Hopf Algebras and Biunitary Matrices

Abstract

Actually to any spin model one can associate a vertex model (this is clear from V. Jones’ initial interpretation – in terms of statistical mechanics – of these objects) and the construction of Hopf algebras from complex Hadamard matrices is a particular case of the construction of Hopf algebras from biunitary matrices. The construction of Hopf algebras from biunitary matrices is a particular case of some more general results from [2], where the most general situation is treated (the biunitarity condition is replaced by a twisted biunitary condition and also the ground field C is replaced by an arbitrary field k) and where the relation with subfactors is also discussed. In this paper we present the (shortest version of the) construction of Hopf algebras from biunitary matrices and we give some examples. The construction is explained in §1 and §2, which are self-contained. In §3 we give examples from [2]. In §4 we present the examples coming from complex Hadamard matrices and we introduce a notion of “magic biunitary”, which should also give other exotic examples. This article is based on my talk “Compact quantum groups, subfactors and the linear algebra of certain commuting squares” given at the Brussels conference “Hopf Algebras

Cite this paper

@inproceedings{Banica1999HopfAA, title={Hopf Algebras and Biunitary Matrices}, author={Teodor Banica}, year={1999} }