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...

Abstract. We construct a bar involution for quantum symmetric pair coideal subalgebras B c,s corresponding to involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras. To this end we give unified presentations of these algebras in terms of generators and relations, extending previous results by G. Letzter and the second-named author. We specify precisely the set of parameters c for which such an intrinsic bar involution exists.

We continue the study of the lower central series L i (A) and its successive quotients B i (A) of a noncommutative associative algebra A,for A a quotient of the free algebra on two or three generators by the two-sided ideal generated by a generic homogeneous element. We prove that it is isomorphic to a certain quotient of Kähler differentials on the non-smooth variety associated to the abelianization of A.

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In this paper, we describe the irreducible representations in category O of the rational Cherednik algebra Hc(G 12 , h) associated to the complex reflection group G 12 with reflection representation h for an arbitary complex parameter c. In particular, we determine the irreducible finite dimensional representations and compute their characters.Next, there is a simple equivalence of categories O c → O −c , enabling us to assume c > 0. Also well known but more involved are the equivalences of categories O 1/d → O r/d for r, d > 0, d = 2, and r and d relatively prime, and O c → O c+1 . These equivalences Φ c,c ′