Quantum symmetric Kac–Moody pairs

By S. Kolb. 2012

258 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fe6d44978cf12c0862df62828e844371d94caea1

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Citations in the corpus

Affine flag varieties and quantum symmetric pairs (2016, unknown): “ ı-version [BLW14, LW15, FL15]. 10 Z.FAN,C.LAI,Y.LI,L.LUO,ANDW.WANG There is also a totally diﬀerent, purely algebraic, construction [BW16] of canonical bases for general quantum symmetric pairs (cf. [Ko14]). That approach does not establish the positivity of canonical bases. 1.3.3. An overview of P”
Geometric Schur Duality of Classical Type (2014, methodology): “ show that (by restricting to one idempotented summand) monomial bases of U˙ ıD εn+1 and Q K ı[D εn+1] are parametrized by two sets which are in natural bijection (by passing to U(gl(N −1))−D 0 using [Le02, K14] and Q K−[D 0], where QK − is the negative half of QK = U˙ (gl(N −1)) constructed in [BLM”
A new approach to Kazhdan-Lusztig theory of type ${\bf B}$ via quantum symmetric pairs (2013, methodology): “ quantum symmetric pairs remains to be developed. The quantum symmetric pairs (U,Uı) and (U,U) are just two examples of general quantum symmetric pairs of ﬁnite or more generally Kac-Moody type (see [Le, Ko]). The general quantum symmetric pairs aﬀord presentations similar to the ones given in this ”
Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality (2013, unknown): “To make the precise connection to [55] and [48] we consider the following quantization of Θ.”