Open Access
2013 Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras
Dmitri I. Panyushev
Algebra Number Theory 7(6): 1505-1534 (2013). DOI: 10.2140/ant.2013.7.1505


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

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

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×JJ is equidimensional, while for all other simple Jordan algebras equidimensionality fails.


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Dmitri I. Panyushev. "Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras." Algebra Number Theory 7 (6) 1505 - 1534, 2013.


Received: 19 September 2012; Accepted: 24 January 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1304.14061
MathSciNet: MR3107571
Digital Object Identifier: 10.2140/ant.2013.7.1505

Primary: 14L30
Secondary: 17B08 , 17B40 , 17C20 , 22E46

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

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 6 • 2013
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