Filtered multiplicative bases of restricted enveloping algebras
A vector space basis \(\mathfrak{B}\) of a finite-dimensional associative algebra \(A\) is called filtered multiplicative if the product of two basis vectors in \(\mathfrak{B}\) is either zero or belongs to \(\mathfrak{B}\) and the basis vectors in \(\mathfrak{B}\) belonging to the Jacobson radical \(\mathfrak{rad}(A)\) of \(A\) form a basis of \(\mathfrak{rad}(A)\). In the paper under review the authors investigate the existence of filtered multiplicative bases for the restricted universal enveloping algebras of certain classes of finite-dimensional \(p\)-nilpotent restricted Lie algebras. Let \(\mathfrak{F}_p\) denote the class of finite-dimensional \(p\)-nilpotent restricted Lie algebras. Then the authors prove that for abelian \(L\in\mathfrak{F}_p\) the restricted universal enveloping algebra \(u(L)\) of \(L\) has a filtered multiplicative basis if and only if \(L\) is a direct sum of cyclic \(p\)-subalgebras. In particular, over perfect ground fields \(u(L)\) has always a filtered multiplicative basis, and contrary to commutative group algebras, commutative restricted universal enveloping algebras need not to have a filtered multiplicative basis. Moreover, it is proved that restricted universal enveloping algebras of certain generalizations of abelian \(p\)-nilpotent restricted Lie algebras, namely powerful restricted Lie algebras and \(p\)-nilpotent restricted Lie algebras of nilpotency class \(2\) over fields of characteristic \(p\neq 2\), do not have a filtered multiplicative basis. The \(2\)-nilpotent three-dimensional Heisenberg algebra shows that the last result fails in characteristic \(2\).
scientific article; zbMATH DE number 6054170