Fusion rules of the lowest weight representations of {\rm {osp}}_q(1|2) at roots of unity: polynomial realization

Abstract

The degeneracy of the lowest weight representations of the quantum superalgebra osp q (1|2) and their tensor products at exceptional values of q is studied. The main features of the structures of the finite dimensional lowest weight representations and their fusion rules are illustrated using realization of group generators as finite-difference operators acting in the space of the polynomials. The complete fusion rules for the decompositions of the tensor products at roots of unity are presented. The appearance of indecomposable representations in the fusions is described using Clebsh-Gordan coefficients derived for general values of q and at roots of unity.In this article we clarify the mentioned aspects in visual form for the lowest weight representations of the quantum superalgebra osp q (1|2) [2,4,8,9,10,15,17,18,20]. The orthosymplectic superalgebras osp(n|2m) (and their quantum deformations) are actual in CFT, in the theory of integrable models, in the string theory (see the works [4,21] and references therein). The change of the representation's spectrum at roots of unity brings to new peculiarities, for instance to new solutions of the Yang-Baxter equations in the theory of 2d integrable models [14,16,17].There is a correspondence between the irreps of the graded algebra osp q (1|2) and the irreps of the algebra sl iq 1/2 (2) [4,15,18,17] at general q, which allows to use some techniques evolved from the study of the quantum deformation of sl(2) [3,4,12,22] for investigation of the super-algebra osp q (1|2). And our approach consists in studying the finite-dimensional representations and their tensor products at general values of q, to find out at the complex q-plane the "singular points" (located on the unit circle) of these representations or their tensor products' decompositions.The finite-difference realization of the group generators, acting on the space of the polynomials, provides clear and compact description and answers the purposes in the best way. The approach of the projection operators gives a simple understanding why degeneracies appear in the decompositions of the representations' tensor products. Being comprehensible enough constructions presented here are adapted especially for the physical applications.The paper is organized as follows: in the second section the algebra generators and the coproduct in the mentioned polynomial realization are presented and in the third section some examples of the representations and their tensor products are considered in details. The discussed patterns illustrate the principal cases, including multiple tensor products of the irreps and indecomposable representations. In the next section an analysis of the fusion rules from the viewpoint of the projector operators is performed. The last sections are devoted to general analysis and conclusions based on the observations of the previous sections, accompanied with explicit proof of the general formulas. In the subsections 5.1, 5.2 we represent indecomposable representations emerging from the tensor product's...

Solutions to the Yang–Baxter equations with symmetry: Lax operators

“…Such solutions to YBE for the cases when q is a root of unity can meet also for the higher dimensional representations (see the subsection 4.2). As it is known from the analysis of the quantum (super-)algebras [20,21], the structure of the set of the non reducible representations and their fusion rules are deformed, when q is a root of unity, so the cases, like to the (4.8) require separate investigation.…”

Series of the solutions to Yang–Baxter equations: Hecke type matrices and descendant R-, L-operators

“…As a yet another confirmation of the mentioned observation could be served the existence of a series of solutions to the YBE with symmetry of the quantum super-algebra osp q (1|2) defined on the spin-irreps, which differs from the known solutions [20,22,26], and the discussion done in the Section 2 of this work demonstrates the exact derivation of this series. The similar solutions (Hecke type R-operators), as it is known, exist for the quantum algebra sl q (2) [8,10,19,29,28], and this reflects the circumstance that there is an explicit correspondence between the representations of the quantum algebras sl q (2) and osp q (1|2), providing that q → i √ q [23,24,26,27]. Then in Section 3 a descendant series of the men-tioned solutions is constructed.…”

“…The set of the composed states, fitting to the truncated tensor products of the spin-irreps, can be built for each case separately. For the simple case n = 2, the normalized (but not orthogonal) states of U r 2 −1 , induced from the initial sublattice, are determined elementary, using the relation (1.5): Note, that at the exceptional values of the deformation parameter of the quantum group (i.e, when q is a root of unity), the specter of the irreducible representations is restricted, higher spin irreps are deforming, and new indecomposable representations are arising [25], and correspondingly, the fusion rules also are deformed, but however in this case also the solutions of YBE defined on the composed states can be found, properly defining the centralisers or the projection operators (see [30,27] and the references therein). As example, at q 4 = 1 the descendant matrix As it is known, by means of the YBE solutions the braid group representations can be realised, and they can be employed to obtain the link and knot invariants [19,21,32].…”

“…Here ⊗ denotes the graded tensor product, and I is a unity operator. The quadratic Casimir operator can be written as c = (q [23,27]. Below we shall use the notation q r = [(−1) r + 1] iπ 4 log q for the factor arising in the case of even dimensional irreps.…”

New solutions to the -invariant Yang–Baxter equations at roots of unity: Cyclic representations

“…For constructing the projectors explicitly at first one has to determine the fusion rules at roots of unity [3]. Using the detailed rules, formulated in [16] for the highest/lowest weight indecomposable representations, in our previous paper [17] by means of the direct construction of the projection operators, we see that the consideration of the highest/lowest weight indecomposable representations even for the simplest case q 4 = 1 gives a large amount of various new solutions.…”