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.

scholarly journals The dynamicalU(n)quantum group

2006 ◽  
Vol 2006 ◽  
pp. 1-30 ◽  
Author(s):  
Erik Koelink ◽  
Yvette Van Norden
Keyword(s):  
Quantum Group ◽  
Matrix Algebra ◽  
Exterior Algebra ◽  
Matrix Elements ◽  
Algebra Representation ◽  
The Matrix ◽  
Quantum Minor ◽  
Quantum Determinant

We study the dynamical analogue of the matrix algebraM(n), constructed from a dynamicalR-matrix given by Etingof and Varchenko. A left and a right corepresentation of this algebra, which can be seen as analogues of the exterior algebra representation, are defined and this defines dynamical quantum minor determinants as the matrix elements of these corepresentations. These elements are studied in more detail, especially the action of the comultiplication and Laplace expansions. Using the Laplace expansions we can prove that the dynamical quantum determinant is almost central, and adjoining an inverse the antipode can be defined. This results in the dynamicalGL(n)quantum group associated to the dynamicalR-matrix. We study a∗-structure leading to the dynamicalU(n)quantum group, and we obtain results for the canonical pairing arising from theR-matrix.

scholarly journals Quantum dynamical -matrix with spectral parameter from fusion

1998 ◽  
Vol 31 (42) ◽  
pp. 8533-8538
Author(s):  
Xu-Dong Luo ◽  
Xing-Chang Song ◽  
Shi-Kun Wang ◽  
Ke Wu
Keyword(s):  
Spectral Parameter ◽  
Dynamical Matrix ◽  
Quantum Dynamical

scholarly journals A categorical reconstruction of crystals and quantum groups at q = 0

2019 ◽  
Vol 70 (3) ◽  
pp. 895-925
Author(s):  
Craig Smith
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Analogous Result ◽  
Crystal Bases ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Monoidal Structure

Abstract The quantum co-ordinate algebra Aq(g) associated to a Kac–Moody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite-dimensional irreducible Uq(g) modules. In this paper, we investigate whether an analogous result is true when q=0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this, we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over Z whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at v=∞.

scholarly journals COHERENT STATES, DISPLACED NUMBER STATES AND LAGUERRE POLYNOMIAL STATES FOR su(1, 1) LIE ALGEBRA

2000 ◽  
Vol 14 (10) ◽  
pp. 1093-1103 ◽  
Author(s):  
XIAO-GUANG WANG
Keyword(s):  
Coherent State ◽  
Lie Algebra ◽  
Hypergeometric Functions ◽  
Laguerre Polynomial ◽  
Optical Systems ◽  
Matrix Elements ◽  
Operator Formalism ◽  
Ladder Operator ◽  
Deformed Oscillator ◽  
The Matrix

The ladder operator formalism of a general quantum state for su(1, 1) Lie algebra is obtained. The state bears the generally deformed oscillator algebraic structure. It is found that the Perelomov's coherent state is a su(1, 1) nonlinear coherent state. The expansion and the exponential form of the nonlinear coherent state are given. We obtain the matrix elements of the su(1, 1) displacement operator in terms of the hypergeometric functions and the expansions of the displaced number states and Laguerre polynomial states are followed. Finally some interesting su(1, 1) optical systems are discussed.

scholarly journals LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION

2008 ◽  
Vol 10 (02) ◽  
pp. 221-260 ◽  
Author(s):  
CHENGMING BAI
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Symmetric Part ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Poisson Lie Groups ◽  
Lagrangian Subalgebras

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

scholarly journals A q-DEFORMATION OF VIRASORO AND U(1) KAC–MOODY ALGEBRAS WITH HOPF STRUCTURE

1999 ◽  
Vol 14 (12) ◽  
pp. 733-743 ◽  
Author(s):  
M. MANSOUR ◽  
E. H. TAHRI
Keyword(s):  
Lie Algebra ◽  
Central Extension ◽  
Baxter Equation ◽  
R Matrix ◽  
Unit Square

Using a general formalism of a q-deformation of an arbitrary Lie algebra, new kinds of q-deformed centerless Virasoro and U(1) Kac–Moody algebras are found. This q-deformation is associated to an R-matrix of unit square satisfying the quantum Yang–Baxter equation and allows a nontrivial Hopf structure. The central extension is also incorporated in this formalism.

Unitary representations of the quantum group SU q (1, 1): II ? Matrix elements of unitary representations and the basic hypergoemetric functions

1990 ◽  
Vol 19 (3) ◽  
pp. 195-204 ◽  
Author(s):  
Tetsuya Masuda ◽  
Katsuhisa Mimachi ◽  
Yoshiomi Nakagami ◽  
Masatoshi Noumi ◽  
Yutaka Saburi ◽  
...  
Keyword(s):  
Quantum Group ◽  
Unitary Representations ◽  
Matrix Elements
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