Representations of the Quantized Function Algebras, 2-Categories and Zamolodchikov Tetrahedra Equation

Kazhdan, David; Soibelman, Yan

doi:10.1007/978-1-4612-0345-2_10

Cited by 10 publications

(12 citation statements)

References 5 publications

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“…In Section 2 we begin by reviewing the type A case following the idea of the paper [18] in 1994. See also [22]. It may be viewed as a formulation of many parallel ideas and results in [24,25,20,7,6] in terms of the representation theory of A q (SL n ) [41,39,40].…”

Section: 2)mentioning

confidence: 99%

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Tetrahedron and 3D reflection equations from quantized algebra of functions

Kuniba¹,

Okado²

2012

J. Phys. A: Math. Theor.

105

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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.

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Section: 2)mentioning

confidence: 99%

“…See [11,34,33,41,39,40] for example. The simplest one is A q (SL 2 ) = t 11 , t 12 , t 21 , t 22 with the relations t 11 t 21 = qt 21 t 11 , t 12 t 22 = qt 22 t 12 , t 11 t 12 = qt 12 t 11 , t 21 t 22 = qt 22 t 21 ,…”

Section: Introductionmentioning

confidence: 99%

Tetrahedron and 3D reflection equations from quantized algebra of functions

Kuniba¹,

Okado²

2012

J. Phys. A: Math. Theor.

105

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“…The underlying idea of relaxing a system of N -simplex equations in the above way, by introducing an objectR, such that consistency imposes the (N + 1)-simplex equation on it, is the "obstruction method" in [25,44,73,74,75,77,82]. Also see [9,50,51,58,93] for a formulation in the setting of 2-categories. Indeed, the obstruction method corresponds to the introduction of laxness ("laxification" [93]).…”

Section: Introductionmentioning

confidence: 99%

Simplex and Polygon Equations

Dimakis

Müller‐Hoissen

2015

SIGMA

103

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Abstract. It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a "mixed order". We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of "polygon equations" realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N -simplex equation to the (N + 1)-gon equation, its dual, and a compatibility equation.

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“…Matrix P (n) = n i, j=1 E i j ⊗ E ji acts on C n ⊗ C n as the permutation of the tensor factors and also provides a solution to (10).…”

Section: An Upper Triangular Quantum Groupmentioning

confidence: 99%

Tetrahedron Equation, Weyl Group, and Quantum Dilogarithm

Bytsko

Volkov

2014

Lett Math Phys

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We derive a family of solutions to the tetrahedron equation using the RTT presentation of a two parametric quantized algebra of regular functions on an upper triangular subgroup of G L(n). The key ingredients of the construction are the longest element of the Weyl group, the quantum dilogarithm function, and central elements of the quantized division algebra of rational functions on the subgroup in question.Mathematics Subject Classification. 20G42, 16K20, 16T25.

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Representations of the Quantized Function Algebras, 2-Categories and Zamolodchikov Tetrahedra Equation

Cited by 10 publications

References 5 publications

Tetrahedron and 3D reflection equations from quantized algebra of functions

Tetrahedron and 3D reflection equations from quantized algebra of functions

Simplex and Polygon Equations

Tetrahedron Equation, Weyl Group, and Quantum Dilogarithm

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