# 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. 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).

## 21 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…

### 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…

### Holomorphic Reflexivity for Locally Finite and Profinite Groups: The Abelian and General Cases

- MathematicsMathematical Notes
- 2022

Akbarov’s theory of holomorphic reflexivity for topological Hopf algebras has been developed in two directions, namely, by the complication of definitions when expanding the scope and by their…

### On holomorphic reflexivity conditions for complex Lie groups

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2021

Abstract 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…

### 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…

### 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…

### Length functions exponentially distorted on subgroups of complex Lie groups

- Mathematics
- 2022

. We introduce a notion of a length function exponentially distorted on a (compactly generated) subgroup of a locally compact group. We prove that for a connected linear complex Lie group there is a…

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

- MathematicsTransactions of the Moscow Mathematical Society
- 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…

### 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. In [11] Yu. N. Kuznetsova made an attempt to construct a generalization of the Pontryagin…

### On continuous duality for Moore groups

- MathematicsJournal of Operator Theory
- 2022

n 2013, Yu.N.~Kuznetsova constructed a duality theory for Moore groups, based on the idea of a continuous envelope of topological algebra and having the advantage over the existing theories that its…

## References

SHOWING 1-10 OF 40 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…

### 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…

### 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…

### 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…

### 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…

### 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…

### 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…

### Cocommutative Hopf algebras with antipode

- Mathematics
- 1967

We shall describe the structure of a certain kind of Hopf algebra over an algebraically closed field k of characteristic p, namely those Hopf algebras whose coalgebra structure is commutative and…