nLab reflection equation algebra

Idea

The role of quantum Yang-Baxter equation in integrable systems is in the presence of certain type of boundary conditions complemented by a reflection equation.

Some formulations

Reflection equation

Given a solution $R$ of the Yang-Baxter equation, the reflection equation (with additive spectral parameter) reads

$$R(u-v) K_1 (u) R(u+v) K_2 (v) = K_2 (v) R(u+v) K_1 (u) R(u-v)$$

Reflection equation algebra

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Related concepts

• quantum inverse scattering method
• quantum symmetric pair

References

The equation is physically motivated and introduced in

• I. V. Cherednik: Factorizing particles on a half-line and root systems, Theoretical and Mathematical Physics 61 (1984) 977-983 [doi:10.1007/BF01038545, doi:10.1007/BF01038545]

See also:

• E. K. Sklyanin, Boundary conditions for integrable quantum systems, Journal of Physics A 21(10) (1988) 2375–2389. doi

• P. P. Kulish, Reflection equation algebras and quantum groups, in: Quantum and Non-Commutative Analysis, Mathematical Physics Studies 16 (1993) 207–220 doi

• P. P. Martin, D. Woodcock, D. Levy, A diagrammatic approach to Hecke algebras of the reflection equation. Journal of Physics A 33(6) (2000) 1265–1296 doi

• Anastasia Doikou, From affine Hecke algebras to boundary symmetries, Nuclear Physics B 725 (2005) 493–530 doi

• Dimitri Gurevich?, Pavel Pyatov, Pavel Saponov, Reflection equation algebra in braided geometry, Journal of Generalized Lie Theory and Applications 2 (2008) No. 3, 162–174 pdf

• Andrey Mudrov, Characters of $U_q(gl(n))$-reflection equation algebra, Lett. Math. Phys. 60:3, 283–291 (2002) doi

• Christian Schwiebert, Extended reflection equation algebras, the braid group on a handlebody and associated link polynomials, J. Math. Physics 02/1994; doi hep-th/9402051; Reflection equation and link polynomials for arbitrary genus solid tori, hep-th/9301023

• Stefan Kolb, J. V. Stokman, Reflection equation algebras, coideal subalgebras, and their centres Sel. Math. New Ser. 15, 621–664 (2009) doi

• J. Donin, P. P. Kulish, A. I. Mudrov, On a universal solution to the reflection equation, Lett. Math. Phys. 63 (2003) 179–194 doi

• Martina Balagović, Stefan Kolb, Universal K-matrix for quantum symmetric pairs, Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, 747 (2016) doi