Vol. 7, No. 6, 2013

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

Dmitri I. Panyushev

Vol. 7 (2013), No. 6, 1505–1534

Let σ1 and σ2 be commuting involutions of a connected reductive algebraic group G with g = Lie(G). Let

g = i,j=0,1g@ij

be the corresponding 2 × 2-grading. If {α,β,γ} = {01,10,11}, then [,] maps g@α × gβ into gγ, and the zero fiber of this bracket is called a 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 GGσ1 is a Hermitian symmetric space of tube type, then one can find three conjugate pairwise commuting involutions σ1, σ2, and σ3 = σ1σ2. In this case, any σ-commuting variety is isomorphic to the commuting variety of the simple Jordan algebra associated with σ1. As an application, we show that if J is the Jordan algebra of symmetric matrices, then the product map J ×J J is equidimensional, while for all other simple Jordan algebras equidimensionality fails.

semisimple Lie algebra, commuting variety, Cartan subspace, quaternionic decomposition, nilpotent orbit, Jordan algebra
Mathematical Subject Classification 2010
Primary: 14L30
Secondary: 17B08, 17B40, 17C20, 22E46
Received: 19 September 2012
Accepted: 24 January 2013
Published: 19 September 2013
Dmitri I. Panyushev
Dobrushin Mathematics Laboratory
Institute for Information Transmission Problems
Russian Academy of Sciences
Bolshoy Karetny per. 19
Moscow, 127994