Classification of Finite-Growth General Kac–Moody Superalgebras

Hoyt, Crystal; Serganova, Vera

doi:10.1080/00927870601115781

Cited by 26 publications

(31 citation statements)

References 8 publications

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“…His first list of inequivalent Cartan matrices (in other words, distinct Z-gradings) for finite dimensional Lie superalgebras g(A) in [29] had gaps; Serganova [47] and (by a different method and only for symmetrizable matrices) van de Leur [53] fixed the gaps and even classified Lie superalgebras of polynomial growth (for the proof in the non-symmetrizable case, announced 20 years earlier, see [26]). Kac also suggested analogs of Dynkin diagrams to graphically encode the Cartan matrices.…”

Section: Step 1: An Overview Of Known Resultsmentioning

confidence: 99%

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Bouarroudj

Grozman

Leites

2009

SIGMA

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Abstract. Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superalgebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described.

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Section: Step 1: An Overview Of Known Resultsmentioning

confidence: 99%

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Bouarroudj

Grozman

Leites

2009

SIGMA

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“…It would be interesting to find a Z-graded contragredient Lie superalgebra in correspondence with the Lie algebra E 6 of Table 6 and to look for applications to supergravity theories. We remark that it cannot be a finitedimensional or (twisted) affine Lie superalgebra, as it follows from the classification of contragredient Lie superalgebras of finite Gelfand-Kirillov growth, see [25,19]. Dynkin diagrams with shape E 6 appeared in the context of almost affine Kac-Moody Lie superalgebras in [8] but no extensive review of their properties is known to us.…”

Section: Comparison With the Lie Algebra Casementioning

confidence: 99%

Classification of maximal transitive prolongations of super-Poincaré algebras

Altomani

Santi

2014

Advances in Mathematics

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Let V be a complex vector space with a non-degenerate symmetric bilinear form and S an irreducible module over the Clifford algebra Cℓ(V ) determined by this form. A supertranslation algebra is a Z-graded Lie superalgebra m = m−2 ⊕m−1, where m−2 = V and m−1 = S⊕· · ·⊕S is the direct sum of an arbitrary number N ≥ 1 of copies of S, whose bracket. We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finitedimensional for dim V ≥ 3 and classify them in terms of super-Poincaré algebras and appropriate Z-gradings of simple Lie superalgebras.the Clifford algebra determined by this form, with its natural Z 2 -grading. We adopt the conventions used in [24], in particular, the product in Cℓ(V ) satisfies vu + uv = −2(v, u)1 for any v, u ∈ V and the natural inclusion of the orthogonal Lie algebra in the Clifford algebra is given by the mapwhere v ∧ u is the anti-symmetric endomorphism (v, ·)u − (u, ·)v. Note that some authors prefer to use different conventions, cf. Deligne's lectures [12].

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“…A proof of the following lemma can be found in [3]. 1. if a ii = 0 and p(i) = 0, then a ij = 0 for all j ∈ I;…”

Section: Contragredient Lie Superalgebrasmentioning

confidence: 99%

“…The results of this paper are a crucial part of the proof for the classification of contragredient Lie superalgebras with finite growth, and in particular, for the classification of finite-growth Kac-Moody superalgebras [2,3]. Previously, such a classification was known only for contragredient Lie superalgebras with either symmetrizable Cartan matrices [12,13], or Cartan matrices with no zeros on the main diagonal, i.e.…”

Section: Introductionmentioning

confidence: 99%