On the Lie structure of the skew elements of a simple superalgebra with superinvolution
Let \(F\) be a field of characteristic \(\neq 2\). A superinvolution in an associative superalgebra \(A\) is a homogeneous linear operator \(a\to a^*\) of zero degree such that \(a^{**}=a,\) and \((ab)^*=(-1)^{ab}b^*a^*\) for any homogeneous elements \(a,b\). Denote by \(K\) the set of all skew elements \(a\in A\), that is \(a^*=-a\). It is clear that \(K\) is a Lie superalgebra. A superinvolution \(*\) is of the first kind if \(*\) acts identically on the center \(Z(A)\). Otherwise \(*\) is of the second kind. Let \(U\) be a Lie ideal of \(K\). If \(*\) is of the second kind then either \(U\subseteq Z(A)\) or \(U=[K,K]\) except for if \(A\) is \(C(2)\) or \(C(3)\). If \(*\) is of the first kind, then \(U\supseteq [K,K]\), except for if \(A\) is \(C(4)\). In particular the following Lie superalgebras are simple: \[ \begin{aligned} & K(\text{Mat}(r| s),\text{osp})=\\& [K(\text{Mat}(r| s),\text{osp}),K(\text{Mat}(r| s),\text{osp})]\text{ for all } r,s\geq 1,\quad s \text{ is even}, \\ & [K(\text{Mat}(r| s),\text{tsp}),K(\text{Mat}(r| s),\text{tsp})]\text{ for all } r\geq 3. \end{aligned} \] Here \[ \begin{pmatrix} A & B \\ C & D \end{pmatrix}^{\text{osp}}= \begin{pmatrix} T_0 & 0\\ 0 & T_1 \end{pmatrix}^{-1} \begin{pmatrix} A & -B\\ C & D \end{pmatrix}^t \begin{pmatrix} T_0 & 0\\ 0 & T_1 \end{pmatrix} \] for some special matrices \(T_0,T_1\) and \[ \begin{pmatrix} A & B \\ C & D \end{pmatrix}^{\text{tsp}}= \begin{pmatrix} D^t & -B^t\\ C^t & A^t \end{pmatrix} . \]
scientific article; zbMATH DE number 1221655