Irreducible Representations Of Uq Research Articles

Invariants of knots and links at roots of unity

We present a comprehensive classification of invariants of knots and links associated with irreducible representations of Uq(slN), when the parameter of quantization q is a root of unity. We demonstrate that, besides the standard colored HOMFLY-PT invariants, which are associated with representations with highest and lowest weights, non-trivial invariants can be associated only with nilpotent representations with parameters. While such invariants were discussed for Uq(sl2) and are known as the ADO invariants, we generalize them to the general Uq(slN) case and discuss their relations with standard invariants at particular values of parameters.

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.

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.

Representations of the cyclically symmetric q-deformed algebra soq(3)

An algebra homomorphism ψ from the nonstandard q-deformed (cyclically symmetric) algebra Uq(so3) to the extension Ûq(sl2) of the Hopf algebra Uq(sl2) is constructed. Not all irreducible representations of Uq(sl2) can be extended to representations of Ûq(sl2). Composing the homomorphism ψ with irreducible representations of Ûq(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 irreducible representations of Uq(so3) when q is not a root of unity. A part of these representations turns into irreducible representations of the Lie algebra so3 when q→1. Representations of the other part have no classical analog. Using the homomorphism ψ it is shown how to construct tensor products of finite-dimensional representations of Uq(so3). Irreducible representations of Uq(so3) when q is a root of unity are constructed. Some of them are obtained from irreducible representations of Ûq(sl2) by means of the homomorphism ψ.

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.

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.

Irreducible Representations of the Quantum Group Uq(m + n) in Uq(m) ⊕ Uq(n) Basis11The project supported by the Excellent Young Teacher's Foundation of the Education Committee of China and National Natural Science Foundation of China

Irreducible representations of Uq(m + n) in Uq(m) ⊕ Uq(n) basis are discussed. The noncanonical basis of Uq(m + n) are expanded in terms of its canonical basis. The expansion coefficients or subduction coefficients of Uq(m + n) ↓ Uq(m) ⊕ Uq(n) are derived from the subduction coefficients of Hecke algebras.