What question did this study set out to answer?

The research aims to explore the structure of a semigroup within a split real connected simple Lie group and to establish conditions for its equality with the group.

March 29, 2026Open Access

Cartan Decompositions and Semigroups of Simple Lie Groups

Key Points

The research aims to explore the structure of a semigroup within a split real connected simple Lie group and to establish conditions for its equality with the group.
Examined Cartan decomposition of the Lie algebra of G.
Evaluated invariance by the adjoint action of sl(2, R).
Characterized properties of the Lie saturate of semigroup S.
Presented necessary and sufficient conditions for semigroup S to equal the group G.
Characterized the Lie saturate properties related to the structure of S.

Abstract

Let G be a split real connected simple Lie group and S a semigroup of G that contains a subgroup G() for an arbitrary root , isomorphic to SL(2, R) .We present a Cartan decomposition of the Lie algebra of G, related to , invariant by the adjoint action of the Lie algebra sl(2, R) that allows to characterize some properties of the Lie saturate of the semigroup S .We give necessary and sufficient conditions for S to be equal to the whole group G.

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