Higman ideals and verlinde-type formulas for hopf algebras

Miriam Cohen; Sara Westreich

doi:10.1007/978-3-0346-0007-1_5

Higman ideals and verlinde-type formulas for hopf algebras

Miriam Cohen, Sara Westreich

Department of Management

Ben-Gurion University of the Negev

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

11 Scopus citations

Abstract

We offer a comprehensive discussion on Verlinde-type formulas for Hopf algebras H over an algebraically closed field of characteristic 0. Some of the results are new and some are known, but are reproved from the point of view of symmetric algebras and the associated Higman (trace) map. We give an explicit form for the central Casimir element of C(H), which is also known to be χad, the character of the adjoint map on H. We then discuss the following variations of the Verlinde formula: (i) Fusion rules for irreducible characters of semisimple Hopf algebras whose character algebras C(H) are commutative. (ii) Structure constants for what we call here conjugacy sums associated to conjugacy classes for these Hopf algebras. (iii) Equality up to rational scalar multiples between the fusion rules of irreducible characters and the structure constants for semisimple factorizable Hopf algebras. (iv) Projective fusion rules for the multiplication of irreducible and indecomposable projective characters for non-semisimple factorizable Hopf algebras.

Original language	English
Title of host publication	Ring and Module Theory
Editors	Toma Albu, Gary F. Birkenmeier, Ali Erdoǧan, Adnan Tercan
Publisher	Springer International Publishing
Pages	91-114
Number of pages	24
ISBN (Print)	9783034600064
DOIs	https://doi.org/10.1007/978-3-0346-0007-1_5
State	Published - 2010
Event	International Conference on Ring and Module Theory, 2008 - Ankara, Turkey Duration: 18 Aug 2008 → 22 Aug 2008

Publication series

Name	Trends in Mathematics
Volume	50
ISSN (Print)	2297-0215
ISSN (Electronic)	2297-024X

Conference

Conference	International Conference on Ring and Module Theory, 2008
Country/Territory	Turkey
City	Ankara
Period	18/08/08 → 22/08/08

Bibliographical note

Publisher Copyright:
© 2010 Springer Basel AG.

Keywords

Characters
Conjugacy sums and classes
Factorizable ribbon Hopf algebras
Fusion rules
Hopf algebras
Projective center
Symmetric algebras
Verlinde formula

Access to Document

10.1007/978-3-0346-0007-1_5

Cite this

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title = "Higman ideals and verlinde-type formulas for hopf algebras",

abstract = "We offer a comprehensive discussion on Verlinde-type formulas for Hopf algebras H over an algebraically closed field of characteristic 0. Some of the results are new and some are known, but are reproved from the point of view of symmetric algebras and the associated Higman (trace) map. We give an explicit form for the central Casimir element of C(H), which is also known to be χad, the character of the adjoint map on H. We then discuss the following variations of the Verlinde formula: (i) Fusion rules for irreducible characters of semisimple Hopf algebras whose character algebras C(H) are commutative. (ii) Structure constants for what we call here conjugacy sums associated to conjugacy classes for these Hopf algebras. (iii) Equality up to rational scalar multiples between the fusion rules of irreducible characters and the structure constants for semisimple factorizable Hopf algebras. (iv) Projective fusion rules for the multiplication of irreducible and indecomposable projective characters for non-semisimple factorizable Hopf algebras.",

keywords = "Characters, Conjugacy sums and classes, Factorizable ribbon Hopf algebras, Fusion rules, Hopf algebras, Projective center, Symmetric algebras, Verlinde formula",

author = "Miriam Cohen and Sara Westreich",

note = "Publisher Copyright: {\textcopyright} 2010 Springer Basel AG.; International Conference on Ring and Module Theory, 2008 ; Conference date: 18-08-2008 Through 22-08-2008",

year = "2010",

doi = "10.1007/978-3-0346-0007-1_5",

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isbn = "9783034600064",

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Cohen, M & Westreich, S 2010, Higman ideals and verlinde-type formulas for hopf algebras. in T Albu, GF Birkenmeier, A Erdoǧan & A Tercan (eds), Ring and Module Theory. Trends in Mathematics, vol. 50, Springer International Publishing, pp. 91-114, International Conference on Ring and Module Theory, 2008, Ankara, Turkey, 18/08/08. https://doi.org/10.1007/978-3-0346-0007-1_5

Higman ideals and verlinde-type formulas for hopf algebras. / Cohen, Miriam; Westreich, Sara.
Ring and Module Theory. ed. / Toma Albu; Gary F. Birkenmeier; Ali Erdoǧan; Adnan Tercan. Springer International Publishing, 2010. p. 91-114 (Trends in Mathematics; Vol. 50).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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KW - Projective center

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Higman ideals and verlinde-type formulas for hopf algebras

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