Central extensions of matrix Lie superalgebras over \(\mathbb{Z}/2\mathbb{Z}\)-graded algebras
The aim of the paper is to provide general results about central extensions of the Lie superalgebra \(\mathfrak{sl}_n(A)\) when \(A\) is a \(\mathbb{Z}/2\mathbb{Z}\)-graded associative superalgebra over a unital commutative ring \(K\). The main result establishes that for \(n \geq 3\), the kernel of the central extension \((\mathfrak{st}_n(A), \varphi)\) of the Lie superalgebra \(\mathfrak{sl}_n(A)\) is isomorphic to \(HC_1(A)\), the first \(\mathbb{Z}/2\mathbb{Z}\)-graded cyclic homology group of the superalgebra \(A\). Here \((\mathfrak{st}_n(A), n \geq 3\), is the Steinberg Lie superalgebra. Finally, for \(n = 3, 4\), is proved that \((\widehat {\mathfrak{st}}_n(A), \pi)\) is the universal central extension of \(\mathfrak{st}_n(A)\), where \(\widehat{\mathfrak{st}}_4(A) = \mathfrak{st}_4(A) \oplus W\) \(\left( W \right.\) is the direct sum of six distinct copies of \(A_2 = A/{\mathcal I}_2, {\mathcal I}_2\) is the two-sided \(\mathbb{Z}/2\mathbb{Z}\)-graded ideal of \(A\) generated by the elements \(2a\) and \(\displaystyle \left. ab-(-1)^{\mid a \mid \mid b \mid} ba , i.e. {\mathcal I}_2 = 2A + A[A,A]\right)\) and \(\displaystyle\widehat{\mathfrak{st}}_3(A) = \mathfrak{st}_3(A) \oplus U\) (\(U\) is the direct sum os six distinct copies of \(A_3=A/{\mathcal I}_3, {\mathcal I}_3 = 3A + A[A,A]\)).
scientific article; zbMATH DE number 6162432
scientific article