Quasi-Hopf $*$-Algebras

By M. D. Gould, T. Lekatsas. 2006

607 citations recorded by the source corpus.

We introduce quasi-Hopf $*$-algebras i.e. quasi-Hopf algebras equipped with a conjugation (star) operation. The definition of quasi-Hopf $*$-algebras proposed ensures that the class of quasi-Hopf $*$-algebras is closed under twisting and additionally, that any Hopf $*$-algebra becomes a quasi-Hopf $*$-algebra via twisting. The basic properties of these algebras are developed. The relationship between the antipode and star structure is investigated. Quasi-triangular quasi-Hopf $*$-algebras are introduced and studied.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/706fa5f8d21f5da7acb65e114ad205f96e835bd9

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Citations in the corpus

The Kashiwara-Vergne conjecture and Drinfeld’s associators (2008, unknown): “In [6] Drinfeld proved the following theorem:”
Non-Abelian Anyons and Interferometry (2007, methodology): “2 D(ZN) The Abelian anyon models derived from the quantum double D(ZN ) of Z (0) N describe certain orbifold CFTs [129, 130], topological ZN gauge theories [131], and also Kitaev’s toric code [62] based on the group ZN .”
On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups (2006, unknown): “(x,g,g−1yg) . (5) For any quasi-Hopf algebra H= (H,∆,ǫ,Φ,α,β,S) and any unit u∈H, we can twist Hby uto obtain a second quasi-Hopf algebra Hu = (H,∆,ǫ,Φ,uα,βu−1,Su), where Su(h) = uS(h)u−1 for h∈H.See [D] for details. We obtain another quasiHopf algebra HF = (H,∆F,ǫ,ΦF,αF,βF,S) via a gauge tranformat”
Quantum permutation groups: a survey (2006, background): “The idea of twisting, originally due to Drinfeld [36], and developed by Doi [34] in the dual framework, is the following one.”
Universal lifting theorem and quasi-Poisson groupoids (2005, background): “It also generalizes Drinfeld’s quasi-Lie bialgebras [8], the classical limit of quasi-Hopf algebras.”
Modular invariants for group-theoretical modular data. I (2009, background): “One of the simplest examples of modular categories are (the categories of representations of) so-called quantum doubles of finite groups [12], also known as (untwisted) group-theoretical modular categories.”
On braided fusion categories I (2009, background): “These principles go back to the article [14], in which Manin pairs (which are more or less the same as Lie quasi-bialgebras) appear as classical limits of quasi-Hopf algebras”
Central invariants and Frobenius–Schur indicators for semisimple quasi-Hopf algebras (2003, background): “The algebra DðGÞ is of interest in orbifold conformal field theory [23], indeed in this context there is a more general object, the twisted quantum double DoðGÞ; that arises naturally [7].”
Noncommutative geometry and gravity (2005, background): “We conclude Section 2 by recalling the construction of the Hopf algebra UΞ [19].”
On finite-dimensional Hopf algebras (2014, background): “Then H (the same algebra with comultiplication ∆ := F∆F) is again a Hopf algebra, named the twisting of H by F [26].”
Anyons and matrix product operator algebras (2015, background): “In this way we indeed recover the standard labeling of dyonic excitations in discrete twisted gauge theories: the flux is labeled by a conjugacy class CA and the charge is labeled by a projective irrep of the centralizer Zg of a representative element g in CA [55, 56].”
Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory (2003, methodology): “It is due to Drinfeld [17], while in [32] and [8] the dual case, that of a quasi-Lie bialgebra, and the more general case of proto-bialgebras (called there “proto-Liebialgebras”) are treated.”
The twisted Drinfeld double of a finite group via gerbes and finite groupoids (2005, background): “The fact that the twisted representation group of a groupoid is the direct sum of the representation groups of its automorphism groups translates in Section 3.3 into the decomposition of the representation group of the Drinfeld double à la DijkgraafPasquier-Roche induction.”
Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward (2022, unknown): “The resulting state will be a twisted quantum double of G [90, 91].”
Etingof–Kazhdan quantization of Lie superbialgebras (2004, methodology): “For example, it has been used in knot theory to relate quantum group invariants and the Kontsevich integral.”