Lie Algebras and Algebras of Associative Type

N. A. Koreshkov

Matematičeskie zametki, Tome 88 (2010) no. 1, pp. 43-52

Résumé

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type $A=\bigoplus_{\alpha\in G}A_\alpha$ graded by some group $G$, over a field of characteristic zero, has a nonzero component $A_1$ (where 1 stands for the identity element of $G$), and $A_1$ is a semisimple associative algebra. Let $B=\bigoplus_{\alpha\in G}B_\alpha$ be a finite-dimensional semisimple Lie algebra over a prime field $F_p$, and let $B$ be graded by a commutative group $G$. If $B=F_p\otimes_{\mathbb Z}A_L$, where $A_L$ is the commutator algebra of a $\mathbb Z$-algebra $A=\bigoplus_{\alpha\in G}A_\alpha$; if $\mathbb Q\otimes_{\mathbb Z}A$ is an algebra of associative type, then the 1-component of the algebra $K\otimes_{\mathbb Z}B$, where $K$ stands for the algebraic closure of the field $F_p$, is the sum of some algebras of the form $\operatorname{gl}(n_i,K)$.

Détail

Keywords: algebra of associative type, graded algebra, finite-dimensional semisimple Lie algebra, finite-dimensional semisimple algebra of associative type.