Central extensions of Lie superalgebras
A finite dimensional Lie superalgebra \(\mathfrak g=\mathfrak g_0\oplus \mathfrak g_1\) is basic classical ifNEWLINENEWLINE(1) \(\mathfrak g\) is a simple Lie superalgebra;NEWLINENEWLINE(2) \(\mathfrak g_0\) is a reductive Lie algebra;NEWLINENEWLINE (3) there exists a non-degenerate bilinear form \(b(x,y)\) on \(\mathfrak g\) such thatNEWLINENEWLINEa) \(\mathfrak g_0,\,\mathfrak g_1\) are orthogonal with respect to \(b(x,y)\);NEWLINENEWLINEb) \(b(x,y)=(-1)^{| x| \,| y| }b(y,x)\) for all homogeneous elements \(x,y\in \mathfrak g\);NEWLINENEWLINEc) \(b([a,b],c)=b(a,[b,c])\) for all \(a,b,c\in \mathfrak g\).NEWLINENEWLINE Suppose that \(\mathfrak g\) is a complex basic classical Lie superalgebra such that if \(\mathfrak g\) is of a type \(D(2,1;a)\) then \(a\in \mathbb Z\setminus \{0,\,-1\}\). It is shown that there exists a \(\mathbb Z\)-form \(\mathfrak g_{\mathbb Z}\) of \(\mathfrak g\).NEWLINENEWLINE For any commutative ring \(k\) put \(g_k=k\otimes_{\mathbb Z}\mathfrak g_{\mathbb Z}\). Suppose that \(A\) is a commutative \(k\)-algebra which is a free \(k\)-module andNEWLINENEWLINE(i) 2 is invertible in \(k\) if \(\mathfrak g\) is of one of types \(A(m,n),\, C(n),\, D(m,n)\);NEWLINENEWLINE(ii) 6 is invertible in \(k\) if \(\mathfrak g\) is of types \(B(m,n),\, F(4),\, G(3)\);NEWLINENEWLINE(iii) \(2a(a+1)\) is invertible in \(k\) if \(\mathfrak g\) is of type \(D(2,1;a)\).NEWLINENEWLINEThen the universal central extension \(\mathfrak g_k(A)\) of \(\mathfrak g_k\otimes A\) has the form NEWLINE\[NEWLINE \mathfrak g_k(A)= \begin{cases} \mathfrak g_k \otimes A\oplus \Omega^1_{A/k}/dA , \quad & \mathfrak g \text{ is not of type } A(n,n), \\ \mathfrak d_k \otimes A\oplus \Omega^1_{A/k}/dA , \quad & \mathfrak g \text{ is of type } A(1,1),\\ \mathfrak(n+1,n+1)_k\oplus \Omega^1_{A/ k}/dA , \quad & \mathfrak g \text{ is of type } A(n,n),\, n>1.\end{cases} NEWLINE\]NEWLINE Here \(\Omega^1_{A/k}/dA\) is the \(A\)-module of Kähler differentials and \(\mathfrak d\) the Lie superalgebra of type \(D(2,1;-1)\).NEWLINENEWLINENote that similar problems were considered independently in a series of papers by \textit{A.V. Mikhalev} and \textit{I. A. Pinchuk}, which is not mentioned in the present paper. In the paper [``Universal central extensions of the matrix Lie superalgebras \({\mathfrak{sl}}(m,n,A)\)'', Contemp. Math. 264, 111--125 (2000;Zbl 1135.17301)], universal central extensions of \(\mathfrak g=\mathfrak(m,n,A)\) are classified for any associative \(k\)-algebra \(A\) where \(k\) is a commutative ring. In [Universal algebra and applications, Proc. of an international seminar, Volgograd, 201--221 (2000), and Proc. Petrovsky seminar, Moscow University, 261--282 (2002)], the authors considered the case of Lie superalgebras \(\mathfrak g\square A\) where \(\mathfrak g\) is a simple Lie superalgebra with a non-degenerate Killing form and \(A\) a commutative \(k\)-algebra.
scientific article; zbMATH DE number 1610944