Cartan Subalgebras of

  • Neeb K
  • Penkov I
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Abstract

Let V be a vector space over a field of characteristic zero and V * be a space of linear functionals on V which separate the points of V . We consider V ⊗ V * as a Lie algebra of finite rank operators on V , and set (V, V * ) := V ⊗ V * . We define a Cartan subalgebra of (V, V * ) as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of (V;V * ) under the assumption that is algebraically closed. A subalgebra of (V, V * ) is a Cartan subalgebra if and only if it equals for some one-dimensional subspaces V j ⊆ V and (V j ) * ⊆ V * with (Vi) * (V j ) = δ ij and such that the spaces . We then discuss explicit constructions of subspaces V j and (V j ) * as above. Our second main result claims that a Cartan subalgebra of (V, V * ) can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra h which coincides with the maximal locally nilpotent h-submodule of (V, V * ), and such that the adjoint representation of is locally finite.

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Neeb, K.-H., & Penkov, I. (2003). Cartan Subalgebras of. Canadian Mathematical Bulletin, 46(4), 597–616. https://doi.org/10.4153/cmb-2003-056-1

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