Generalization of A.~I.~Mal'tsev problem on commutativa subalgebras for Chevalley algebras

V. M. Levchuk ; G. S. Suleimanova

Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 231-240

Résumé

In 1945 A. I. Mal'tsev investigated the problem on description of abelian subgroups of largest dimension in complex simple Lie groups. This problem's arisen from the theorem of I. Schur: The largest dimension of abelian subgroups of the group $SL(n,\mathbb{C})$ equals to $[n^2/4]$ and abelian subgroups of such dimension for $n>3$ are transformed by automorphisms into each other. A. I. Mal'tsev solved his problem by the reduction to complex Lie algebras. In Cartan – Killing theory semisimple complex Lie algebras are classified making use of the classification of root systems in Euclidean space $V$. A Chevalley algebra $\mathcal{L}_\Phi(K)$ is associated with the indecomposable root system $\Phi$ and with the field $K$; the base of the Chevalley algebra consists of the base of certain abelian self-normalized subalgebra $H$ and of the elements $e_r$ $(r\in \Phi)$ with $H$-invariant subspace $Ke_r$. The elements $e_r$ $(r\in\Phi^+)$ form a base of niltriangular subalgebra $N\Phi(K)$. Methods of A. I. Mal'tsev were developed for the solving of the problem on large abelian subgroups in finite Chevalley groups. In this article we use the worked out methods for the reduction of A. I. Mal'cev theorem for the Chevalley algebras. We investigate the problems: (A) to describe commutative subalgebras of largest dimension in a Chevalley algebra $\mathcal{L}_\Phi(K)$ over arbitrary field $K$. (B) to describe commutative subalgebras of largest dimension in subalgebra $N\Phi(K)$ of the Chevalley algebra $\mathcal{L}_\Phi(K)$ over arbitrary field $K$. In this article we give the description of all commutative subalgebras of largest dimension in subalgebra $N\Phi(K)$ of classical type over arbitrary field $K$ up to automorphisms of algebra $\mathcal{L}_\Phi(K)$ and of subalgebra $N\Phi(K)$.

Détail

Keywords: Chevalley algebra, commutative subalgebra, niltriangular subalgebra.