New simple infinite-dimensional Lie algebras of characteristic 0
Following recent developments in constructing infinite-dimensional simple Lie algebras [\textit{N. Kawamoto}, Hiroshima Math. J. 16, 417-426 (1986; Zbl 0607.17008)] the author defines generalized Cartan type algebras \(W^*\), \(S^*\), \(H^*\), \(K^*\). For the simplest case \(W^*\) the construction is described by the formula: \[ \bigl[(x^\alpha)_{ij} (x^\beta)_j\bigr]=\beta_i (x^{\alpha +\beta})_j -\alpha_j (x^{\alpha +\beta})_i \] where \((x^\alpha)_i\), \(i=1, \dots, n\) form a basis in \(A= \oplus^n_{i=1} B\) and each \(B\) is generated by ``monomials'' \(x^\alpha\) and \(\alpha\) is a multi-index \(\alpha= (\alpha_1, \dots, \alpha_n)\) and each \(\alpha_j\) belongs to its own additive group \(\Delta_j\) which is an additive subgroup in the ground field \(F\). All the constructed algebras are proved to be simple.
infinite-dimensional simple Lie algebras
scientific article; zbMATH DE number 954784