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443 KiB

Open AccessArticle

Frobenius–Schur Indicator for Categories with Duality

by Kenichi Shimizu

Axioms 2012, 1(3), 324-364; https://doi.org/10.3390/axioms1030324 - 23 Oct 2012

Cited by 2 | Viewed by 3750

We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” [...] Read more.

We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we establish twisted versions of the Frobenius–Schur theorem for various algebraic objects. We also give several applications to the quantum SL₂. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

285 KiB

Open AccessCommunication

The Hecke Bicategory

by Alexander E. Hoffnung

Axioms 2012, 1(3), 291-323; https://doi.org/10.3390/axioms1030291 - 9 Oct 2012

Cited by 2 | Viewed by 4811

Abstract

We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest [...] Read more.

We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

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387 KiB

Open AccessArticle

The Sum of a Finite Group of Weights of a Hopf Algebra

by Apoorva Khare

Axioms 2012, 1(3), 259-290; https://doi.org/10.3390/axioms1030259 - 5 Oct 2012

Cited by 1 | Viewed by 5158

Abstract

Motivated by the orthogonality relations for irreducible characters of a finite group, we evaluate the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements. We then discuss results for products of skew-primitive elements. Examples [...] Read more.

Motivated by the orthogonality relations for irreducible characters of a finite group, we evaluate the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements. We then discuss results for products of skew-primitive elements. Examples include groups, (quantum groups over) Lie algebras, the small quantum groups of Lusztig, and their variations (by Andruskiewitsch and Schneider). Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

183 KiB

Open AccessArticle

Hopf Algebra Symmetries of an Integrable Hamiltonian for Anyonic Pairing

by Jon Links

Axioms 2012, 1(2), 226-237; https://doi.org/10.3390/axioms1020226 - 20 Sep 2012

Cited by 4 | Viewed by 5816

Abstract

Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of [...] Read more.

Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of its coupling parameters. While the integrable structure of the model relates to the well-known six-vertex solution of the Yang–Baxter equation, the Hopf algebra symmetries are not in terms of the quantum algebra Uq(sl(2)). Rather, they are associated with the Drinfel’d doubles of dihedral group algebras D(Dn). Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

318 KiB

Open AccessArticle

Bundles over Quantum RealWeighted Projective Spaces

by Tomasz Brzeziński and Simon A. Fairfax

Axioms 2012, 1(2), 201-225; https://doi.org/10.3390/axioms1020201 - 17 Sep 2012

Cited by 8 | Viewed by 6704

Abstract

The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd [...] Read more.

The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

200 KiB

Open AccessArticle

From Coalgebra to Bialgebra for the Six-Vertex Model: The Star-Triangle Relation as a Necessary Condition for Commuting Transfer Matrices

by Jeffrey R. Schmidt

Axioms 2012, 1(2), 186-200; https://doi.org/10.3390/axioms1020186 - 27 Aug 2012

Cited by 2 | Viewed by 5895

Abstract

Using the most elementary methods and considerations, the solution of the star-triangle condition (a2+b2-c2)/2ab = ((a’)^{^2}+(b’)^{^2}-(c’))^{^2}/2a’b’ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. [...] Read more.

Using the most elementary methods and considerations, the solution of the star-triangle condition (a2+b2-c2)/2ab = ((a’)^{^2}+(b’)^{^2}-(c’))^{^2}/2a’b’ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices T(a, b, c) and T(a’, b’, c’). Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

159 KiB

Open AccessArticle

The Duality between Corings and Ring Extensions

by Florin F. Nichita and Bartosz Zielinski

Axioms 2012, 1(2), 173-185; https://doi.org/10.3390/axioms1020173 - 10 Aug 2012

Cited by 2 | Viewed by 5551

Abstract

We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional [...] Read more.

We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional algebras and coalgebra. Both these duality extensions have some similarities with the Pontryagin-van Kampen duality theorem. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

278 KiB

Open AccessArticle

Quasitriangular Structure of Myhill–Nerode Bialgebras

by Robert G. Underwood

Axioms 2012, 1(2), 155-172; https://doi.org/10.3390/axioms1020155 - 24 Jul 2012

Cited by 2 | Viewed by 4868

Abstract

In computer science the Myhill–Nerode Theorem states that a set L of words in a finite alphabet is accepted by a finite automaton if and only if the equivalence relation ∼L, defined as x ∼L y if and only if xz ∈ [...] Read more.

In computer science the Myhill–Nerode Theorem states that a set L of words in a finite alphabet is accepted by a finite automaton if and only if the equivalence relation ∼L, defined as x ∼L y if and only if xz ∈ L exactly when yz ∈ L, ∀z, has finite index. The Myhill–Nerode Theorem can be generalized to an algebraic setting giving rise to a collection of bialgebras which we call Myhill–Nerode bialgebras. In this paper we investigate the quasitriangular structure of Myhill–Nerode bialgebras. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

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194 KiB

Open AccessCommunication

Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions

by Michiel Hazewinkel

Axioms 2012, 1(2), 149-154; https://doi.org/10.3390/axioms1020149 - 16 Jul 2012

Cited by 2 | Viewed by 6215

Abstract

Let NSymm be the Hopf algebra of non-commutative symmetric functions (in an infinity of indeterminates): . It is shown that an associative algebra A with a Hasse-Schmidt derivation ) on it is exactly the same as an NSymm module algebra. The primitives of [...] Read more.

Let NSymm be the Hopf algebra of non-commutative symmetric functions (in an infinity of indeterminates): . It is shown that an associative algebra A with a Hasse-Schmidt derivation ) on it is exactly the same as an NSymm module algebra. The primitives of NSymm act as ordinary derivations. There are many formulas for the generators in terms of the primitives (and vice-versa). This leads to formulas for the higher derivations in a Hasse-Schmidt derivation in terms of ordinary derivations, such as the known formulas of Heerema and Mirzavaziri (and also formulas for ordinary derivations in terms of the elements of a Hasse-Schmidt derivation). These formulas are over the rationals; no such formulas are possible over the integers. Many more formulas are derivable. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

413 KiB

Open AccessArticle

Valued Graphs and the Representation Theory of Lie Algebras

by Joel Lemay

Axioms 2012, 1(2), 111-148; https://doi.org/10.3390/axioms1020111 - 4 Jul 2012

Cited by 2 | Viewed by 6090

Abstract

Quivers (directed graphs), species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and [...] Read more.

Quivers (directed graphs), species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K -species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

541 KiB

Open AccessCommunication

Gradings, Braidings, Representations, Paraparticles: Some Open Problems

by Konstantinos Kanakoglou

Axioms 2012, 1(1), 74-98; https://doi.org/10.3390/axioms1010074 - 15 Jun 2012

Cited by 5 | Viewed by 6599

Abstract

A research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings and braided group structures present in the various parastatistical algebraic [...] Read more.

A research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings and braided group structures present in the various parastatistical algebraic models. The second part of the proposal aims at refining and utilizing a previously published methodology for the study of the Fock-like representations of the parabosonic algebra, in such a way that it can also be directly applied to the other parastatistics algebras. Finally, in the third part, a couple of Hamiltonians is proposed, suitable for modeling the radiation matter interaction via a parastatistical algebraic model. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

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