HOCHSCHILD (CO)HOMOLOGY OF ℤ2×ℤ2-GALOIS COVERINGS OF QUANTUM EXTERIOR ALGEBRAS

2008 ◽  
Vol 78 (1) ◽  
pp. 35-54
Author(s):  
HOU BO ◽  
XU YUNGE
Keyword(s):  
Characteristic Zero ◽  
Hochschild Homology ◽  
Cyclic Homology ◽  
Exterior Algebra ◽  
Cohomology Groups ◽  
Galois Covering ◽  
Bimodule Resolution ◽  
Galois Coverings ◽  
Minimal Projective Bimodule Resolution ◽  
Algebra Over A Field

AbstractLet Aq=k〈x,y〉/(x2,xy+qyx,y2) be the quantum exterior algebra over a field k with $\mathrm {char}\,k\neq 2$, and let Λq be the ℤ2×ℤ2-Galois covering of Aq. In this paper the minimal projective bimodule resolution of Λq is constructed explicitly, and from it we can calculate the k-dimensions of all Hochschild homology and cohomology groups of Λq. Moreover, the cyclic homology of Λq can be calculated in the case where the underlying field is of characteristic zero.

scholarly journals A new description of equivariant cohomology for totally disconnected groups

2008 ◽  
Vol 1 (3) ◽  
pp. 431-472 ◽  
Author(s):  
Christian Voigt
Keyword(s):  
Equivariant Cohomology ◽  
Simplicial Complexes ◽  
Cyclic Homology ◽  
Cohomology Theory ◽  
Cohomology Groups ◽  
Natural Class ◽  
New Approach ◽  
Totally Disconnected

AbstractWe consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the theory of Baum and Schneider generalizes cosheaf homology we finally see that all these approaches to equivariant cohomology for totally disconnected groups are closely related.

scholarly journals Pro unitality and pro excision in algebraic K-theory and cyclic homology

2018 ◽  
Vol 2018 (736) ◽  
pp. 95-139 ◽  
Author(s):  
Matthew Morrow
Keyword(s):  
Short Proof ◽  
Hochschild Homology ◽  
Cyclic Homology ◽  
Strong Form ◽  
Noetherian Rings ◽  
General Base ◽  
Continuity Properties ◽  
Suitable Sense ◽  
K Theory

AbstractThe purpose of this paper is to study pro excision in algebraicK-theory and cyclic homology, after Suslin–Wodzicki, Cuntz–Quillen, Cortiñas, and Geisser–Hesselholt, as well as continuity properties of André–Quillen and Hochschild homology. A key tool is first to establish the equivalence of various pro Tor vanishing conditions which appear in the literature.This allows us to prove that all ideals of commutative, Noetherian rings are pro unital in a suitable sense. We show moreover that such pro unital ideals satisfy pro excision in derived Hochschild and cyclic homology. It follows hence, and from the Suslin–Wodzicki criterion, that ideals of commutative, Noetherian rings satisfy pro excision in derived Hochschild and cyclic homology, and in algebraicK-theory.In addition, our techniques yield a strong form of the pro Hochschild–Kostant–Rosenberg theorem; an extension to general base rings of the Cuntz–Quillen excision theorem in periodic cyclic homology; a generalisation of the Feĭgin–Tsygan theorem; a short proof of pro excision in topological Hochschild and cyclic homology; and new Artin–Rees and continuity statements in André–Quillen and Hochschild homology.

scholarly journals ELEMENTARY INVARIANTS FOR CENTRALIZERS OF NILPOTENT MATRICES

2009 ◽  
Vol 86 (1) ◽  
pp. 1-15 ◽  
Author(s):  
JONATHAN BROWN ◽  
JONATHAN BRUNDAN
Keyword(s):  
Lie Algebra ◽  
Characteristic Zero ◽  
Universal Enveloping Algebra ◽  
General Linear ◽  
Enveloping Algebra ◽  
Nilpotent Matrix ◽  
Nilpotent Matrices ◽  
Algebra Over A Field

AbstractWe construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.

ON FINITE BRANCHED UNIFORMIZATIONS OF THE PROJECTIVE PLANE

2013 ◽  
Vol 24 (02) ◽  
pp. 1350017
Author(s):  
A. MUHAMMED ULUDAĞ ◽  
CELAL CEM SARIOĞLU
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Free Product ◽  
Plane Curve ◽  
Complex Projective Plane ◽  
Galois Covering ◽  
Cyclic Groups ◽  
Galois Coverings ◽  
Complex Projective ◽  
Branching Index

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

scholarly journals Identities of the left-symmetric Witt algebras

2016 ◽  
Vol 26 (02) ◽  
pp. 435-450 ◽  
Author(s):  
Daniyar Kozybaev ◽  
Ualbai Umirbaev
Keyword(s):  
Lie Algebras ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Witt Algebra ◽  
Symmetric Algebras ◽  
Algebra Over A Field

Let [Formula: see text] be the polynomial algebra over a field [Formula: see text] of characteristic zero in the variables [Formula: see text] and [Formula: see text] be the left-symmetric Witt algebra of all derivations of [Formula: see text] [D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math. 4(3) (2006) 323–357]. We describe all right operator identities of [Formula: see text] and prove that the set of all algebras [Formula: see text], where [Formula: see text], generates the variety of all left-symmetric algebras. We also describe a class of general (not only right operator) identities for [Formula: see text].

scholarly journals Higher orbital integrals, Shalika germs, and the Hochschild homology of Hecke algebras

2001 ◽  
Vol 26 (3) ◽  
pp. 129-160 ◽  
Author(s):  
Victor Nistor
Keyword(s):  
Hecke Algebras ◽  
Hochschild Homology ◽  
Cyclic Homology ◽  
Detailed Calculation ◽  
Orbital Integrals ◽  
Compactly Supported ◽  
Shalika Germs ◽  
Definition Of ◽  
Compactly Supported Functions

We give a detailed calculation of the Hochschild and cyclic homology of the algebra𝒞c∞(G)of locally constant, compactly supported functions on a reductivep-adic groupG. We use these calculations to extend to arbitrary elements the definition of the higher orbital integrals introduced by Blanc and Brylinski (1992) for regular semi-simple elements. Then we extend to higher orbital integrals some results of Shalika (1972). We also investigate the effect of the “induction morphism” on Hochschild homology.

scholarly journals Galois coverings of one-sided bimodule problems

2021 ◽  
Vol 14 (2) ◽  
pp. 93-116
Author(s):  
Vyacheslav Babych ◽  
Nataliya Golovashchuk
Keyword(s):  
Universal Covering ◽  
Cell Complex ◽  
Finiteness Conditions ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Complex Theory ◽  
Galois Covering ◽  
Geometric Methods ◽  
Galois Coverings ◽  
Dimensional Cell

Applying geometric methods of 2-dimensional cell complex theory, we construct a Galois covering of a bimodule problem satisfying some structure, triangularity and finiteness conditions in order to describe the objects of finite representation type. Each admitted bimodule problem A is endowed with a quasi multiplicative basis. The main result shows that for a problem from the considered class having some finiteness restrictions and the schurian universal covering A', either A is schurian, or its basic bigraph contains a dotted loop, or it has a standard minimal non-schurian bimodule subproblem.

scholarly journals Linearity defect and regularity over a Koszul algebra

2009 ◽  
Vol 104 (2) ◽  
pp. 205 ◽  
Author(s):  
Kohji Yanagawa
Keyword(s):  
Complete Intersection ◽  
Polynomial Ring ◽  
Commutative Algebra ◽  
Exterior Algebra ◽  
Finitely Generated ◽  
Koszul Algebra ◽  
Algebra Over A Field

Let $A = \bigoplus_{i\in \mathsf{N}}A_i$ be a Koszul algebra over a field $K = A_0$, and $*\operatorname{mod} A$ the category of finitely generated graded left $A$-modules. The linearity defect $\mathrm{ld}_A(M)$ of $M \in *\operatorname{mod} A$ is an invariant defined by Herzog and Iyengar. An exterior algebra $E$ is a Koszul algebra which is the Koszul dual of a polynomial ring. Eisenbud et al. showed that $\mathrm{ld}_E(M) < \infty$ for all $M \in *\operatorname{mod} E$. Improving this, we show that the Koszul dual $A^!$ of a Koszul commutative algebra $A$ satisfies the following. Let $M \in *\operatorname{mod} A^!$. If $\{\dim_K M_i \mid i \in {\mathsf Z}\}$ is bounded, then $\mathrm{ld}_{A^!}(M) < \infty$. If $A$ is complete intersection, then $\mathrm{reg}_{A^!}(M) < \infty$ and $\mathrm{ld}_{A^!}(M) < \infty$ for all $M \in *\operatorname{mod} A^!$. If $E=\bigwedge \langle y_1, \ldots, y_n\rangle$ is an exterior algebra, then $\mathrm{ld}_E(M)\leq c^{n!} 2^{(n-1)!}$ for $M \in *\operatorname{mod} E$ with $c := \max \{\dim_K M_i \mid i \in{\mathsf Z}\}$.

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