Tetrahedron and 3D reflection equations from quantized algebra of functions

Abstract

Soibelman's theory of quantized function algebra A q (SL n ) 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 A q (Sp 2n ) and obtain the intertwiner K corresponding to the quartic Coxeter relation. Together with the previously known 3-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 the type B and F 4 cases is also given.

Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation

“…For (3.10) see [16] where the definitions of R and R abc ijk are identical with this paper. The second relation means R abc ijk = R cba kji .…”

“…It was also given in [2] and the two were identified in [16]. It defines a vertex model on a cubic lattice whose edges are assigned with spin variables in Z ≥0 and vertices with polynomial Boltzmann weights in q.…”

Tetrahedron equation and quantumRmatrices for infinite-dimensional modules of $U_q(A^{(1)}_1)$ and $U_q (A^{(2)}_2)$

“…Here the bracket means the evaluation with respect to the 3rd component. Denoting (2.7) just by R(z) ∈ End(F ⊗ F ), it is also convenient to introducě 8) where P (u ⊗ v) = v ⊗ u is the linear operator exchanging the components and ̺(z) is an arbitrary scalar function. Then the Yang-Baxter equation takes another familiar form:…”

“…The indices of R and L signify the components of the tensor product on which these operators act non trivially. Viewed as an equation on R, (5.4) is equivalent [8] to the intertwining relation of the irreducible representations of the quantized coordinate ring A q (sl 3 ) [6] in the sense that the both lead to the same solution given in (2.11) up to an overall normalization. 5 The eigenvalues of M 2,2 d (q) with odd d is not directly derivable from (3.9) since the matrix consists of the sub-matricesŘ ǫ 1 ,ǫ 2 (z) with ǫ 1 ǫ 2 = −1 which correspond, due to (4.2), to the situation α = β in (3.9).…”

“…We apply the reduction scheme based on the special boundary vectors [10] to the solution R of the tetrahedron equation studied in [6,4,3,12,8] 1 . It acts on the tensor cube of the q-oscillator Fock space F = m≥0 Q(q)|m .…”