Abstract. The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac-Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras. They give rise to triangular decompositions, including a quantum analog of the Iwasawa decomposition, and they can be written explicitly in terms of generators and relations. Moreover, their centers and their specializations are determined. The constructions follow G. Letzter's theory of quantum symmetric pairs for semisimple Lie algebras. The main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras due to Kac and Wang. The resulting theory comprises various classes of examples which have previously appeared in the literature, such as q-Onsager algebras and the twisted q-Yangians introduced by Molev, Ragoucy, and Sorba.

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.

It is shown that quantized irreducible flag manifolds possess a canonical q-analogue of the de Rham complex. Generalizing the well-known situation for the standard Podleś' quantum sphere this analogue is obtained as the universal differential calculus of a distinguished first order differential calculus. The corresponding differential d can be written as a sum of differentials ∂ and∂. The universal differential calculus corresponding to the first order differential calculi d, ∂, and∂ are given in terms of generators and relations. Relations to well-known quantized exterior algebras are established. The dimensions of the homogeneous components are shown to be the same as in the classical case. The existence of a volume form is proven.

We construct braid group actions on coideal subalgebras of quantized enveloping algebras which appear in the theory of quantum symmetric pairs. In particular, we construct an action of the semidirect product of Z n and the classical braid group in n strands on the coideal subalgebra corresponding to the symmetric pair (sl 2n (C), sp 2n (C)). This proves a conjecture by Molev and Ragoucy. We expect similar actions to exist for all symmetric simple complex Lie algebras. The given actions are inspired by Lusztig's braid group action on quantized enveloping algebras and are defined explicitly on generators. Braid group and algebra relations are verified with the help of the package Quagroup within the computer algebra program GAP.