Special odd Lie superalgebras in prime characteristic
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 nonsingular invariant form.
scientific article; zbMATH DE number 6040083
scientific article