Tetrahedron Equation and Quantum R Matrices for Spin Representations of $${B^{(1)}_n}$$ B n ( 1 ) , $${D^{(1)}_n}$$ D n ( 1 ) and $${D^{(2)}_{n+1}}$$ D n + 1 ( 2 )

Abstract

(2) n+1 ATSUO KUNIBA AND SERGEY SERGEEV Dedicated to Professor Vladimir Bazhanov on the occasion of his sixtieth birthday. AbstractIt is known that a solution of the tetrahedron equation generates infinitely many solutions of the Yang-Baxter equation via suitable reductions. In this paper this scheme is applied to an oscillator solution of the tetrahedron equation involving bosons and fermions by using special 3d boundary conditions. The resulting solutions of the Yang-Baxter equation are identified with the quantum R matrices for the spin representations of B

Tetrahedron equation and quantumRmatrices for q-oscillator representations

“…(Proposition 2 corresponds to taking the classical part to be U q (C n ).) As far as χ 1 (z)| and |χ 1 (1) are concerned, the above correspondence agrees with the observation made in [11,Remark 7.2] on the similar result concerning a 3d L operator. With regard to χ 2 (z)| and |χ 2 (1) , the relevant affine Lie algebras A (2) 2n and C (1) n in this paper are the subalgebras of B 4.…”

“…This paper is a summary and supplement of the recent result [9] by the authors, which is motivated by the earlier works [13,2,11]. The tetrahedron equation (1) [14] is a three dimensional generalization of the Yang-Baxter equation [1].…”

“…The tetrahedron equation (1) [14] is a three dimensional generalization of the Yang-Baxter equation [1]. In [11] a new prescription was proposed to reduce it to the Yang-Baxter equation R 1,2 R 1,3 R 2,3 = R 2,3 R 1,3 R 1,2 by using the special boundary vectors defined by (3) and (10). Applied to a particular solution of the tetrahedron equation (3d L operator [2]), the reduction was shown [11] to give the quantum R matrices for the spin representations [12].…”

“…We recall the prescription which produces an infinite family of solutions to the Yang-Baxter equation from a solution to the tetrahedron equation based on special boundary vectors [11].…”

Tetrahedron and 3D reflection equations from quantized algebra of functions

“…The tetrahedron equation is more challenging but many efforts and results continue to emerge until today. See [3,23,32,5,30,21,24,12,18,25,37,19,36,35,20,7,6,29] for example. In this paper we study the tetrahedron equation and its generalizations based on representation theory of quantized algebra of functions.…”