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

scholarly journals Unitary spherical representations of Drinfeld doubles

2018 ◽  
Vol 2018 (742) ◽  
pp. 157-186 ◽  
Author(s):  
Yuki Arano
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Complete Classification ◽  
Unitary Representations ◽  
Compact Lie Group ◽  
Drinfeld Double ◽  
Quantum Analogue ◽  
Property T ◽  
Simply Connected

Abstract We study irreducible spherical unitary representations of the Drinfeld double of the q-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of \mathrm{SU}_{q}(3) , we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group \mathrm{SU}_{q}(2n+1) has property (T), which also implies central property (T) of the dual of \mathrm{SU}_{q}(2n+1) .

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)).

su(2) -expansion of the Lorentz algebra so(3,1 )

2013 ◽  
Vol 91 (8) ◽  
pp. 589-598 ◽  
Author(s):  
Rutwig Campoamor-Stursberg ◽  
Hubert de Guise ◽  
Marc de Montigny
Keyword(s):  
Coherent State ◽  
Unitary Representations ◽  
Iwasawa Decomposition ◽  
Matrix Formalism ◽  
Matrix Elements ◽  
Infinite Dimensional ◽  
Lorentz Algebra ◽  
Finite Dimensional

We exploit the Iwasawa decomposition to construct coherent state representations of [Formula: see text], the Lorentz algebra in 3 + 1 dimensions, expanded on representations of the maximal compact subalgebra [Formula: see text]. Examples of matrix elements computation for finite dimensional and infinite-dimensional unitary representations are given. We also discuss different base vectors and the equivalence between these different choices. The use of the [Formula: see text]-matrix formalism to truncate the representation or to enforce unitarity is discussed.

Unitary representations of the quantum group SU q (1,1): Structure of the dual space ofU q (sl(2))

1990 ◽  
Vol 19 (3) ◽  
pp. 187-194 ◽  
Author(s):  
Tetsuya Masuda ◽  
Katsuhisa Mimachi ◽  
Yoshiomi Nakagami ◽  
Masatoshi Noumi ◽  
Yutaka Saburi ◽  
...  
Keyword(s):  
Quantum Group ◽  
Dual Space ◽  
Unitary Representations

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 Matrix elements and duality for type 2 unitary representations of the Lie superalgebragl(m|n)

2015 ◽  
Vol 56 (12) ◽  
pp. 121703 ◽  
Author(s):  
Jason L. Werry ◽  
Mark D. Gould ◽  
Phillip S. Isaac
Keyword(s):  
Unitary Representations ◽  
Matrix Elements

Understanding the q-Factors in Quantum Group Symmetry

1997 ◽  
Vol 52 (1-2) ◽  
pp. 59-62
Author(s):  
L. C. Biedenharnf ◽  
K. Srinivasa Rao
Keyword(s):  
Characteristic Feature ◽  
Quantum Group ◽  
Quantum Groups ◽  
Group Symmetry ◽  
Explicit Calculation ◽  
Matrix Elements ◽  
Structural Symmetry ◽  
Symmetry Properties ◽  
Q Factors ◽  
Tensor Operators

AbstractA characteristic feature of quantum groups is the occurrence of q-factors (factors of the form qk, k ∈ ℝ), which implement braiding symmetry. We show how the q-factors in matrix elements of elementary q-tensor operators (for all Uq(n)) may be evaluated, without explicit calculation, directly from structural symmetry properties.

scholarly journals The Haagerup property for locally compact quantum groups

2016 ◽  
Vol 2016 (711) ◽  
Author(s):  
Matthew Daws ◽  
Pierre Fima ◽  
Adam Skalski ◽  
Stuart White
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Unitary Representations ◽  
Compact Groups ◽  
Locally Compact ◽  
Haagerup Property ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups ◽  
Dual Quantum

AbstractThe Haagerup property for locally compact groups is generalised to the context of locally compact quantum groups, with several equivalent characterisations in terms of the unitary representations and positive-definite functions established. In particular it is shown that a locally compact quantum group 𝔾 has the Haagerup property if and only if its mixing representations are dense in the space of all unitary representations. For discrete 𝔾 we characterise the Haagerup property by the existence of a symmetric proper conditionally negative functional on the dual quantum group

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