33 research outputs found
Quantum symmetric pairs and representations of double affine Hecke algebras of type
We build representations of the affine and double affine braid groups and
Hecke algebras of type , based upon the theory of quantum symmetric
pairs . In the case , our constructions provide a
quantization of the representations constructed by Etingof, Freund and Ma in
arXiv:0801.1530, and also a type generalization of the results in
arXiv:0805.2766.Comment: Final version, to appear in Selecta Mathematic
On centralizer algebras for spin representations
We give a presentation of the centralizer algebras for tensor products of
spinor representations of quantum groups via generators and relations. In the
even-dimensional case, this can be described in terms of non-standard
q-deformations of orthogonal Lie algebras; in the odd-dimensional case only a
certain subalgebra will appear. In the classical case q = 1 the relations boil
down to Lie algebra relations
Coideal Quantum Affine Algebra and Boundary Scattering of the Deformed Hubbard Chain
We consider boundary scattering for a semi-infinite one-dimensional deformed
Hubbard chain with boundary conditions of the same type as for the Y=0 giant
graviton in the AdS/CFT correspondence. We show that the recently constructed
quantum affine algebra of the deformed Hubbard chain has a coideal subalgebra
which is consistent with the reflection (boundary Yang-Baxter) equation. We
derive the corresponding reflection matrix and furthermore show that the
aforementioned algebra in the rational limit specializes to the (generalized)
twisted Yangian of the Y=0 giant graviton.Comment: 21 page. v2: minor correction
Quasitriangular coideal subalgebras of Uq(g) in terms of generalized Satake diagrams
© 2020 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits use, distribution and reproduction in any medium, provided the original work is properly cited.Let (Formula presented.) be a finite-dimensional semisimple complex Lie algebra and (Formula presented.) an involutive automorphism of (Formula presented.). According to Letzter, Kolb and Balagović the fixed-point subalgebra (Formula presented.) has a quantum counterpart (Formula presented.), a coideal subalgebra of the Drinfeld–Jimbo quantum group (Formula presented.) possessing a universal (Formula presented.) -matrix (Formula presented.). The objects (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.) can all be described in terms of Satake diagrams. In the present work, we extend this construction to generalized Satake diagrams, combinatorial data first considered by Heck. A generalized Satake diagram naturally defines a semisimple automorphism (Formula presented.) of (Formula presented.) restricting to the standard Cartan subalgebra (Formula presented.) as an involution. It also defines a subalgebra (Formula presented.) satisfying (Formula presented.), but not necessarily a fixed-point subalgebra. The subalgebra (Formula presented.) can be quantized to a coideal subalgebra of (Formula presented.) endowed with a universal (Formula presented.) -matrix in the sense of Kolb and Balagović. We conjecture that all such coideal subalgebras of (Formula presented.) arise from generalized Satake diagrams in this way.Peer reviewe
Properties of generalized univariate hypergeometric functions
Based on Spiridonov's analysis of elliptic generalizations of the Gauss
hypergeometric function, we develop a common framework for 7-parameter families
of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric
functions. In each case we derive the symmetries of the generalized
hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic)
and of type E_6 (trigonometric) using the appropriate versions of the
Nassrallah-Rahman beta integral, and we derive contiguous relations using
fundamental addition formulas for theta and sine functions. The top level
degenerations of the hyperbolic and trigonometric hypergeometric functions are
identified with Ruijsenaars' relativistic hypergeometric function and the
Askey-Wilson function, respectively. We show that the degeneration process
yields various new and known identities for hyperbolic and trigonometric
special functions. We also describe an intimate connection between the
hyperbolic and trigonometric theory, which yields an expression of the
hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric
hypergeometric functions.Comment: 46 page
Translation functors and decomposition numbers for the periplectic Lie superalgebra
We study the category of finite-dimensional integrable representations of the periplectic Lie superalgebra . We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter on the category by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for resembling those for . Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of . We also prove that indecomposable projective modules in this category are multiplicity-free
Translation functors and decomposition numbers for the periplectic Lie superalgebra
We study the category of finite-dimensional integrable representations of the periplectic Lie superalgebra . We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter on the category by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for resembling those for . Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of . We also prove that indecomposable projective modules in this category are multiplicity-free
Translation functors and decomposition numbers for the periplectic Lie superalgebra
We study the category of finite-dimensional integrable representations of the periplectic Lie superalgebra . We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter on the category by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for resembling those for . Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of . We also prove that indecomposable projective modules in this category are multiplicity-free
Generalized q-Onsager Algebras and Dynamical K-matrices
A procedure to construct -matrices from the generalized -Onsager
algebra \cO_{q}(\hat{g}) is proposed. This procedure extends the intertwiner
techniques used to obtain scalar (c-number) solutions of the reflection
equation to dynamical (non-c-number) solutions. It shows the relation between
soliton non-preserving reflection equations or twisted reflection equations and
the generalized -Onsager algebras. These dynamical -matrices are
important to quantum integrable models with extra degrees of freedom located at
the boundaries: for instance, in the quantum affine Toda field theories on the
half-line they yield the boundary amplitudes. As examples, the cases of
\cO_{q}(a^{(2)}_{2}) and \cO_{q}(a^{(1)}_{2}) are treated in details