# Hochschild And Cyclic Homology Of Quantum Groups

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### CURRICULUM VITAE

abelian groups Padova (Italia). 2000 On the classification of finite dimensional Hopf algebras, la conferinţa: Quantum groups, Manchester (Marea Britanie). 2007 Hochschild and cyclic homology, la conferinţa: Algebra, Topology and Geometry, ICTAMI, Piteşti.

### THE CYCLIC-HOMOLOGY CHERN-WEIL HOMOMORPHISM FOR PRINCIPAL

nilpotent extension (row extension), and prove that their cyclic-homology groups are isomorphic. In the proof, we use a chain homotopy invariance of complexes computing Hochschild, and hence cyclic homology, for arbi-trary row extensions. In the context of the cyclic-homology Chern-Weil ho-

### Cyclic Homology Handout

Hence, Hochschild homology extends to a functor kALG → AB. V. Cyclic Homology Given a cyclic module C , there is a classical complex arising from calculations in group homology: C n o1−t n o N n o1−t o N where N = 1 + t + t2 + + tn is the norm map. A group action on a space X is a continuous map G×X −→ X.

### Miscellaneous Notes

Pairing bar cochains to give Hochschild cochains. Misc 30 Calculations in a space of connections. Free loop space of BU. Misc 31 On rm, at modules. Misc 2003 32 Module calculations. Misc 33 On quivers and quantum groups. Gelfand functors. Misc 34 Cyclic cohomology and anomalies. Misc 35 Topological Markov chains. Modules over a ring where A2 = A. 2

### ASSIST.PROF. SERKAN SÜTLÜ Işık University Faculty of Arts and

Title: Hopf-cyclic cohomology of the Connes-Moscovici Hopf algebras * Focus Program on Noncommutative Geometry and Quantum Groups (Theme Week on Quantum Groups and Hopf-cyclic Homology), Fields Institute, June 10th - June 14th, 2013, Toronto, ON, CANADA Title: On the characteristic classes of symplectic foliations via Hopf-cyclic cocycles

### HOCHSCHILD (CO)HOMOLOGY OF 2× 2-GALOIS COVERINGS OF QUANTUM

Keywords and phrases: Hochschild (co)homology, quantum exterior algebra, Galois covering. 1. Introduction Let 3 be a ﬁnite-dimensional algebra (associative with identity) over a ﬁeld k. We consider the enveloping algebra 3e =3op ⊗k 3. For a ﬁnitely-generated bimodule 3X3, the ith Hochschild homology and cohomology of 3 with coefﬁcients in

### BIRS Workshop on Noncommutative Geometry

through topological Hochschild and cyclic homology, the idempotent conjecture for group algebras, the theory of quantum groups and Hopf algebras and their homological invariants); number theory (Connes new approach to the Riemann hypothesis, relations between Hecke algebras and quantum

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plane. Its quantum algebra was constructed in [DH2]. We compute the K-groups, the periodic cyclic homology of these quantum algebras and the corresponding Chern-Connes characters. In order to obtain these results, we use the methods from [DKT1]-[DKT2] and the methods and

### Hochschild and cyclic homology of the quantum multiparametric

which will serve to compute the desired homology groups. The cyclic homology groups of L, are then deduced using Connes s exact sequence. 1. Main results We state the two theorems which give explicitely the Hochschild and the cyclic ho- mology of the quantum torus.

### TRACING CYCLIC HOMOLOGY PAIRINGS UNDER TWISTING OF GRADED

Our goal is to understand: (1) the cyclic cohomology of such algebras as abstract linear spaces, and (2) the pairing with K-groups / cyclic homology groups for deformed algebras, in terms of those for the original algebra and the cohomology of G. For the case of the group algebra of G= Z2(which

### Homology and cohomology of associative algebras. A concise

to elements of the Hochschild homology groups, de Rham cohomology to cyclic homology, vector bundles to elements of the algebraic K 0-group, and the Chern character of a vector bundle to the algebraic Chern character of Section 8. An interesting feature of these invariants is that they are related to the trace map.

### HOCHSCHILD HOMOLOGY IN A BRAIDED TENSOR CATEGORY

sentations of a quantum group). If m = mR2 , then A is both a left and right module over the braided tensor product Ae = A§A0v , where Aop is simply A equipped with the opposite multiplication map wop = mR. Moreover, there is an explicit chain complex computing the braided Hochschild homology HR(A) = To^'^A, A).

### Cyclic homology, S -equivariant Floer cohomology, and Calabi

B, and the corresponding S1-equivariant homology theories are called cyclic homology groups. A relationship between the Hochschild homology of the Fukaya category and Floer homology is provided by the so-called open-closed string map [A] (1.1) OC : CH (F) !CF+n(M): Our main result is about the compatibility of OC with C (S1) actions. Namely, we

### THE HOMOLOGY OF ALGEBRAS OF PSEUDO-DIFFERENTIAL SYMBOLS AND

in a geometric way, the Hochschild homology of quantum algebras based on conic symplectic manifolds. In §3, we use the same ideas to calculate the cyclic homology of these algebras. In §4, we apply these methods to calculate the Hochschild and cyclic homology of the algebra of pseudo-diﬀerential symbols, and we derive the

### Excision theory in the dihedral and reflexive (co)homology of

(co)homology theory of associative algebras. That is, for such an extension, we obtain a six-term exact sequence in the dihedral cohomology. Also, we present and prove the relation between cyclic and dihedral cohomology of algebras and some examples. Subjects: Advanced Mathematics; Applied Mathematics; Foundations & Theorems

### Hochschild and cyclic homology of a family of Auslander algebras

and hence the cyclic homology of n is: (HC2p(n) = kQ0 ˘= kn 2 HC2p+1(n) = 0 8p 0: Remark 3.6 There doesn t seem to be any connection between these results and those for n. Indeed, the Hochschild and cyclic homologies for the Taft algebras are given as follows (see [S] for the Hochschild homology and [T, T1] for the cyclic homology): (HH0(n) = kn

### HochschildCochainsasaFrobeniusAlgebra arXiv:1409.4825v1 [math

coproduct on the chain complex for Hochschild homology, and there is a non-degenerate pairing h, ion Hochschild cocahins satisfying hα β, γi= hα, β γi. Moreover, the cochain complex for group cohomology under the simplicial cup product occurs as a subalgebra of the Hochschild cochain complex under the Gerstenhaber product.

### Hochschild and cyclic homology of -algebras

are generalizations of the Hochschild homology and cyclic homology of algebras and of the superalgebras. Finally we calculate some Hochschild homology groups of the quaternionic algebra H. M.S.C. 2000: 13D03, 55N35, 13N15. Key words: noncommutative geometry, Hochschild and cyclic homology, ‰-algebras, quaternionic algebra. 1 Introduction

### dspace.nbuv.gov.ua

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 041, 27 pages Cyclic Homology and Quantum Orbits Tomasz MASZCZYK and Serkan SUTL U Institute of M

### Atabey Kaygun - Anasayfa

Cyclic and Hochschild (co)-homology of algebras, coal- Kaygun and S. Sutl¨ u,¨ A characteristic map for compact quantum groups. Journal of Homotopy and Related

### Cyclic-Homology Chern Weil Theory for Families of Principal

used as a tool in computing Hochschild and cyclic homology and the Chern character [3,39,41], we use our Theorem 2.1 in calculations in the homotopy category of chain complexes.

### Sheel Ganatra: Curriculum Vitae

Sheel Ganatra 4 January 24, 2014, Columbia Symplectic Geometry, Gauge Theory, and Categoriﬁcation Seminar, Cyclic homology and S1-equivariant symplectic cohomology October 7-11, 2013, University of Hamburg, Fukaya categories, Hochschild homology, and topology ﬁeld theory

### Hochschild Cohomology of Algebras: Structure and Applications

Many of the deep results on the structure of Hochschild homology and cohomology involve methods and results particular to a given ﬁeld. It is one of the constant challenges to understand the unifying principles or possibilities of transfer from one area to another. We mention as one example the Hodge decomposition

### Summer School on String Topology and Hochschild Homology

Madsen have introduced Topological Hochschild and Topological cyclic ho-mology [1]. These two theories have been a wonderful technical tool to com-pute some K-groups [9]. Recently borrowing ideas from rational homotopy theory Hess and Rognes have given algebraic models for computing Topological Hocschild and cyclic homology.

### Cyclic Homology of Differential Operators, the Virasoro

elementary observation from cyclic homology theory: any cyclic 1 -cocycle (or equivalently antisymmetric Hochschild 1 -cocycle) ψ on an associative algebra A, i.e. an antisymmetric bilinear form ψ on A such that for all α 0, a l9 a 2 in A we have φ(α 0 α l5 α 2) - ιp(a 0, a±a 2} + ιp(a 2 a 0, α^ = 0 ,

### Curriculum Vitae Mihai D. Staic - IMAR

Co-PI for the grant: Hopf algebras, cyclic homology and monoidal categories 2009-2011 Supported by the Romanian National University Research Council. V. PUBLICATIONS 1. Samuel Carolus, Mihai D. Staic, G-Algebra Structure on the Higher Order Hochschild Cohomology H S2 (A;A) arXiv:1804.05096, submitted (2018). 2.

### Hochschild Cohomology of Algebras: Structure and Applications

Title: Hochschild (co)homology of quantum complete intersections Abstract: This is joint work with Karin Erdmann. We construct a minimal projective bimodule resolution for ﬁnite dimensional quantum complete intersections of codimension 2. Then we use this resolution to compute the Hochschild homology and cohomology for such an algebra.

### ORBIFOLD COHOMOLOGY AS PERIODIC CYCLIC HOMOLOGY

cyclic homology groups of exact categories. In Sec. 3, we prove Theorem 1.1. In Sec. 4, we show how an equivalence of derived categories implies equality of (orb-ifold) cohomology groups and also give some examples in which this equivalence is known. Finally, in Sec. 5, we give a conjecture about the singular case and a conjec-

### www.cambridge.org

Journal of the Inst. of Math. Jussieu (2004) 3(1), 17 68 c Cambridge University Press 17 DOI:10.1017/S1474748004000027 Printed in the United Kingdom CYCLIC

### Hochschild and Cyclic Homology of Quantum Groups

quantum groups is likely necessary to understand all these. A study of non-commutative differential geometry will provide much needed insight into the geometrical and topological properties of quantum groups. The main phenomenon concerning the Hochschild and cyclic homology of quantum groups is the following. As is well-known, any deformation

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plane. Its quantum algebra was constructed in [DH2]. We compute the K-groups,the periodic cyclic homology of these quantum algebras and the corresponding Chern-Connescharacters. In order to obtain these results, we use the methods from [DKT1]-[DKT2]and the methods and

### Math 7350: Differential Graded Algebras and Differential

Math 7350: Differential Graded Algebras and Differential Graded Categories Taught by Yuri Berest Notes by David Mehrle [email protected] Cornell University

### Introduction to Quantum Toric Geometry (1st Lecture)

Introduction to Quantum Toric Geometry (1st Lecture) ERNESTO LUPERCIO - CENTER FOR RESEARCH AND ADVANCED STUDIES (CINVESTAV), MEXICO CITY. JOINT WORK WITH LUDMIL KATZARKOV, LAURENT MEERSSEMAN AND

### Articles and Preprints by A. Kaygun

Excision in Hopf Cyclic Homology (withM.Khalkhali) K-Theory Vol.37 (1-2),2006. 6. Hopf Modules and Noncommutative Differential Geometry (withM.Khalkhali) Letters in Mathematical PhysicsVol.76 (1),2006. 7. Hopf-Hochschild (co)homology of module algebras Homology, Homotopy andApplications Vol.9(2),2007. 8. Bivariant Hopf cyclic cohomology (withM

### Some Remarks About the Cyclic Homology of Skew PBW

ring) consists of abelian groups HC n(B), n 0. If k is a ﬁeld with characteristic zero, these groups are the homology groups of the quotient of the Hochschild complex by the action of the ﬁnite cyclic groups; this is the reason for the term cyclic The notation HC was for Homologie de Connes , but soon became Homologie

### www.mn.uio.no

K-Theory 26: 101 137, 2002. c 2002 Kluwer Academic Publishers. Printed in the Netherlands. 101 The Connes Moscovici Approach to Cyclic Cohomology for Compact Quantum Groups J.

### Tom Hadﬁeld February 18, 2008 arXiv:math/0403201v2 [math.QA

Hochschild and cyclic homology and cohomology of the underlying algebra of quantum SL(2). The results are markedly diﬀerent from the untwisted case. Both twisted Hochschild and cyclic homology and cohomology are ﬁnite dimen-sional in all degrees (we exhibit the generators), whereas in the untwisted case the lowest degrees are inﬁnite

### IC/2003/56 abdus salam educational, scientific and cultural

K-groups, the periodic cyclic homology of these quantum algebras and the corresponding Chern-Connes characters. The groups SL(2, R) has a family of elliptic hyperboloid orbits, a family of two-fold elliptic hyperboloid orbits, upper and lower elliptic half-cones, and the origin as a one-point orbit. The