• Corpus ID: 233220003

The bar involution for quantum symmetric pairs -- hidden in plain sight

@inproceedings{Kolb2021TheBI,
  title={The bar involution for quantum symmetric pairs -- hidden in plain sight},
  author={Stefan Kolb},
  year={2021}
}
  • S. Kolb
  • Published 13 April 2021
  • Mathematics, Physics
We show that all quantum symmetric pair coideal subalgebras Bc of Kac-Moody type have a bar involution for a suitable choice of parameters c. The proof relies on a generalized notion of quasi K-matrix. The proof does not involve an explicit presentation of Bc in terms of generators and relations. 
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References

SHOWING 1-10 OF 19 REFERENCES
A SERRE PRESENTATION FOR THE QUANTUM GROUPS
Let (U, ) be a quasi-split quantum symmetric pair of arbitrary Kac–Moody type, where “quasi-split” means the corresponding Satake diagram contains no black node. We give a presentation of the group
The bar involution for quantum symmetric pairs
We construct a bar involution for quantum symmetric pair coideal subalgebras $B_{\mathbf{c},\mathbf{s}}$ corresponding to involutive automorphisms of the second kind of symmetrizable Kac-Moody
Factorisation of quasi K-matrices for quantum symmetric pairs
The theory of quantum symmetric pairs provides a universal K-matrix which is an analog of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal
Symmetric Pairs for Quantized Enveloping Algebras
Abstract Let θ be an involution of a semisimple Lie algebra g , let g θ denote the fixed Lie subalgebra, and assume the Cartan subalgebra of g has been chosen in a suitable way. We construct a
NAZAROV-WENZL ALGEBRAS, COIDEAL SUBALGEBRAS AND CATEGORIFIED SKEW HOWE DUALITY
We describe how certain cyclotomic Nazarov-Wenzl algebras occur as endomorphism rings of projective modules in a parabolic version of BGG category O of type D. Furthermore we study a family of
Introduction to Quantum Groups
We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra
Quasitriangular coideal subalgebras of Uq(g) in terms of generalized Satake diagrams
Let g be a finite‐dimensional semisimple complex Lie algebra and θ an involutive automorphism of g . According to Letzter, Kolb and Balagović the fixed‐point subalgebra k=gθ has a quantum counterpart
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