Hecke symmetries
type: Item
instance of: scholarly article
title: Hecke symmetries
zbMATH Open document ID: 0963.16038
DOI: 10.1016/S0022-4049(99)00145-0
published in: Journal of Pure and Applied Algebra
publication date: 2000-11-21T00:00:00Z
review text: Let \(K\) be an algebraically closed field of characteristic zero and let \(V\) be a finite-dimensional vector space over \(K\). A Hecke symmetry is an invertible closed operator \(R\colon V\otimes V\to V\otimes V\), satisfying the braid equation and the Hecke equation: \((R+1)(R-q)=0\), for some \(q\in K\) (\(q\) is assumed not to be a root of \(1\), unless \(q=1\)). In a first part, this paper contains a very nice review of the main results in the study of Hecke symmetries and the associated Hopf (bi-) algebras, with emphasis on their representation theory. These results concern mainly the class of even (and odd) Hecke symmetries. In a second part, non-even Hecke symmetries are studied. New results are given; in particular, it is shown that the quantum rank of a Hecke symmetry is a \(q\)-integer. \textit{M. Takeuchi, D. Tambara}'s example [Hokkaido Math. J. 21, No. 3, 409-419 (1992; Zbl 0763.17017), Proc. Japan Acad., Ser. A 67, No. 8, 267-269 (1991; Zbl 0753.17032)] is discussed, serving as an illustration to open questions and conjectures.
reviewed by: Sonia Natale
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16W35
zbMATH DE Number: 1517296
zbMATH Keywords: Hecke operators
zbMATH Keywords: Hopf algebras
zbMATH Keywords: braid equation
author: Phùng Hô Hai
MaRDI profile type: Publication
cites work: The quantum group structure associated with non-linearly extended Virasoro algebras
cites work: Representations of Hecke Algebras of General Linear Groups
cites work: Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence
cites work: Q4892544
cites work: Q3351567
cites work: Q4864684
cites work: Covariant differential complexes on quantum linear groups
cites work: A \(q\)-analogue of \(U(\mathfrak{gl}(N+1))\), Hecke algebra, and the Yang-Baxter equation
cites work: Two dual classes of bialgebras related to the concepts of “quantum group” and “quantum lie algebra”
cites work: Hopf algebras and vector symmetries
cites work: Quantum supergroups of \(\text{GL}(n| m)\) type: Differential forms, Koszul complexes, and Berezinians
cites work: Q3947818
cites work: Glueing operation for R-matrices, quantum groups and link-invariants of Hecke type
cites work: Q4040073
cites work: Multiparametric quantum deformation of the general linear supergroup
cites work: Koszul property and Poincaré series of matrix bialgebras of type \(A_ n\)
cites work: On the structure of quantum super groups \(\text{GL}_q(m|n)\)
cites work: Poincaré series of quantum spaces associated to Hecke operators
cites work: Ribbon graphs and their invariants derived from quantum groups
cites work: Q4838907
cites work: A new one-parameter family of \(2\times{}2\) matrix bialgebras
cites work: A new one-parameter family of \(2 \times{} 2\) quantum matrices
Recommended article: Q3481970
Recommended article: Hecke symmetries: an overview of Frobenius properties
Recommended article: Q3463429
Recommended article: Q4356241
Recommended article: Hecke symmetries and characteristic relations on reflection equation algebras
Recommended article: Hecke-symmetry and rational period functions
Recommended article: Hecke algebras and symplectic reflection algebras
Recommended article: Q4308253
Recommended article: Symmetric cyclotomic Hecke algebras
Recommended article: Q4405607
description: scientific article; zbMATH DE number 1517296