Central extension of quadratic Lie algebras and its relation to dihedral homology (Q1911782)

Let \(k\) be a commutative ring and \(A\) a \(k\)-algebra and suppose that \(A\) has a \(k\)-algebra involution \(a\mapsto \overline {a}\). Fix an element \(\lambda\) in the center of \(A\) which satisfies \(\lambda \overline {\lambda} =1\). Let \(\max= \{a\in A\mid a=- \lambda \overline {a}\}\) and let \(\min= \{a- \lambda \overline {a}\mid a\in A\}\). A form parameter \(\Lambda \subset A\) is a \(k\)-submodule such that \(\min \subset \Lambda \subset \max\); and \(a \Lambda \overline {a} \subset \Lambda\) for all \(a\in A\). In particular, max and min are form parameters. Given the triple \((A, \lambda, \Lambda)\), the authors define the general quadratic Lie algebra \(gq_{2n} (\Lambda)\) to be the \(2n \times 2n\) block \(n\times n\) matrices \(\left( \begin{smallmatrix} M &Q\\ R &-M \end{smallmatrix} \right)\), where \(Q\in M_n (\Lambda)\) and \(R\in M_n (\overline {\Lambda})\). Then they define the special quadratic Lie algebra \(sq_{2n} (A)\) to be the commutator subalgebra of \(gq_{2n} (\Lambda)\) (they prove that the commutator subalgebra is independent of the choice of form parameter). In the paper, the authors construct a universal central extension of \(sq_{2n} (A)\) and use it to calculate cohomology: they are able to calculate the homology groups \(H_1 (gq_{2n} (\Lambda), k)\) for \(n\geq 3\) and \(H_1 (sq_{2n} (A), k)\) for \(n\geq 5\).