On density of polynomials in algebra of holomorphic functions of exponential type on linear Lie group

O. Yu. Aristov

Ufa mathematical journal, Tome 16 (2024) no. 2, pp. 76-80

Résumé

The question of the density of the algebra of polynomials (regular functions) in the algebra of holomorphic functions of exponential type on a complex Lie group arose in the study of duality for Hopf algebras of holomorphic functions. It was shown by the author in [J . Lie Theory, 29:4, 1045–1070 (2019)] that the answer is affirmative in the connected linear case. However, the argument is quite involved and here we present a short proof. It contains two ingredients. The first is the existences of a finite–dimensional faithful holomorphic representation with closed range. To prove it, we use an approach developed by Ðjoković. The second is a lower bound for the norm of a one–parameter matrix subgroup, which is based on some elementary linear algebra consideration. The rest of the proof is close to the original one and uses a decomposition of the group into a semidirect product of a simply connected solvable and linearly complex reductive factors.

Détail

Keywords: linear group, holomorphic function of exponential type, submultiplicative weight.
Mots-clés : complex Lie group

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