Tetrahedron equation and generalized quantum groups

Abstract

We construct 2 n -families of solutions of the Yang-Baxter equation from n-products of threedimensional R and L operators satisfying the tetrahedron equation. They are identified with the quantum R matrices for the Hopf algebras known as generalized quantum groups. Depending on the number of R's and L's involved in the product, the trace construction interpolates the symmetric tensor representations of U q (A (1) n−1 ) and the anti-symmetric tensor representations of U −q −1 (A (1) n−1 ), whereas a boundary vector construction interpolates the q-oscillator representation of U q (D (2) n+1 ) and the spin representation of U −q −1 (D (2) n+1 ). The intermediate cases are associated with an affinization of quantum superalgebras. 1 The formula for it on p194 in [23] contains a misprint unfortunately. Eq. (2.2) here is a correction of it.

Integrable Lattice Spin Models from Supersymmetric Dualities

“…The reader might wonder whether the gauge/YBE could be used for three-dimensional lattice models. Actually, there are a lot of attempts to extend the idea of integrability to three- [102][103][104] and higher-dimensional generalization [105,106] of lattice models. The condition of commutativity for the transfer matrices in the three-dimensional case takes the form of the so-called tetrahedron equation by Zamolodchikov [107].…”

Integrable Structure of Multispecies Zero Range Process

“…The underlying algebra has been identified with a generalized quantum group. We shall present these results with a brief background based on [36].…”

“…The formula (2.4) is due to [43]. See Section 6 and [33,36] for a further explanation and generalization. For recent progress on evaluating the sum (2.3), we refer to [9].…”

“…, 0). These R matrices have been obtained from the special solutions to the tetrahedron equation by a certain reduction [36]. The underlying algebra has been identified with a generalized quantum group.…”

“…In Section 6 we extend the factorization (2.8) to the R matrices for a class of generalized quantum groups. The result is presented with some background connected to the tetrahedron equation [36]. Section 7 is a short summary.…”