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Sheaves of non-commutative smooth and holomorphic functions associated with the non-abelian two-dimensional Lie algebra

Oleg Aristov

Abstract

Dosi and, quite recently, the author showed that, on the character space of a nilpotent Lie algebra, there exists a sheaf of Fréchet--Arens--Michael algebras (of noncommutative holomorphic functions in the complex case and of noncommutative smooth functions in the real case). We construct similar sheaves (both versions, holomorphic and smooth) on a special space of representations for the Lie algebra of the group of affine transformations of the real line (which is the simplest nonnilpotent solvable Lie algebra).

Sheaves of non-commutative smooth and holomorphic functions associated with the non-abelian two-dimensional Lie algebra

Abstract

Dosi and, quite recently, the author showed that, on the character space of a nilpotent Lie algebra, there exists a sheaf of Fréchet--Arens--Michael algebras (of noncommutative holomorphic functions in the complex case and of noncommutative smooth functions in the real case). We construct similar sheaves (both versions, holomorphic and smooth) on a special space of representations for the Lie algebra of the group of affine transformations of the real line (which is the simplest nonnilpotent solvable Lie algebra).

Paper Structure

This paper contains 2 sections, 11 theorems, 31 equations.

Table of Contents

  1. Algebras of triangular matrices
  2. Locally defined algebras of noncommutative functions and sheaves

Key Result

Proposition 1

If $n\in\mathbb{Z}_+$, $p,m\in\mathbb{N}$ and $\underline{K}\in\mathcal{C}^m_p$, then $\mathrm T_{p}(C^n(\underline{K}))$ is a Banach $\mathbb{R}$-algebra of polynomial growth.

Theorems & Definitions (19)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 9 more