# Holomorphic functions of exponential type and duality for stein groups with algebraic connected component of identity

@article{Akbarov2008HolomorphicFO, title={Holomorphic functions of exponential type and duality for stein groups with algebraic connected component of identity}, author={S. Akbarov}, journal={Journal of Mathematical Sciences}, year={2008}, volume={162}, pages={459-586} }

We suggest a generalization of Pontryagin duality from the category of commutative, complex Lie groups to the category of (not necessarily commutative) Stein groups with algebraic connected component of identity. In contrast to the other similar generalizations, in our approach the enveloping category consists of Hopf algebras (in a proper symmetrical monoidal category).

#### 17 Citations

A duality for Moore groups

- Mathematics
- 2009

We suggest a new generalization of Pontryagin duality from the category of Abelian locally compact groups to a category which includes all Moore groups, i.e. groups whose irreducible representations… Expand

Holomorphic duality for countable discrete groups

- Mathematics
- 2020

In 2008, the author proposed a version of duality theory for (not necessarily, Abelian) complex Lie groups, based on the idea of using the Arens-Michael envelope of topological algebra and having an… Expand

On holomorphic reflexivity conditions for complex Lie groups.

- Mathematics
- 2020

We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components,… Expand

Duality for SIN-groups

- Mathematics
- 2009

We suggest a generalization of Pontryagin duality from the category of Abelian locally compact groups to a category which includes all SIN-groups, i.e. groups with a base of invariant neighborhoods… Expand

C∞(M) as a smooth envelope of its subalgebras

- Mathematics
- 2015

A smooth envelope of a topological algebra is introduced, and the following result is announced: the smooth envelope of a given subalgebra A in C∞(M) coincides with C∞(M) if and only if A has the… Expand

Arens–Michael envelopes of nilpotent Lie algebras, holomorphic functions of exponential type, and homological epimorphisms

- Mathematics
- 2018

Our aim is to give an explicit description of the Arens-Michael envelope for the universal enveloping algebra of a finite-dimensional nilpotent complex Lie algebra. It turns out that the… Expand

Holomorphic functions of exponential type on connected complex Lie groups

- Mathematics
- 2019

Holomorphic functions of exponential type on a complex Lie group $G$ (introduced by Akbarov) form a locally convex algebra, which is denoted by $\cO_{exp}(G)$. Our aim is to describe the structure of… Expand

Kernels and cokernels in the category of augmented involutive stereotype algebras

- Mathematics
- 2020

We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras: (1) this category has kernels and cokernels, (2) the cokernel is preserved under th...

On continuous duality for Moore groups

- Mathematics
- 2018

In this paper we correct the errors of Yu.~N.~Kuznetsova's paper on the continuous duality for Moore groups.

Kernel and cokernel in the category of augmented involutive stereotype algebras

- Mathematics
- 2018

We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras: 1) the morphisms of the augmented involutive stereotype algebras have kernels and… Expand

#### References

SHOWING 1-10 OF 38 REFERENCES

Several Complex Variables with Connections to Algebraic Geometry and Lie Groups

- Mathematics
- 2002

Selected problems in one complex variable Holomorphic functions of several variables Local rings and varieties The Nullstellensatz Dimension Homological algebra Sheaves and sheaf cohomology Coherent… Expand

Kac Algebras and Duality of Locally Compact Groups

- Mathematics
- 1992

1. Co-Involutive Hopf-Von Neumann Algebras.- 2. Kac Algebras.- 3. Representations of a Kac Algebra Dual Kac Algebra.- 4. Duality Theorems for Kac Algebras and Locally Compact Groups.- 5. The Category… Expand

Theory Of Stein Spaces

- Mathematics
- 1979

A. Sheaf Theory.- B. Cohomology Theory.- I. Coherence Theory for Finite Holomorphic Maps.- II. Differential Forms and Dolbeault Theory.- III. Theorems A and B for Compact Blocks ?m.- IV. Stein… Expand

Locally compact quantum groups

- Mathematics
- 2000

These lecture notes are intended as an introduction to the theory of locally compact quantum groups that are studied in the framework of operator algebras, i.e. C*-algebras and von Neumann algebras.… Expand

Quantum groups: a path to current algebra

- Mathematics
- 2007

Introduction 1. Revision of basic structures 2. Duality between geometry and algebra 3. The quantum general linear group 4. Modules and tensor products 5. Cauchy modules 6. Algebras 7. Coalgebras and… Expand

Quantum Groups

- Mathematics
- 1994

Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups… Expand

The Theory of Topological Commutative Groups

- Mathematics
- 1934

The purpose of the present paper is to make an exhaustive investigation into the structure of continuous, locally compact, commutative groups, satisfying the second axiom of countability.2 1.

Nonunimodular Ring Groups and Hopf-Von Neumann Algebras

- Mathematics
- 1974

A number of authors have introduced ring groups as objects generalizing locally compact groups. An analogue of the Pontrjagin principle of duality holds for ring groups. In this paper we introduce a… Expand

A guide to quantum groups

- Mathematics
- 1994

Introduction 1. Poisson-Lie groups and Lie bialgebras 2. Coboundary Poisson-Lie groups and the classical Yang-Baxter equation 3. Solutions of the classical Yang-Baxter equation 4. Quasitriangular… Expand