Actions of Lie superalgebras on reduced rings. (Q930357)

The authors consider actions of finite-dimensional Hopf algebras on reduced and graded-reduced algebras and study the question of when the subalgebra of invariants is non-zero. The first result says that if \(R\) is a graded-reduced ring of characteristic \(p>2\) acted on by a finitely generated restricted \(K\)-Lie superalgebra \(L\), where \(K\) is a subring of \(C_0\) and \(C\) is the extended center of \(R\), then \(R^L\) is non-zero. This result has the following corollaries. Let \(R\) be a reduced algebra over a field \(K\) of characteristic \(p>2\) acted on by a finite-dimensional restricted \(K\)-Lie superalgebra \(L\) and let \(H=u(L)\#G\), where \(G\) is the group of order \(2\) with the natural action on \(L\). Then \(A^H\neq 0\), for every non-zero \(H\)-stable subalgebra \(A\) of \(R\). In the second corollary they assume that \(R\) is a reduced algebra over a field \(K\) of characteristic \(p>2\) acted on by a finite-dimensional restricted \(K\)-Lie superalgebra \(L\) and \(H=u(L)\#G\), where \(G\) is the group of order \(2\) with the natural action on \(L\). Then if \(R^H\) satisfies a polynomial identity of degree \(d\), then \(R\) satisfies a polynomial identity of degree \(dN\), where \(N\) is the dimension of \(H\).