Constructing simple Lie superalgebras from associative graded algebras
The purpose of the paper under review is to generalize to the supercase the classical result of Herstein that for any simple associative algebra \(A\) with centre \(Z\), the Lie algebra \([A,A]/([A,A]\cap Z)\) is also simple, unless \([A:Z]=4\) and \(A\) has characteristic 2. Section 1 gives the necessary background on \((\varepsilon,G)\)-Lie color superalgebras, where \(G\) is an abelian group, \(k\) is the base field and \(\varepsilon:G\times G\to k^{\ast}\) is an anti-symmetric bicharacter. Section 2 studies \(\varepsilon\)-Lie ideals of an associative \(G\)-graded algebra \(A\). Some of the results have been shown in more general Hopf algebra setting in [\textit{Yu. Bakhturin, D. Fischman} and \textit{S. Montgomery}, Isr. J. Math. 96, 27-48 (1996; Zbl 0874.16024)], but now the author establishes them with considerably simpler arguments involving essentially the \(G\)-grading. In Section 3 the main theorem on Lie simplicity is a graded analogue of the theorem of Herstein, provided that certain odd elements satisfy an additional hypothesis. A number of examples is given. The main applications are in the construction of simple Lie superalgebras arising from the Weyl algebras \({\mathbb A}_n\). In Section 4 the author shows that \([{\mathbb A}_n,{\mathbb A}_n]/([{\mathbb A}_n,{\mathbb A}_n]\cap k)\), \(n\geq 1\), is a simple Lie superalgebra. Unlike the ungraded case, she establishes that \({\mathbb A}_1=k\oplus [{\mathbb A}_1,{\mathbb A}_1]\). Section 5 gives several examples when the grading group \(G\) is different from \({\mathbb Z}_2\). If \(G=({\mathbb Z}_2)^n\), then the \((\varepsilon,G)\)-Lie color algebra \([{\mathbb A}_n,{\mathbb A}_n]\) is simple and again \({\mathbb A}_n= k\oplus [{\mathbb A}_n,{\mathbb A}_n]\). Finally, in Section 6 the author deals with the 2-cocycle twist \(L^{\sigma}\) of an \((\varepsilon,G)\)-Lie color algebra \(G\) discovered in [\textit{M. Scheunert}, J. Math. Phys. 20, 712-720 (1979; Zbl 0423.17003)] and shows that the superstructure of \({\mathbb A}_n\) obtained by the Scheunert construction is quite different from the structure using the supercommutator.
scientific article; zbMATH DE number 1088616
scientific article