Grassmannian-parameterized solutions to direct-sum polygon and simplex equations

@article{Dimakis2020GrassmannianparameterizedST,
  title={Grassmannian-parameterized solutions to direct-sum polygon and simplex equations},
  author={Aristophanes Dimakis and Igor G. Korepanov},
  journal={arXiv: Mathematical Physics},
  year={2020}
}
We consider polygon and simplex equations, of which the simplest nontrivial examples are pentagon (5-gon) and Yang--Baxter (2-simplex), respectively. We examine the general structure of (2n+1)-gon and 2n-simplex equations in direct sums of vector spaces. Then we provide a construction for their solutions, parameterized by elements of the Grassmannian Gr(n+1,2n+1). 

Figures from this paper

Set-theoretical solutions of simplex equations

The n-simplex equation (n-SE) was introduced by A. B. Zamolodchikov as a generalization of the Yang–Baxter equation, which is the 2-simplex equation in these terms. In the present paper we suggest

Odd-gon relations and their cohomology

A cohomology theory for “odd polygon” relations—algebraic imitations of Pachner moves in dimensions 3, 5, . . . —is constructed. Manifold invariants based on polygon relations and nontrivial polygon

Hierarchies of compatible maps and integrable difference systems

We present two non-equivalent hierarchies of non-Abelian 3D−compatible maps and we provide their Lax pair formulation. These hierarchies are naturally associated with integrable difference systems

On non-abelian quadrirational Yang–Baxter maps

We introduce four non-equivalent lists of families of non-abelian quadrirational Yang–Baxter maps, the so-called F , H , K and Λ lists. We provide the canonical form of the generic map in each list,

Heptagon relation in a direct sum

  • I. Korepanov
  • Mathematics
    St. Petersburg Mathematical Journal
  • 2022
An Ansatz is proposed for the heptagon relation, that is, an algebraic imitation of the five-dimensional Pachner move 4–3. The formula in question is realized in terms of matrices acting in a direct

Quadratic heptagon cohomology

A cohomology theory is proposed for the recently discovered heptagon relation—an algebraic imitation of a 5-dimensional Pachner move 4–3. In particular, ‘quadratic cohomology’ is introduced, and it

Obituary: Aristophanes Dimakis

The theoretical physicist and mathematician Aristophanes Dimakis passed away on July 8, 2021, at the age of 68, in Athens, Greece. We briefly review his life, career and scientific achievements. We

Tetrahedron maps, Yang–Baxter maps, and partial linearisations

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and Yang–Baxter maps, which are set-theoretical solutions to the quantum Yang–Baxter

Heptagon relations parameterized by simplicial 3-cocycles

A piecewise linear (PL) manifold triangulation can be transformed into any other triangulation by means of a sequence of Pachner moves [7, 6]. Algebraic imitations of such moves are often called

References

SHOWING 1-10 OF 24 REFERENCES

Simplex and polygon equations. SIGMA

  • Symmetry, Integrability and Geometry: Methods and Applications,
  • 2015

Functional tetrahedron equation

We describe a method for constructing classical integrable models in a (2+1)-dimensional discrete spacetime based on the functional tetrahedron equation, an equation that makes the symmetries of a

Non-commutative birational maps satisfying Zamolodchikov equation, and Desargues lattices

We present new solutions of the functional Zamolodchikov tetrahedron equation in terms of birational maps in totally non-commutative variables. All the maps originate from Desargues lattices, which

Heptagon relation in a direct sum

  • I. Korepanov
  • Mathematics
    St. Petersburg Mathematical Journal
  • 2022
An Ansatz is proposed for the heptagon relation, that is, an algebraic imitation of the five-dimensional Pachner move 4–3. The formula in question is realized in terms of matrices acting in a direct

Introduction to Superanalysis

1. Grassmann Algebra.- 2. Superanalysis.- 3. Linear Algebra in Z2-Graded Spaces.- 4. Supermanifolds in General.- 5. Lie Superalgebras.- 1. Lie Superalgebras.- 2. Lie Supergroups.- 3. Laplace-Casimir

Polynomial-valued constant hexagon cohomology

Hexagon relations are algebraic realizations of four-dimensional Pachner moves. `Constant' -- not depending on a 4-simplex in a triangulation of a 4-manifold -- hexagon relations are proposed, and

Nonconstant hexagon relations and their cohomology

A construction of hexagon relations—algebraic realizations of four-dimensional Pachner moves—is proposed. It goes in terms of “permitted colorings” of 3-faces of pentachora (4-simplices), and its

DISTRIBUTIVE GROUPOIDS IN KNOT THEORY

A sequence of new knot invariants is constructed by using the relationship between the theory of distributive groupoids and knot theory.Bibliography: 3 titles.