scholarly journals Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang–Baxter equation

2006 ◽  
Vol 47 (10) ◽  
pp. 103511 ◽  
Author(s):  
K. A. Dancer ◽  
P. S. Isac ◽  
J. Links
Keyword(s):  
Finite Group ◽  
Spectral Parameter ◽  
Group Algebras ◽  
Baxter Equation ◽  
Parameter Dependent

scholarly journals ON TYPE I QUANTUM AFFINE SUPERALGEBRAS

1995 ◽  
Vol 10 (23) ◽  
pp. 3259-3281 ◽  
Author(s):  
GUSTAV W. DELIUS ◽  
MARK D. GOULD ◽  
JON R. LINKS ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Spectral Parameter ◽  
Lie Superalgebras ◽  
Baxter Equation ◽  
Type I ◽  
Free Fermion ◽  
General Technique ◽  
Finite Dimensional ◽  
Quantum Deformations ◽  
Free Fermion Model ◽  
Parameter Dependent

The type I simple Lie superalgebras are sl(m|n) and osp(2|2n). We study the quantum deformations of their untwisted affine extensions Uq[sl(m|n)(1)] and Uq[osp(2|2n)(1)]. We identify additional relations between the simple generators (“extra q Serre relations”) which need to be imposed in order to properly define Uq[sl(m|n)(1)] and Uq[osp(2|2n)(1)]. We present a general technique for deriving the spectral-parameter-dependent R matrices from quantum affine superalgebras. We determine the R matrices for the type I affine superalgebra Uq[sl(m|n)(1)] in various representations, thereby deriving new solutions of the spectral-parameter-dependent Yang-Baxter equation. In particular, because this algebra possesses one-parameter families of finite-dimensional irreps, we are able to construct R matrices depending on two additional spectral-parameter-like parameters, providing generalizations of the free fermion model.

scholarly journals Quantum double finite group algebras and their representations

1993 ◽  
Vol 48 (2) ◽  
pp. 275-301 ◽  
Author(s):  
M.D. Gould
Keyword(s):  
Finite Group ◽  
Explicit Construction ◽  
Unitary Representations ◽  
Group Algebras ◽  
Baxter Equation ◽  
Irreducible Matrix ◽  
Matrix Representations ◽  
Quantum Double ◽  
Induced Representations ◽  
A Finite Group

The quantum double construction is applied to the group algebra of a finite group. Such algebras are shown to be semi-simple and a complete theory of characters is developed. The irreducible matrix representations are classified and applied to the explicit construction of R-matrices: this affords solutions to the Yang-Baxter equation associated with certain induced representations of a finite group. These results are applied in the second paper of the series to construct unitary representations of the Braid group and corresponding link polynomials.

scholarly journals TWISTED QUANTUM AFFINE ALGEBRAS AND SOLUTIONS TO THE YANG-BAXTER EQUATION

1996 ◽  
Vol 11 (19) ◽  
pp. 3415-3437 ◽  
Author(s):  
GUSTAV W. DELIUS ◽  
MARK D. GOULD ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Lie Algebras ◽  
Spectral Parameter ◽  
Baxter Equation ◽  
Affine Lie Algebras ◽  
Quantum Affine Algebras ◽  
Enveloping Algebras ◽  
Quantized Enveloping Algebras ◽  
Affine Algebras ◽  
Parameter Dependent

We construct spectral-parameter-dependent R matrices for the quantized enveloping algebras of twisted affine Lie algebras. These give new solutions to the spectral-parameter-dependent quantum Yang-Baxter equation.

Gaussian (N, z)-generalized Yang-Baxter operators

2016 ◽  
pp. 105-114
Author(s):  
Eric Rowell
Keyword(s):  
Bell States ◽  
Entangled States ◽  
Unitary Matrix ◽  
Baxter Equation ◽  
Careful Study ◽  
Standard Measurement ◽  
Measurement Basis ◽  
Parameter Dependent

We find unitary matrix solutions R˜(a) to the (multiplicative parameter-dependent) (N, z)-generalized Yang-Baxter equation that carry the standard measurement basis to m-level N-partite entangled states that generalize the 2-level bipartite entangled Bell states. This is achieved by a careful study of solutions to the Yang-Baxter equation discovered by Fateev and Zamolodchikov in 1982.

scholarly journals ON THE MORITA REDUCED VERSIONS OF SKEW GROUP ALGEBRAS OF PATH ALGEBRAS

2020 ◽  
Vol 71 (3) ◽  
pp. 1009-1047
Author(s):  
Patrick Le Meur
Keyword(s):  
Finite Group ◽  
Monoidal Category ◽  
Path Algebra ◽  
Group Algebras ◽  
Explicit Formulas ◽  
Morita Equivalent ◽  
Path Algebras ◽  
A Finite Group ◽  
Skew Group Algebras ◽  
Skew Group Algebra

Abstract Let $R$ be the skew group algebra of a finite group acting on the path algebra of a quiver. This article develops both theoretical and practical methods to do computations in the Morita-reduced algebra associated to $R$. Reiten and Riedtmann proved that there exists an idempotent $e$ of $R$ such that the algebra $eRe$ is both Morita equivalent to $R$ and isomorphic to the path algebra of some quiver, which was described by Demonet. This article gives explicit formulas for the decomposition of any element of $eRe$ as a linear combination of paths in the quiver described by Demonet. This is done by expressing appropriate compositions and pairings in a suitable monoidal category, which takes into account the representation theory of the finite group.

scholarly journals The EllipticGL(n)Dynamical Quantum Group as an𝔥-Hopf Algebroid

2009 ◽  
Vol 2009 ◽  
pp. 1-41 ◽  
Author(s):  
Jonas T. Hartwig
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Spectral Parameter ◽  
Exterior Algebra ◽  
Baxter Equation ◽  
Matrix Elements ◽  
Elliptic Solution ◽  
Hopf Algebroid ◽  
Quantum Dynamical ◽  
Hopf Algebroids

Using the language of𝔥-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group,ℱell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra𝔰𝔩n. We apply the generalized FRST construction and obtain an𝔥-bialgebroidℱell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the𝔥-Hopf algebroidℱell(GL(n)).

scholarly journals Multiparametric and Colored Extensions of the Quantum Group GLq(N) and the Yangian Algebra Y(glN) Through a Symmetry Transformation of the Yang–Baxter Equation

1997 ◽  
Vol 12 (05) ◽  
pp. 945-962 ◽  
Author(s):  
B. Basu-Mallick ◽  
P. Ramadevi ◽  
R. Jagannathan
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Spectral Parameter ◽  
General Relation ◽  
Symmetry Transformation ◽  
Baxter Equation ◽  
Color Parameter ◽  
R Matrix ◽  
General Symmetry ◽  
Yangian Algebra

Inspired by Reshetikhin's twisting procedure to obtain multiparametric extensions of a Hopf algebra, a general "symmetry transformation" of the "particle conserving" R-matrix is found such that the resulting multiparametric R-matrix, with a spectral parameter as well as a color parameter, is also a solution of the Yang–Baxter equation (YBE). The corresponding transformation of the quantum YBE reveals a new relation between the associated quantized algebra and its multiparametric deformation. As applications of this general relation to some particular cases, multiparametric and colored extensions of the quantum group GL q(N) and the Yangian algebra Y(glN) are investigated and their explicit realizations are also discussed. Possible interesting physical applications of such extended Yangian algebras are indicated.

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