Special odd Lie superalgebras in prime characteristic
type: Item
instance of: scholarly article
title: Special odd Lie superalgebras in prime characteristic
zbMATH Open document ID: 1270.17010
author: Wende Liu
author: Jixia Yuan
published in: Science China. Mathematics
publication date: 2012-05-31T00:00:00Z
review text: In the paper simple special odd Hamiltonian and contact Lie superalgebras of characteristic \(p>3\) are considered. Note that simple special odd Hamiltonian Lie superalgebra \(\mathfrak{H}(n,\underline{t})\) corresponds to the supercase \((n,n)\). It is the second term of the derived series of the intersection of odd Hamiltonian Lie superalgebra and the special Lie superalgebra, \(\underline{t}\) is an \(n\)-tuple of positive integers defining a commutative divided powers algebra \(\mathcal{O}(n,\underline{t}).\) The simple special odd contact Lie superalgebra \(\mathfrak{K}(n,\lambda ,\underline{t})\) corresponds to the supercase \((n,n+1)\). It is the second term of the derived series of the subalgebra \(\mathfrak{K}''(n,\lambda ,\underline{t})\) consisting of elements \(a\) of the odd contact superalgebra such that \(\text{div}_{\lambda}(a)=0\). Here divergence \(\text{div}_{\lambda}\) depends on a parameter \(\lambda \in F\). The main point of the paper is to find all simple special odd Lie superalgebras for \(p>3\), \(n\geq 3\) which possess a nontrivial invariant bilinear form. It is proved that any simple special odd Hamiltonian Lie superalgebra admits such a form. A special odd contact Lie superalgebra \(\mathfrak{K}(n,\lambda ,\underline{t})\) possesses a bilinear nonsingular invariant form iff \(n\neq 0\pmod p\), \(\lambda= -\frac{2}{n}\). The methods applied are analogous to those which were developed for modular Lie algebras of Cartan type. It is well known that in the case when a Lie (super) algebra \(L\) admits a bilinear nonsingular invariant form, the group \(H^2(L)\) is isomorphic to the space of outer skew-symmetric derivations of \(L\) (see p. 102 of [\textit{G. B. Seligman}, Modular Lie algebras. Berlin etc.: Springer (1967; Zbl 0189.03201)]). Using the information on derivations of simple special odd Lie superalgebras obtained earlier the authors give the description of \(H^2(L)\) for simple Lie superalgebras under consideration which admit a no…
reviewed by: Michael Kuznetsov
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B65
zbMATH DE Number: 6040083
zbMATH Keywords: Lie superalgebra
zbMATH Keywords: associative form
zbMATH Keywords: cohomology
MaRDI profile type: Publication
full work available at URL: s11425-011-4316-4
OpenAlex ID: W1995361558
cites work: Central extensions and \(H^ 1(L, L^*)\) of the graded Lie algebras of Cartan type
cites work: Q3758099
cites work: Dual Space Derivations and H2(L, F) of Modular Lie Algebras
cites work: The Associative Forms of the Graded Cartan Type Lie Algebras
cites work: The Cartan-Type Modular Lie SuperalgebraKO
cites work: Lie superalgebras
cites work: Classification of infinite-dimensional simple linearly compact Lie superalgebras
cites work: FINITE-DIMENSIONAL SPECIAL ODD HAMILTONIAN SUPERALGEBRAS IN PRIME CHARACTERISTIC
cites work: The derivation algebra of the Cartan-type Lie superalgebra \(HO\).
cites work: The associative forms of the graded Cartan type Lie superalgebras
cites work: Derivation Algebra Der(H) and Central Extensions of Lie Superalgebras
cites work: Second Cohomology of the Modular Lie Superalgebra of Cartan Type K
DOI: 10.1007/S11425-011-4316-4
description: scientific article; zbMATH DE number 6040083