Tetrahedron equation and generalized quantum groups

Atsuo Kuniba, M Okado, Sergey Sergeev

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Abstract

We construct 2n-families of solutions of the YangBaxter equation from nproducts of three-dimensional 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 1's and L's involved in the product, the trace construction interpolates the symmetric tensor representations of Uq (An-1(1)) and the antisymmetric tensor representations of U-q-1 (An-1(1)), whereas a boundary vector construction interpolates the q-oscillator representation of Uq (Dn 1) (2) + and the spin representation of Uq (Dn+1(2)) . The intermediate cases are associated with an affinization of quantum superalgebras.
Original languageEnglish
Pages (from-to)1-29
Number of pages29
JournalJournal of Physics A: Mathematical and Theoretical
Volume48
Issue number30
DOIs
Publication statusPublished - 2015

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Triangular pyramid
Quantum Groups
tetrahedrons
Tensors
Tensor
Interpolate
Yang-Baxter Equation
Superalgebra
R-matrix
Antisymmetric
Hopf Algebra
Algebra
Trace
tensors
Three-dimensional
Operator
algebra
oscillators
operators
products

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Kuniba, Atsuo ; Okado, M ; Sergeev, Sergey. / Tetrahedron equation and generalized quantum groups. In: Journal of Physics A: Mathematical and Theoretical. 2015 ; Vol. 48, No. 30. pp. 1-29.
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Tetrahedron equation and generalized quantum groups. / Kuniba, Atsuo; Okado, M; Sergeev, Sergey.

In: Journal of Physics A: Mathematical and Theoretical, Vol. 48, No. 30, 2015, p. 1-29.

Research output: Contribution to journalArticle

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T1 - Tetrahedron equation and generalized quantum groups

AU - Kuniba, Atsuo

AU - Okado, M

AU - Sergeev, Sergey

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AB - We construct 2n-families of solutions of the YangBaxter equation from nproducts of three-dimensional 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 1's and L's involved in the product, the trace construction interpolates the symmetric tensor representations of Uq (An-1(1)) and the antisymmetric tensor representations of U-q-1 (An-1(1)), whereas a boundary vector construction interpolates the q-oscillator representation of Uq (Dn 1) (2) + and the spin representation of Uq (Dn+1(2)) . The intermediate cases are associated with an affinization of quantum superalgebras.

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KW - tetrahedron equation

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