Tetrahedron and 3D reflection equations from quantized algebra of functions

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 45; no. 46; pp. 465206 - 27
• Main Authors: Kuniba, Atsuo, Okado, Masato
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 23.11.2012; IOP
• Subjects: Algebra; Combinatorial analysis; Exact sciences and technology; Functions (mathematics); Mathematical analysis; Physics; Polynomials; quantized algebra of function; Reflection; reflection equation; Representations; tetrahedron equation; Tetrahedrons; three dimension; Three dimensional; R matrix; Tetrahedron; K matrix; Matrix elements; Function algebra
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/45/46/465206

Soibelman's theory of quantized function algebra Aq(SLn) provides a representation theoretical scheme to construct a solution of the Zamolodchikov tetrahedron equation. We extend this idea originally due to Kapranov and Voevodsky to Aq(Sp2n) and obtain the intertwiner K corresponding to the quartic Coxeter relation. Together with the previously known three-dimensional (3D) R matrix, the K yields the first ever solution to the 3D analogue of the reflection equation proposed by Isaev and Kulish. It is shown that matrix elements of R and K are polynomials in q and that there are combinatorial and birational counterparts for R and K. The combinatorial ones arise either at q = 0 or by tropicalization of the birational ones. A conjectural description for type B and F4 cases is also given.