Superalgebra Uq Research Articles

From quantum loop superalgebras to super Yangians

The goal of this paper is to generalize a statement by Drinfeld, asserting that Yangians can be constructed as limit forms of the quantum loop algebras, to the super case. We establish a connection between quantum loop superalgebra and super Yangian of the general linear Lie superalgebra glM|N in RTT type presentation. In particular, we derive the Poincaré-Birkhoff-Witt(PBW) theorem for the quantum loop superalgebra Uq(LglM|N). Additionally, we investigate the application of the same argument to twisted super Yangian of the ortho-symplectic Lie superalgebra. For this purpose, we introduce the twisted quantum loop superalgebra as a one-sided coideal of Uq(LglM|2n) with respect to the comultiplication.

Crystal base of the negative half of the quantum superalgebra [formula omitted

We construct a crystal base of Uq(gl(m|n))−, the negative half of the quantum superalgebra Uq(gl(m|n)). We give a combinatorial description of the associated crystal Bm|n(∞), which is equal to the limit of the crystals of the (q-deformed) Kac modules K(λ). We also construct a crystal base of a parabolic Verma module X(λ) associated with the subalgebra Uq(gl(0|n)), and show that it is compatible with the crystal base of Uq(gl(m|n))− and the Kac module K(λ) under the canonical embedding and projection of X(λ) to Uq(gl(m|n))− and K(λ), respectively.

On generators and defining relations of quantum affine superalgebra Uq(𝔰𝔩̂m|n)

Two presentations of quantum affine superalgebras were introduced by Yamane in [On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. Res. Inst. Math. Sci. 35 (1999) 321–390], which were called Drinfeld–Jimbo realization and Drinfeld realization. Drinfeld realization contains infinite sequences of generators and relations. In this paper, we consider the Drinfeld realization of quantum affine superalgebra [Formula: see text] associated to type [Formula: see text] and define a simple algebra [Formula: see text] generated by only a finite part of these sequences of quantum affine superalgebra [Formula: see text]. We show that the algebra [Formula: see text] is isomorphic to the quantum affine superalgebra [Formula: see text]. Using the above isomorphism, we prove there exists an isomorphism between the two realizations.

Boson-Fermion correspondence, QQ-relations and Wronskian solutions of the T-system

It is known that there is a correspondence between representations of superalgebras and ordinary (non-graded) algebras. Keeping in mind this type of correspondence between the twisted quantum affine superalgebra Uq(gl(2r|1)(2)) and the non-twisted quantum affine algebra Uq(so(2r+1)(1)), we proposed, in the previous paper [1], a Wronskian solution of the T-system for Uq(so(2r+1)(1)) as a reduction (folding) of the Wronskian solution for the non-twisted quantum affine superalgebra Uq(gl(2r|1)(1)). In this paper, we elaborate on this solution, and give a proof missing in [1]. In particular, we explain its connection to the Cherednik-Bazhanov-Reshetikhin (quantum Jacobi-Trudi) type determinant solution known in [2]. We also propose Wronskian-type expressions of T-functions (eigenvalues of transfer matrices) labeled by non-rectangular Young diagrams, which are quantum affine algebra analogues of the Weyl character formula for so(2r+1). We show that T-functions for spinorial representations of Uq(so(2r+1)(1)) are related to reductions of T-functions for asymptotic typical representations of Uq(gl(2r|1)(1)).

New R-matrices with non-additive spectral parameters and integrable models of strongly correlated fermions

We present a general formula for constructing R-matrices with non-additive spectral parameters associated with a type-I quantum superalgebra. The spectral parameters originate from two one-parameter families of inequivalent finite-dimensional irreducible representations of the quantum superalgebra upon which the R-matrix acts. Applying to the quantum superalgebra Uq(gl(2|1)), we obtain the explicit expression for the Uq(gl(2|1))− invariant R-matrix which is of non-difference form in spectral parameters. Using this R­ matrix we derive a new two-parameter integrable model of strongly correlated electrons with pure imaginary pair hopping terms.

Howe duality for quantum queer superalgebras

We establish a new Howe duality between a pair of quantum queer superalgebras (Uq−1(qn), Uq(qm)). The key ingredient is the construction of a non-commutative analogue Aq(qn,qm) of the symmetric superalgebra S(Cmn|mn) with the use of quantum coordinate queer superalgebra. It turns out that this superalgebra is equipped with a Uq−1(qn)⊗Uq(qm)-supermodule structure that admits a multiplicity-free decomposition. We also show that the (Uq−1(qn),Uq(qm))-Howe duality implies the Sergeev-Olshanski duality.

A note on q-oscillator realizations of Uq(gl(M|N)) for Baxter Q-operators

We consider asymptotic limits of q-oscillator (or Heisenberg) realizations of Verma modules over the quantum superalgebra Uq(gl(M|N)), and obtain q-oscillator realizations of the contracted algebras proposed in [1]. Instead of factoring out the invariant subspaces, we make reduction on generators of the q-oscillator algebra, which gives a shortcut to the problem. Based on this result, we obtain explicit q-oscillator representations of a Borel subalgebra of the quantum affine superalgebra Uq(glˆ(M|N)) for Baxter Q-operators.

Commutation relations of vertex operators for Uq(sl^(M|N))

We consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra Uq(sl^(M|N)) by using bosonization. We show that vertex operators give a representation of the graded Zamolodchikov-Faddeev algebra by direct computation. Invertibility relations of type-II vertex operators for N > M are very similar to those of type-I for M > N.

Reflection K-matrices for a nineteen vertex model with [formula omitted] symmetry

We derive the solutions of the boundary Yang–Baxter equation associated with a supersymmetric nineteen vertex model constructed from the three-dimensional representation of the twisted quantum affine Lie superalgebra Uq[osp(2|2)(2)]. We found three classes of solutions. The type I solution is characterized by three boundary free-parameters and all elements of the corresponding reflection K-matrix are different from zero. In the type II solution, the reflection K-matrix is even (every element of the K-matrix with an odd parity is null) and it has only one boundary free-parameter. Finally, the type III solution corresponds to a diagonal reflection K-matrix with two boundary free-parameters.

Quantum walled Brauer–Clifford superalgebras

We introduce a new family of superalgebras, the quantum walled Brauer–Clifford superalgebras BCr,s(q). The superalgebra BCr,s(q) is a quantum deformation of the walled Brauer–Clifford superalgebra BCr,s and a super version of the quantum walled Brauer algebra. We prove that BCr,s(q) is the centralizer superalgebra of the action of Uq(q(n)) on the mixed tensor space Vqr,s=Vq⊗r⊗(Vq⁎)⊗s when n≥r+s, where Vq=C(q)(n|n) is the natural representation of the quantum enveloping superalgebra Uq(q(n)) and Vq⁎ is its dual space. We also provide a diagrammatic realization of BCr,s(q) as the (r,s)-bead tangle algebra BTr,s(q). Finally, we define the notion of q-Schur superalgebras of type Q and establish their basic properties.

Asymptotic representations and q-oscillator solutions of the graded Yang–Baxter equation related to Baxter Q-operators

We consider a class of asymptotic representations of the Borel subalgebra of the quantum affine superalgebra Uq(glˆ(M|N)). This is characterized by Drinfeld rational fractions. In particular, we consider contractions of Uq(gl(M|N)) in the FRT formulation and obtain explicit solutions of the graded Yang–Baxter equation in terms of q-oscillator superalgebras. These solutions correspond to L-operators for Baxter Q-operators. We also discuss an extension of these representations to the ones for contracted algebras of Uq(glˆ(M|N)) by considering the action of renormalized generators of the other side of the Borel subalgebra. We define model independent universal Q-operators as the supertrace of the universal R-matrix and write universal T-operators in terms of these Q-operators based on shift operators on the supercharacters. These include our previous work on Uq(slˆ(2|1)) case [1] in part, and also give a cue for the operator realization of our Wronskian-like formulas on T- and Q-functions in [2,3].

Free field realization of quantum affine superalgebra $U_q(\widehat{{sl}}(N|1))$Uq(sl̂(N|1))

We construct a free field realization of the quantum affine superalgebra \documentclass[12pt]{minimal}\begin{document}$U_q(\widehat{sl}(N|1))$\end{document}Uq(sl̂(N|1)) for an arbitrary level \documentclass[12pt]{minimal}\begin{document}$k \in {\mathbb C}$\end{document}k∈C.

Orthosymplectic quantum function superalgebras OSP q (2l+1|2n)

By the R-matrix of orthosymplectic quantum superalgebra Uq(osp(2l+1|2n)) in the vector representation, we establish the corresponding quantum Hopf superalgebra OSPq(2l + 1|2n). Furthermore, it is shown that OSPq(2l + 1|2n) is coquasitriangular.

ON THE Uq(osp(1|2n)) AND U-q(so(2n + 1)) UNCOLORED QUANTUM LINK INVARIANTS

Let L be a link and [Formula: see text] its link invariant associated with the vector representation of the quantum (super)algebra Uq(A). Let FL(r, s) be the Kauffman link invariant for L associated with the Birman–Wenzl–Murakami algebra BWMf(r, s) for complex parameters r and s and a sufficiently large rank f. For an arbitrary link L, we show that [Formula: see text] and [Formula: see text] for each positive integer n and all sufficiently large f, and that [Formula: see text] and [Formula: see text] are identical up to a substitution of variables. For at least one class of links FL(-r, -s) = FL(r, s) implying [Formula: see text] for these links.

Basic hypergeometric functions and covariant spaces for even-dimensional representations of Uq[osp(1/2)

Representations of the quantum superalgebra Uq[osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch–Gordan decomposition for the superalgebra Uq[osp(1/2)] in which the representations having no classical counterparts are incorporated. Formulae for these Clebsch–Gordan coefficients are derived, and is observed that they may be expressed in terms of the Q-Hahn polynomials. We next investigate representations of the quantum supergroup OSpq(1/2) which are not well defined in the classical limit. Employing the universal -matrix, the representation matrices are obtained explicitly, and found to be related to the little Q-Jacobi polynomials. Characteristically, the relation Q = −q is satisfied in all cases. Using the Clebsch–Gordan coefficients derived here, we construct new noncommutative spaces that are covariant under the coaction of the even-dimensional representations of the quantum supergroup OSpq(1/2).

Universal T-matrix, representations of OSpq(1∕2) and little Q-Jacobi polynomials

We obtain a closed form expression of the universal T-matrix encapsulating the duality between the quantum superalgebra Uq[osp(1∕2)] and the corresponding supergroup OSpq(1∕2). The classical q→1 limit of this universal T-matrix yields the group element of the undeformed OSpq(1∕2) supergroup. The finite dimensional representations of the quantum supergroup OSpq(1∕2) are readily constructed employing the above-mentioned universal T-matrix and the known finite dimensional representations of the dually related deformed Uq[osp(1∕2)] superalgebra. Proceeding further, we derive the product law, the recurrence relations, and the orthogonality of the representations of the quantum supergroup OSpq(1∕2). It is shown that the entries of these representation matrices are expressed in terms of the little Q-Jacobi polynomials with Q=−q. Two mutually complementary singular maps of the universal T-matrix on the universal R-matrix are also presented.

Eigenvalues of Casimir invariants for Uq[osp(m∣n)

For each quantum superalgebra Uq[osp(m∣n)] with m>2, an infinite family of Casimir invariants is constructed. This is achieved by using an explicit form for the Lax operator. The eigenvalue of each Casimir invariant on an arbitrary irreducible highest weight module is also calculated.

Link Invariants Associated with Gauge Equivalent Solutions of the Yang–Baxter Equation: The One-Parameter Family of Minimal Typical Representations of Uq[gl(2|1)

In this paper we investigate the construction of state models for link invariants using representations of the braid group obtained from various gauge choices for a solution of the trigonometric Yang–Baxter equation. Our results show that it is possible to obtain invariants of regular isotopy (as defined by Kauffman) which may not be ambient isotopic. We illustrate our results with explicit computations using solutions of the trigonometric Yang–Baxter equation associated with the one-parameter family of minimal typical representations of the quantum superalgebra Uq[gl(2|1)]. We have implemented MATHEMATICA code to evaluate the invariants for all prime knots up to 10 crossings.

A Poincaré–Birkhoff–Witt commutator lemma for Uq[gl(m|n)

We present and prove in detail a Poincaré–Birkhoff–Witt commutator lemma for the quantum superalgebra Uq[gl(m|n)].

Deformed CliffordClq(n m) and orthosymplecticUq[osp(2n 1 2m)] superalgebras and their root of unity representations

It is shown that the Clifford superalgebra Cl(n|m) generated by m pairs of Bose operators (odd elements) anticommuting with n pairs of Fermi operators (even elements) can be deformed to Clq(n|m) such that the latter is a homomorphic image of the quantum superalgebra Uq[osp(2n + 1|2m)]. The Fock space F(n|m) of Clq(n|m) is constructed. For q being a root of unity (q = exp(i?l/k)) q-bosons (and q-fermions) are operators acting in a finite-dimensional subspace Fl/k(n|m) of F(n|m). Each Fl/k(n|m) is turned through the above-mentioned homomorphism into an irreducible (root of unity) Uq[osp(2n + 1|2m)] module. For q being a primitive root of unity (l = 1) the corresponding representation is unitary. The module F1/k(n|m) is decomposed into a direct sum of irreducible Uq[sl(m|n)] submodules. The matrix elements of all Cartan?Weyl elements of Uq[sl(m|n)] are given within each such submodule.