Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras

be the corresponding $ℤ_{2} \times ℤ_{2}$ -grading. If ${α, β, γ} = {01, 10, 11}$ , then $[,]$ maps $g_{@ α} \times g_{β}$ into $g_{γ}$ , and the zero fiber of this bracket is called a $\vec{σ}$ -commuting variety. The commuting variety of $g$ and commuting varieties related to one involution are particular cases of this construction. We develop a general theory of such varieties and point out some cases, when they have especially good properties. If $G ∕ G^{σ_{1}}$ is a Hermitian symmetric space of tube type, then one can find three conjugate pairwise commuting involutions $σ_{1}$ , $σ_{2}$ , and $σ_{3} = σ_{1} σ_{2}$ . In this case, any $\vec{σ}$ -commuting variety is isomorphic to the commuting variety of the simple Jordan algebra associated with $σ_{1}$ . As an application, we show that if $J$ is the Jordan algebra of symmetric matrices, then the product map $J \times J \to J$ is equidimensional, while for all other simple Jordan algebras equidimensionality fails.

Keywords

semisimple Lie algebra, commuting variety, Cartan subspace, quaternionic decomposition, nilpotent orbit, Jordan algebra

Mathematical Subject Classification 2010

Primary: 14L30

Secondary: 17B08, 17B40, 17C20, 22E46

Milestones

Received: 19 September 2012

Accepted: 24 January 2013

Published: 19 September 2013

Authors

Dmitri I. Panyushev
Dobrushin Mathematics Laboratory Institute for Information Transmission Problems Russian Academy of Sciences Bolshoy Karetny per. 19 Moscow, 127994 Russia

Vol. 7, No. 6, 2013

Dmitri I. Panyushev

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors