Universal K-matrix for quantum symmetric pairs

Abstract

Let g be a symmetrizable Kac-Moody algebra and let Uq(g) denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras Bc,s of Uq(g) have a universal K-matrix if g is of finite type. By a universal K-matrix for Bc,s we mean an element in a completion of Uq(g) which commutes with Bc,s and provides solutions of the reflection equation in all integrable Uq(g)-modules in category O. The construction of the universal K-matrix for Bc,s bears significant resemblance to the construction of the universal R-matrix for Uq(g). Most steps in the construction of the universal K-matrix are performed in the general Kac-Moody setting.In the late nineties T. tom Dieck and R. Häring-Oldenburg developed a program of representations of categories of ribbons in a cylinder. Our results show that quantum symmetric pairs provide a large class of examples for this program.2010 Mathematics Subject Classification. 17B37; 81R50.For the symmetric pairs (sl 2N , s(gl N × gl N )) and (sl 2N +1 , s(gl N × gl N +1 )) the construction of K coincides with the construction of the B c,s -module homomorphisms T M in [BW13] up to conventions. The longest element w 0 induces a diagram automorphism τ 0 of g and of U q (g). Any U q (g)-module M can be twisted by an algebra automorphism ϕ : U q (g) → U q (g) if we define u⊲m = ϕ(u)m for all u ∈ U q (g), m ∈ M . We denote the resulting twisted module by M ϕ . We show in Corollary 7.7 that the element K defines a B c,s -module isomorphismfor all finite-dimensional U q (g)-modules M . Alternatively, this can be written asThe construction of the bar involution for B c,s , the intertwiner X, and the B c,smodule homomorphism K are three expected key steps in the wider program of canonical bases for quantum symmetric pairs proposed in [BW13]. The existence of the bar involution was explicitly stated without proof and reference to the parameters in [BW13, 0.5] and worked out in detail in [BK15]. Weiqiang Wang has informed us that he and Huanchen Bao have constructed X and K ′ M independently in the case X = ∅, see [BW15].In the final Section 9 we address the crucial problem to determine the coproduct ∆(K) in U (2) . The main step to this end is to determine the coproduct of the quasi K-matrix X in Theorem 9.4. Even for the symmetric pairs (sl 2N , s(gl N × gl N )) and (sl 2N +1 , s(gl N × gl N +1 )), this calculation goes beyond what is contained in [BW13]. It turns out that if τ τ 0 = id then the coproduct ∆(K) is given by formula (1.1). Hence, in this case K is a universal K-matrix as defined above for the coideal subalgebra B c,s . If τ τ 0 = id then we obtain a slight generalization of the properties (1) and (2) of a universal K-matrix. Motivated by this observation we introduce the notion of a ϕ-universal K-matrix for B if ϕ is an automorphism of a braided bialgebra H and B is a right coideal subalgebra, see Section 4.3. With this terminology it hence turns out in Theorem 9.5 that in general K is a τ τ 0 -universal Kmatrix for B c,s . The f...

Multiparameter quantum Schur duality of type B

“…On the other hand, there is a notion of quantum symmetric pair, (U, U ı ), where U ı is a coideal subalgebra of U. The algebra U ı allows for some freedom of choices of parameters; see Letzter [Le99] (also see [BK15]). We make a particular choice of the parameters for U ı in this paper depending on p and q.…”

Quiver varieties and symmetric pairs

“…As we learnt from Weiqiang Wang, algebraic K-matrix for quantum symmetric pairs of type AIII/IV first appeared in [BW13]. The relationship between the Kmatrix in this section and the algebraic ones in [BaK16] is not clear. 5.6.…”

“…A new canonical basis was constructed for certain tensor modules of U ′ q (k) in [BW13], for idempotented U ′ q (k) in [BKLW,LW15], and finally a general theory of canonical basis for k of any type was obtained in [BW16]. These works have inspired many developments in various directions, such as categorification [BSWW] and K-matrix [BaK16]. In the work [BKLW], there is a geometric realization of U ′ q (k) of type AIII/AIV without black vertices by using n-step isotropic flag varieties, in the spirit of Beilinson, Lusztig and MacPherson's influential work [BLM].…”

A topological origin of quantum symmetric pairs

“…Recall that various versions of the reflection equation appeared in the works [Nou96,NS95,NDS97,Dij96]. It was only later realised by M. Balagovic and S. Kolb, through their algebraic construction of universal K-matrices, that there is a general framework of twisted reflection equations which unifies all different reflection equations [BK,Remark 9.7]. For a quasi-triangular Hopf algebra H with coideal subalgebra B one fixes an additional Hopf algebra involution φ of H, such that (φ ⊗ φ)(R) = R, called the twist.…”

“…From such a character they construct a quantum symmetric pair, streamlining the approach of Noumi-Sugitani-Dijkhuizen [KS09]. Furthermore, M. Balagovic and S. Kolb showed that any pair in Letzter's classification carries a canonical solution to a reflection equation, called a universal K-matrix [BK,Kol17]. Such solutions can be used for applications to low-dimensional topology in the spirit of the celebrated Reshetikhin-Turaev knot invariants [tD98,tDHO98].…”