Representations Of Uq Research Articles

Braid group and 𝑞-Racah polynomials

The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of U q ( s l 2 ) U_q(\mathfrak {sl}_2) are considered. It is shown that these intermediate Casimirs are related by a conjugation involving braid group representations. Consequently, the entries of the braid group matrices are explicitly given in terms of the q q -Racah polynomials which appear as 6 j 6j -symbols in the Racah problem for U q ( s l 2 ) U_q(\mathfrak {sl}_2) . Formulas for these polynomials are derived from the algebraic relations satisfied by the braid group representations.

Equivalence of a tangle category and a category of infinite dimensional 𝑈_{𝑞}(𝔰𝔩₂)-modules

It is very well known that if V V is the simple 2 2 -dimensional representation of U q ( s l 2 ) \mathrm {U}_q(\mathfrak {sl}_2) , the category of representations V ⊗ r V^{\otimes r} , r = 0 , 1 , 2 , … r=0,1,2,\dots , is equivalent to the Temperley-Lieb category T L ( q ) \mathrm {TL}(q) . Such categorical equivalences between tangle categories and categories of representations are rare. In this work we give a family of new equivalences by extending the above equivalence to one between the category of representations M ⊗ V ⊗ r M\otimes V^{\otimes r} , where M M is a projective Verma module of U q ( s l 2 ) \mathrm {U}_q(\mathfrak {sl}_2) and the type B B Temperley-Lieb category T L B ( q , Q ) \mathbb {TLB}(q,Q) , realised as a subquotient of the tangle category of Freyd, Yetter, Reshetikhin, Turaev and others.

Yokota type invariants derived from non-integral highest weight representations of 𝒰q(sl2)

We define invariants for colored oriented spatial graphs by generalizing CM invariants [F. Costantino and J. Murakami, On [Formula: see text] quantum [Formula: see text]-symbols and their relation to the hyperbolic volume, Quantum Topol. 4 (2013) 303–351], which were defined via non-integral highest weight representations of [Formula: see text]. We apply the same method used to define Yokota’s invariants, and we call these invariants Yokota type invariants. Then, we propose a volume conjecture of the Yokota type invariants of plane graphs, which relates to volumes of hyperbolic polyhedra corresponding to the graphs, and check it numerically for some square pyramids and pentagonal pyramids.

Yangians, quantum loop algebras, and abelian difference equations

Let g \mathfrak {g} be a complex, semisimple Lie algebra, and Y ℏ ( g ) Y_\hbar (\mathfrak {g}) and U q ( L g ) U_q(L\mathfrak {g}) the Yangian and quantum loop algebra of g \mathfrak {g} . Assuming that ℏ \hbar is not a rational number and that q = e π i ℏ q= e^{\pi i\hbar } , we construct an equivalence between the finite-dimensional representations of U q ( L g ) U_q(L\mathfrak {g}) and an explicit subcategory of those of Y ℏ ( g ) Y_\hbar (\mathfrak {g}) defined by choosing a branch of the logarithm. This equivalence is governed by the monodromy of the abelian, additive difference equations defined by the commuting fields of Y ℏ ( g ) Y_\hbar (\mathfrak {g}) . Our results are compatible with q q -characters, and apply more generally to a symmetrizable Kac-Moody algebra g \mathfrak {g} , in particular to affine Yangians and quantum toroïdal algebras. In this generality, they yield an equivalence between the representations of Y ℏ ( g ) Y_\hbar (\mathfrak {g}) and U q ( L g ) U_q(L\mathfrak {g}) whose restriction to g \mathfrak {g} and U q g U_q\mathfrak {g} , respectively, are integrable and in category O \mathcal {O} .

A class of finite-dimensional representations of U q (sl2)

In order to study a class of finite-dimensional representations of Uq(sl2), we deal with the quotient algebra Uq(m, n, b) of quantum group Uq(sl2) with relations Kr = 1, Emr = b, Fnr = 0 in this paper, where q is a root of unity. The algebra Uq(m, n, b) is decomposed into a direct sum of indecomposable (left) ideals. The structures of indecomposable projective representations and their blocks are determined.

Quantum (sln, ∧V n ) link invariant and matrix factorizations

AbstractIn this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum (sln, ∧Vn) link invariant, where∧Vnis the set of fundamental representations ofUq(sln). In the case of an oriented link diagram composed of[k, 1]-crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general[i,j]-crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.

Elliptic central characters and blocks of finite dimensional representations of quantum affine algebras

The category of finite dimensional (type 1) representations of a quantum affine algebra U q ( g ^ ) U_q(\widehat {{\mathfrak g}}) is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition in a direct sum of indecomposable subcategories (blocks). We define the elliptic central character of a finite dimensional (type 1) representation of U q ( g ^ ) U_q(\widehat {{\mathfrak g}}) and show that the block decomposition of this category is parametrized by these elliptic central characters.

Automatic construction of explicit R matrices for the one-parameter families of irreducible typical highest weight [formula omitted] representations of Uq[ gl( m| n)

We detail the automatic construction of R matrices corresponding to (the tensor products of) the ( 0 ̇ m| α ̇ n) families of highest-weight representations of the quantum superalgebras U q [ gl( m| n)]. These representations are irreducible, contain a free complex parameter α, and are 2 mn -dimensional. Our R matrices are actually (sparse) rank 4 tensors, containing a total of 2 4 mn components, each of which is in general an algebraic expression in the two complex variables q and α. Although the constructions are straightforward, we describe them in full here, to fill a perceived gap in the literature. As the algorithms are generally impracticable for manual calculation, we have implemented the entire process in Mathematica; illustrating our results with U q [ gl(3|1)].

The embeddingU'q(so3)⊂Uq(sl3)

We study the embedding U'q(so3)?Uq(sl3), where Uq(sl3) is a well known Drinfeld-Jimbo quantum algebra and the algebra U'q(so3) is the cyclically symmetric q-deformation of the universal enveloping algebra U(so3) of the Lie algebra so3 which is not a Drinfeld-Jimbo quantum algebra. Finite-dimensional irreducible representations of Uq(sl3) are decomposed into irreducible representations of U'q(so3). An explicit expression for the matrix of the transition from the Gel'fand-Tsetlin basis for Uq(sl3) to the bases of irreducible representations of U'q(so3) is calculated for representations of Uq(sl3) with highest weights (l,0,0). Entries of this matrix are expressed in terms of products of dual q-Krawtchouk polynomials and dual q-Hahn polynomials. Expressions for representation operators of Uq(sl3) in the U'q(so3) basis are given.

Representations of the Quantum Algebra U q (u n , 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U q (u n, 1) of the quantized universal enveloping algebra U q (gl n + 1). We investigate the principal series of representations of U q (u n, 1) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U q (u n, 1) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U q (u n, 1) has finite-dimensional irreducible *-representations.

Some Tensor Product Representations of U q (s $${\hat l}{\kern 1pt} $$ 2) at Roots of Unity

When q is a root of unity, a triangular decomposition of Uq(s\({\hat l}{\kern 1pt} \)2) is given and irreducibility conditions concerning some tensor product representations of Uq(s\({\hat l}{\kern 1pt} \)2) are presented. Their connection with physics is also pointed out.

Commutation relations of vertex operators related with the spin representation of Uq(Dn

A new scheme to study the XXZ spin chain in massive regime is introduced by Davies et al in [1]. In this scheme, which is called symmetry approach, the space of states is identified with some Uq(A\_ ) -module. Under this identification, the XXZ Hamiltonian can be constructed on this module as an operator which commutes with the action of Uq(A '). Furthermore, the row transfer matrix and creation (annihilation) operators can be constructed by using vertex operators. In a similar way to [1], the higher spin chain [2] and the models related to the vector representation ofUq(A^) [3], Uq(B^ U q ( D ( n } ) [5] are studied. In this paper, we consider the model related with the spin representation of Uq(Dn ). In physics the model that we consider here is explained as follows. It is a one dimensional quantum spin chain model constructed from the spin representation of Uq(Dn )• The Hamiltonian acts on the space of the infinite tensor product

Representations of the cyclically symmetric q-deformed algebra Uq(so3)

An algebra homomorphism ψ from the nonstandard q-deformed (cyclically symmetric) algebra Uq(so3) to the extension Uq(sl2) of the Hopf algebra Uq(sl2) is constructed. Not all irreducible representations (IR) of Uq(sl2) can be extended to representations of Uq(sl2). Composing the homomorphism ψ with irreducible representations of Uq(sl2) we obtain representations of Uq(so3). Not all of these representations of Uq(so3) are irreducible. Reducible representations of Uq(so3) are decomposed into irreducible components. In this way we obtain all IR of Uq(so3) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so3 when q → 1.

REPRESENTATIONS OF LIE ALGEBRAS FROM REPRESENTATIONS OF QUANTUM GROUPS

In paper [*] (P. Moylan: Czech. J. Phys., Vol. 47 (1997), p. 1251) we gave an explicit embedding of the three dimensional Euclidean algebra ɛ(2) into a quantum structure associated with Uq(so(2, 1)). We used this embedding to construct skew symmetric representations of ɛ(2) out of skew symmetric representations of Uq(so(2, 1)). Here we consider generalizations of the results in [*] to a more complicated quantum group, which is of importance to physics. We consider Uq(so(3, 2)), and we show that, for a particular representation, namely the “Rac” representation, many of the results in [*] carry over to this case. In particular, we construct representations of so(3, 2), P(2, 2), the Poincare algebra in 2+2 dimensions, and the Poincare algebra out of the Rac representation of Uq(so(3, 2)). These results may be of interest to those working on exploiting representations of Uq(so(3, 2)), like the Rac, as an example of kinematical confinement for particle constituents such as the quarks.

The q -boson realizations of the quantum groups U q ( B n )

We give explicit realization for the quantum enveloping algebras U q(B n). In these formulae the generators of the algebra are expressed by means of 2n−1 canonical q-boson pairs and one auxiliary representation of U q(B n−1)

Nonstandard U q (so 3 ) and U q (so 4 ): tensor products of representations, oscillator realizations and roots of unity

Nonstandard q-deformed algebras U q(so3) and U q(so4), which can be embedded into U q(sl3) and U q(sl4) and are coideals in them, are considered. It is shown how to multiply finite dimensional representations of U q(so3) when q is positive. Homomorphisms from U q(so3) and U q(so4) to the q-oscillator algebras are given. By making use of these homomorphisms, irreducible representations of U q(so3) and U q(so4) for q equal to a root of unity are obtained.

Bosonization of Admissible Representation of Uq(sl2) at level -1/2

We construct a representation of Uq(sl2) at level -1/2 by using the bosonic Fock spaces. The irreducible modules are obtained as the kernel of a certain operator, in contrast to the construction by Feingold and Frenkel for q = 1 where such a procedure is not necessary. We also bosonize the q-vertex operators associated with the vector representation.

Free field representation of osp(2|1) and Uq( osp(2|1)) and N = 1 ( q-)superstring correlation functions

The free field realization of irreducible representations of osp(2|1) is constructed, by a unified and systematic scheme. The q-analog of this unified scheme is used to construct q-free field realization of irreducible representations of U q ( osp(2|1)). By using these realization, the two-point function of N = 1 superconformal ( q-superconformal) model based on osp(2|1) ( U q ( osp(2|1))) symmetry have been calculated.

Finite-dimensional representations of the quantum group GLp,q(2) using the exponential map from Up,q(gl(2))

Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra Up,q(gl(2)) to the quantum group GLp,q(2), we show how the (2j+1)-dimensional representations of GLp,q(2) can be obtained by 'exponentiating' the well known (2j+1)-dimensional representations of Up,q(gl(2)) for j=1, 3/2, ...; j= 1/2 corresponds to the defining two-dimensional T-matrix. The earlier results on the finite-dimensional representations of GLq(2) and SLq(2) (or SUq(2)) are obtained when p=q. Representations of Uq,q(2)(q in C mod R) and Uq(2)(q in R mod (0)) are also considered. The structure of the Clebsch-Gordan matrix for Up,q(gl(2)) is studied. The same Clebsch-Gordan coefficients are applicable in the reduction of the direct product representations of the quantum group GLp,q(2).

The q-boson realizations of the quantum groups Uq(B2) and Uq(C2)

We give explicit realization of formulae of canonical realization for the quantum enveloping algebrasUq(B2)~Uq(so(5)) andUq(C2)~Uq(sp(4)). In these formulae the generators of the algebra are expressed by means of 3 canonicalq-boson pairs and one auxiliary representation ofUq(gl(2)).