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Manin triples and non-degenerate anti-symmetric bilinear forms on Lie superalgebras in characteristic 2. (English) Zbl 1523.17038

In the paper under review the authors define and investigate the concepts of Manin triples and Lie bisuperalgebra structures on a given finite-dimensional Lie superalgebra \(\mathfrak{g}=\mathfrak{g}_{\overline{0}} \oplus\mathfrak{g}_{\overline{1}}\) over a field of characteristic 2 in relation to the existence of certain non-degenerate invariant anti-symmetric bilinear forms on \(\mathfrak{g}\). Note that already the definition of a Lie superalgebra in characteristic 2 is more complicated than in other characteristics. Namely, in addition to a Lie bracket on the even part \(\mathfrak{g}_{\overline{0}}\) and a symmetric left and right action of \(\mathfrak{g}_{\overline{0}}\) on the odd part \(\mathfrak{g}_{\overline{1}}\), the superbracket \([-,-]_{\mathfrak{g}_{ \overline{1}}\times\mathfrak{g}_{\overline{1}}}\) on the odd part is a bilinear function arising from a quadratic function (the so-called squaring) \(s:\mathfrak{g}_{\overline{1}}\to\mathfrak{g}_{\overline{0}}\) from the odd part to the even part via \([x,y]=s(x+y)-s(x)-s(y)\). Similarly, the definition of a Lie bisuperalgebra is more involved in characteristic 2. Finally, in characteristic 2 the definition of anti-symmetric bilinear forms is more subtle. By considering the two square matrices in the diagonal of the Gram matrix of the bilinear form, one can define two types of such bilinear forms, namely, \(\overline{0}\)-anti-symmetric and \(\overline{1}\)-anti-symmetric bilinear forms.
Recall that Lie bialgebras give rise to Manin triples and vice versa. Namely, V. G. Drinfel’d [J. Sov. Math. 41, No. 2, 898–915 (1988; Zbl 0641.16006)] proved that a complex Lie algebra \(\mathfrak{g}\) admits a Lie bialgebra structure exactly when \(\mathfrak{g}\) and its linear dual \(\mathfrak{g}^*\) form a Manin triple \((\mathfrak{h}, \mathfrak{g},\mathfrak{g}^*)\) meaning that the (outer) direct sum \(\mathfrak{h}:=\mathfrak{g}\oplus\mathfrak{g}^*\) of vector spaces is a Lie algebra that admits a non-degenerate invariant symmetric bilinear form with respect to which \(\mathfrak{g}\) and \(\mathfrak{g}^*\) are isotropic. For fields of characteristic \(\ne 2\) Drinfeld’s result has been generalized to Lie superalgebras by G. I. Ol’shanskii [Lett. Math. Phys. 24, No. 2, 93–102 (1992; Zbl 0766.17017)]. If in characteristic 2 the bilinear form on \(\mathfrak{h}\) is required to be a non-degenerate invariant \(\overline{0}\)-anti-symmetric even bilinear form and the definition of a Lie bisuperalgebra is amended accordingly, then the authors can show that again the notions of a Lie bisuperalgebra and a Manin triple are equivalent.
Because of what has been said above, the paper is quite technical. The main goal of the authors is to construct Lie bisuperalgebras via admissible classical \(r\)-matrices, Manin triples, or matched pairs by assuming the existence of certain \(\overline{0}\)- (and \(\overline{1}\)-)anti-symmetric invariant bilinear forms on the underlying Lie superalgebra. The latter enable the authors to obtain admissible \(r\)-matrices and Manin triples inductively by the procedure of double extension. The paper closes with a list of several interesting open problems.

MSC:

17B50 Modular Lie (super)algebras
17B62 Lie bialgebras; Lie coalgebras
17B99 Lie algebras and Lie superalgebras
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References:

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