Algebra & Number Theory

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

Dmitri I. Panyushev

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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.

Article information

Algebra Number Theory, Volume 7, Number 6 (2013), 1505-1534.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
Secondary: 17B08: Coadjoint orbits; nilpotent varieties 17B40: Automorphisms, derivations, other operators 17C20: Simple, semisimple algebras 22E46: Semisimple Lie groups and their representations

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


Panyushev, Dmitri I. Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras. Algebra Number Theory 7 (2013), no. 6, 1505--1534. doi:10.2140/ant.2013.7.1505.

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