1}, \\ X̃_{1}^{+} = \frac{1}{t}ξ_{1}ξ_{3}η_{2}η_{3}, & X̃_{2}^{+} = \frac{1}{t}ξ_{1}ξ_{2}η_{1}η_{3}, & X̃_{3}^{+} = \frac{1}{t}ξ_{1}ξ_{2}ξ_{3}η_{1}, & X̃_{0}^{+} = η_{1}η_{2}η_{3} \end{matrix}

The generators X_{i}^{±} for i ≠ 0, clearly, generate 𝔬(6) while all of them generate 𝔬𝔰𝔭(6|2). One expects the same relations between them, but the other generators interfere and the final result is as follows (we skip the superscript):

[X_{2}, X_{3}] = 0, [X_{2}, X_{0}] = 0, [X_{3}, X_{0}] = 0, [X_{0}, X_{0}] = 0;

(adX_{1})^{2}X_{2} = 0; (adX_{1})^{2}X_{3} = 0, (adX_{1})^{2}X_{0} = 0, (adX_{2})^{2}X_{1} = 0, (adX_{3})^{2}X_{1} = 0;

[X_{1}, X̃_{0}] = 0, [X_{1}, X̃_{1}] = 0, [X_{2}, X̃_{0}] = 0, [X_{2}, X̃_{2}] = 0,
[X_{2}, X̃_{3}] = 0, [X_{3}, X̃_{2}] = 0, [X_{3}, X̃_{3}] = 0, [X_{0}, X̃_{0}] = 0
[X̃_{0}, X̃_{0}] = 0, [X̃_{0}, X̃_{1}] = 0, [X̃_{0}, X̃_{2}] = 0,
[X̃_{1}, X̃_{2}] = 0, [X̃_{1}, X̃_{3}] = 0, [X̃_{2}, X̃_{3}] = 0;
[X_{2}, X̃_{1}] + [X_{1}, X̃_{2}] = 0, [X_{3}, X̃_{1}] + [X_{1}, X̃_{3}] = 0;
(adX_{3})^{2}X̃_{0} − 2[X_{0}, X̃_{3}] = 0;

[X̃_{0}, [X_{0}, X̃_{3}]] = 0, [X̃_{1}, [X_{1}, X_{2}]] = 0, [X̃_{1}, [X_{1}, X_{3}]] = 0, [X̃_{1}, [X_{1}, X_{0}]] = 0,

[X̃_{2}, [X_{1}, X_{2}]] = 0, [X̃_{3}, [X_{1}, X_{3}]] = 0,
[X̃_{3}, [X_{3}, X̃_{0}]] = 0, [X̃_{3}, [X_{0}, X̃_{2}]] = 0;
(adX̃_{1})^{2}X_{0} = 0, (adX̃_{2})^{2}X_{0} = 0, (adX̃_{3})^{2}X_{0} = 0, (adX̃_{3})^{2}X̃_{0} = 0;

[[X_{1}, X_{2}], [X_{3}, X̃_{0}]] + 2[X̃_{2}, [X_{0}, X̃_{1}]] + [X_{0}, [X_{1}, X̃_{2}]] + [X̃_{2}, [X_{1}, X_{0}]] = 0;

[[X_{1}, X_{2}], [X_{0}, X̃_{2}]] = 0, [[X_{1}, X_{3}], [X_{0}, X̃_{3}]] = 0,
[[X_{1}, X̃_{2}], [X_{0}, X̃_{2}]] = 0, [[X_{1}, X̃_{3}], [X_{0}, X̃_{3}]] = 0,
[[X_{1}, X̃_{2}], [X_{3}, X̃_{0}]] − [X̃_{2}, [X_{0}, X̃_{1}]] = 0, [[X_{0}, X̃_{3}], [X̃_{2}, [X_{0}, X̃_{1}]] = 0,
[[X_{0}, X̃_{3}], [X̃_{2}, [X_{1}, X_{0}]] + [[X_{0}, X̃_{3}], [X̃_{0}, [X_{1}, X̃_{2}]] = 0,
2[[X_{1}, X̃_{2}], [X_{0}, X̃_{3}]] + 2[X̃_{3}, [X_{0}, [X_{1}, X_{2}]]]+
[[X_{1}, X_{3}], [X_{0}, X̃_{2}]] − [[X_{1}, X_{2}], [X_{0}, X̃_{3}]] = 0;
2[[X_{1}, X̃_{3}], [X_{0}, X̃_{2}]] − 2[X̃_{3}, [X_{0}, [X_{1}, X_{2}]]]+
[[X_{1}, X_{2}], [X_{0}, X̃_{3}]] − [[X_{1}, X_{3}], [X_{0}, X̃_{2}]] = 0.

§4. Presentation of vectorial Lie superalgebras: an overview of open problems

4.1. Vectorial Lie superalgebras with polynomial coefficients and loops with values in them Lemma . ([ShV]) There are embeddings of Lie superalgebras such that their negative parts as ℤ-graded Lie superalgebras coincide:

\begin{matrix}𝔰𝔩(n + 1|m) ⊂ 𝔳𝔢𝔠𝔱(n|m), 𝔬𝔰𝔭(m|2n + 2) ⊂ 𝔨(2n + 1|m), \\ 𝔭𝔢(n + 1) ⊂ 𝔪(n), 𝔰𝔭𝔢(n + 1) ⊂ 𝔰𝔪(n). \end{matrix}

This is already something: it remains to establish presentation of 𝔑^{+} only. The rest of the known information is gathered in the following statement, cf. [HP]. The cases when the Lie superalgebra is not simple are marked by a + sign; we disregard them in this paper.

Theorem . The following table lists the restrictions for the relations of the subalgebra 𝔑^{+} of a simple vectorial Lie superalgebra to be “pure” (blank space) or “dirty” (marked by a dot).

𝔳𝔢𝔠𝔱(n|m)  𝔰𝔳𝔢𝔠𝔱(n|m)
\begin{aligned}m \\ & + \\ 3 & + \\ 2 & + \\ 1 & + & • \\ 0 & + & • & • \\ & 0 & 1 & 2 & 3 & … & n \end{aligned}  \begin{aligned}m \\ & + & + \\ & + & + \\ 3 & + & + \\ 2 & + & + \\ 1 & + & + \\ 0 & + & + & • \\ & 0 & 1 & 2 & 3 & … & n \end{aligned}

𝔥(2n|m)  𝔨(2n + 1|m)
\begin{aligned}m \\ & + \\ & + \\ 3 & + \\ 2 & + \\ 1 & + & • \\ 0 & + & • & • \\ & 0 & 1 & 2 & 3 & … & n \end{aligned}  \begin{aligned}m \\ 3 \\ 2 & • \\ 1 & • \\ 0 & • & • \\ & 0 & 1 & 2 & 3 & … & n \end{aligned}

𝔰𝔳𝔢𝔠𝔱^{∘}(n)  𝔰𝔥(n)
\begin{matrix}+ & + & • \\ 0 & 1 & 2 & 3 & 4 & … & n \end{matrix}  \begin{matrix}+ & + & + & + \\ 0 & 1 & 2 & 3 & 4 & … & n \end{matrix}

𝔪(n)  𝔟_{λ}(n), 𝔩𝔢(n), 𝔰𝔩𝔢(n)
\begin{matrix}• \\ 0 & 1 & 2 & 3 & 4 & … & n \end{matrix}  \begin{matrix}+ & • \\ 0 & 1 & 2 & 3 & 4 & … & n \end{matrix}

𝔳𝔢𝔠𝔱(0|n)^{(1)}, 𝔰𝔥(2n)^{(2)}  𝔰𝔳𝔢𝔠𝔱(0|n)^{(1)}
\begin{matrix}+ \\ 0 & 1 & 2 & 3 & 4 & … & n \end{matrix}  \begin{matrix}+ & + \\ 0 & 1 & 2 & 3 & 4 & … & n \end{matrix}

𝔰𝔥(n)^{(1)}
+  +  +
0  1  2  3  4  …  n

4.2. Twisted loops

There remains to compute the relations in the following twisted loop superalgebras which have no Cartan matrix: see [LSS]: 𝔭𝔰𝔩(n|n)_{Π}^{(2)}; 𝔭𝔰𝔩(n|n)_{Π∘(−st)}^{(2)}; 𝔭𝔰𝔮(n)_{Π}^{(4)}.

4.3. Stringy and vectorial superalgebras: other bases

We have only computed the relations in several cases. There remains to compute the relations in the cases marked by a thick dot (and a cross) in Table 4.1 and for the exceptional algebras [Sh5] (for the consistent gradings thereof this is done in [GLS2]; for the other W-gradings it is desirable, at least, for completeness). It is desirable to compute them for all simple stringy superalgebras or, at least, for the distinguished algebras that admit nontrivial central extensions most often used in applications, cf. [GLS1], and for other bases of vectorial algebras and superalgebras.

References

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P. Grozman

Department of Mathematics,

University of Stockholm,

Roslagsv. 101, Kräftriket hus 6, S-106 91, Stockholm,

Sweden

D. Leites mleites@matematik.su.se

Department of Mathematics,

University of Stockholm,

Roslagsv. 101, Kräftriket hus 6, S-106 91, Stockholm,

Sweden

E. Poletaeva elena@maths.lth.se

Department of Mathematics,

Lund University,

Sweden