# 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).

## 9 Citations

### Set-theoretical solutions of simplex equations

- Mathematics
- 2022

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

- Mathematics
- 2022

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

- Mathematics
- 2022

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

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2022

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

- MathematicsSt. 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

- Mathematics
- 2021

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

- Physics
- 2021

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

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2021

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

- Mathematics
- 2021

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…

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