Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras

  title={Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras},
  author={Alexei Yul’evich Pirkovskii},
  journal={Transactions of the Moscow Mathematical Society},
  • A. Pirkovskii
  • Published 2008
  • Mathematics
  • Transactions of the Moscow Mathematical Society
We describe and investigate Arens–Michael envelopes of associative algebras and their homological properties. We also introduce and study analytic analogs of some classical ring-theoretic constructs: Ore extensions, Laurent extensions, and tensor algebras. For some finitely generated algebras, we explicitly describe their Arens–Michael envelopes as certain algebras of noncommutative power series, and we also show that the embeddings of such algebras in their Arens–Michael envelopes are… Expand
Arens–Michael envelopes of nilpotent Lie algebras, holomorphic functions of exponential type, and homological epimorphisms
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 theExpand
Holomorphically finitely generated algebras
We introduce and study holomorphically finitely generated (HFG) Frechet algebras, which are analytic counterparts of affine (i.e., finitely generated) $\mathbb C$-algebras. Using a theorem of O.Expand
Homological dimensions and Van den Bergh isomorphisms for nuclear Fréchet algebras
We prove the equation for every nuclear Frechet-Arens-Michael algebra of finite weak bidimension, where is the weak global dimension and the weak bidimension of . Assuming that has a projectiveExpand
Maurer-Cartan Moduli and Theorems of Riemann-Hilbert Type
We study Maurer–Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notionExpand
Flat cyclic Frchet modules, amenable Frchet algebras, and approximate identities
Let A be a locally m-convex Fr\'echet algebra. We give a necessary and sufficient condition for a cyclic Fr\'echet A-module X=A_+/I to be strictly flat, generalizing thereby a criterion of HelemskiiExpand
Dual Fréchet algebras: Connes amenability and (σwc)-virtual diagonals
A relation between Connes amenability of dual Banach algebras and the existence of ([Formula: see text])-virtual diagonals was investigated by Volker Runde. In this paper, we first introduce andExpand
On holomorphic reflexivity conditions for complex Lie groups
  • O. Aristov
  • Mathematics
  • Proceedings of the Edinburgh Mathematical Society
  • 2021
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 manyExpand
We survey some results on homological dimensions of the algebraic, complex analytic, and smooth quantum tori. Our main theorem states, in par- ticular, that the smooth and the complex analyticExpand
The Arens-Michael envelope of a smash product
Given a Hopf algebra H and an H-module algebra A, we explicitly describe the Arens-Michael envelope of the smash product A#H in terms of the Arens-Michael envelope of H and a certain completion of A.Expand
The Relation “Commutator Equals Function” in Banach Algebras
Abstract The relation $$xy-yx=h(y)$$ , where $$h$$ is a holomorphic function, occurs naturally in the definitions of some quantum groups. To attach a rigorous meaning to the right-hand side of thisExpand


Arens-Michael enveloping algebras and analytic smash products
Let g be a finite-dimensional complex Lie algebra, and let U(g) be its universal enveloping algebra. We prove that if?(g), the Arens-Michael envelope of U(g) is stably flat over U(g) (i.e., if theExpand
Algebra extensions and nonsingularity
This paper is concerned with a notion of nonsingularity for noncommutative algebras, which arises naturally in connection with cyclic homology. Let us consider associative unital algebras over theExpand
Let An (n = 2, 3, . . . , or n = ∞) be the noncommutative disc algebra, and On (resp. Tn) be the Cuntz (resp. Toeplitz) algebra on n generators. Minimal joint isometric dilations for families ofExpand
Quantized coordinate rings and related noetherian algebras
This paper contains a survey of some ring-theoretic aspects of quantized coordinate rings, with primary focus on the prime and primitive spectra. For these algebras, the overall structure of theExpand
Embeddings of derived categories of bornological modules
Let A be an algebra with a countable basis and let B be, say, a Frechet algebra that contains A as a dense subalgebra. This embedding induces a functor from the derived category of B-modules to theExpand
The algebraic structure of non-commutative analytic Toeplitz algebras
The non-commutative analytic Toeplitz algebra is the wot– closed algebra generated by the left regular representation of the free semigroup on n generators. We develop a detailed picture of theExpand
Banach algebras and automatic continuity
Banach algebras combine algebraic and analytical aspects: it is the interplay of these structures that gives the subject its fascination. This volume expounds the general theory of Banach algebras,Expand
Relative homological algebra
Introduction. The main purpose of this paper is to draw attention to certain functors, exactly analogous to the functors "Tor" and "Ext" of Cartan-Eilenberg [2], but applicable to a module theoryExpand
Holomorphic deformation of Hopf algebras and applications to quantum groups
Abstract In this article we propose a new and so-called holomorphic deformation scheme for locally convex algebras and Hopf algebras. Essentially we regard converging power series expansions of aExpand
Noncommutative disc algebras for semigroups
We study noncommutative disc algebras associated to the free product of discrete subsemigroups of R+. These algebras are associated to generalized Cuntz algebras, which are shown to be simple andExpand