Simple Lie algebras of special type
Let \(K\) be an arbitrary field, \(A\) a commutative \(K\)-algebra, \(\Delta\) a nonzero \(K\)-vector space of commuting \(K\)-derivations of \(A\), and \(W\) the Lie subalgebra \(A\otimes_K\Delta= A\Delta\). The divergence map \(\operatorname {div}: W\rightarrow A\) is defined to be the \(K\)-linear extension of div\((a\delta)=\delta(a)\) for all \(a\in A\), \(\delta\in \Delta\). The kernel \(S\) of div is a Lie subalgebra of \(W\), called a special type Lie algebra. In the article under review, the authors study \(S\) from a ring-theoretic point of view and obtain sufficient conditions for the simplicity of \([S,S]\). These conditions are somewhat technical and cumbersome. But the results are effective in proving the simplicity of a number of interesting examples.
scientific article; zbMATH DE number 1448637