arXiv subjects

Anastasia Doikou

Publications and source records attributed to Anastasia Doikou.

At least 19 recordsLinked to original sources

arXiv2026

Non-combinatorial involutive braidings: the quantum algebra $\mathfrak{gl}_{k,m}$

We investigate involutive, non-combinatorial solutions of the braid equation, viewing them as special deformations of the permutation map. Utilizing these solutions, we identify the associated quantum algebra and introduce it as the $\mathfrak{gl}_{k,m}$ Yangian. This newly derived Yangian is distinct from the known Yangian of the general linear Lie superalgebra; crucially, as a Hopf algebra, it possesses the standard tensor product algebra structure. The underlying algebra $\mathfrak{gl}_{k,m}$ is also introduced as a novel structure and constitutes a subalgebra of the Yangian. We then construct specific highest-weight modules of $\mathfrak{gl}_{k,m}$ that simultaneously yield the eigenstates of spin-chain-like ``Hamiltonians'', which are defined as the sum of the generators of the $A$-type braid group. Furthermore, we study the highest-weight representations and the corresponding combinatorial bases for $\mathfrak{gl}_{1,1}$, explicitly linking them to specific shapes of Young tableaux.

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arXiv2026

Braided finite automata and representation theory

We introduce classical and non-deterministic finite automata associated with representations of the braid group. After briefly reviewing basic definitions on finite automata, Coxeter's groups and the associated word problem, we turn to the Artin presentation of the braid group and its quotients. We present various representations of the braid group as deterministic or non-deterministic finite state automata and discuss connections with $q$-Dicke states, as well as Lusztig and crystal bases. We propose the study of the eigenvalue problem of the $\mathfrak{U}_q(\mathfrak{gl}_n)$ invariant spin-chain like ``Hamiltonian'' as a systematic means for constructing canonical bases for irreducible representations of $\mathfrak{U}_q(\mathfrak{gl}_n).$ This is explicitly proven for the algebra $\mathfrak{U}_q(\mathfrak{gl}_2).$ Special braid representations associated with self-distributive structures are also studied as finite automata. These finite state automata organize clusters of eigenstates of these braid representations.

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arXiv2026

Parametric reflection maps: an algebraic approach

We study solutions of the parametric set-theoretic reflection equation from an algebraic perspective by employing recently derived generalizations of the familiar shelves and racks, called parametric (p)-shelves and racks. Generic invertible solutions of the set-theoretic reflection equation are also obtained by a suitable parametric twist. The twist leads to considerably simplified constraints compared to the ones obtained from general set-theoretic reflections. In this context, novel algebraic structures of (skew) p-braces that generalize the known (skew) braces and are suitable for the parametric Yang-Baxter equation are introduced. The p-rack Yang-Baxter and reflection operators as well as the associated algebraic structures are defined setting up the frame for formulating the p-rack reflection algebra.

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arXiv2026

Parametric set-theoretic Yang-Baxter equation: p-racks, solutions & quantum algebras

The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. The first step towards this objective is the introduction of certain generalizations of the familiar shelves and racks called parametric (p)-shelves and racks. These objects satisfy a parametric self-distributivity condition and lead to solutions of the Yang-Baxter equation. Novel, non-reversible solutions are obtained from p-shelf/rack solutions by a suitable parametric twist, whereas all reversible set-theoretic solutions are reduced to the identity map via a parametric twist. The universal algebras associated to both p-rack and generic parametric, set-theoretic solutions are next presented and the corresponding universal R-matrices are derived. The admissible universal Drinfel'd twist is constructed allowing the derivation of the general set-theoretic universal R-matrix. By introducing the concept of a parametric coproduct we prove the existence of a parametric co-associativity. We show that the parametric coproduct is an algebra homomorphism and the universal R-matrices satisfy intertwining relations with the algebra coproducts

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arXiv2025

Quantum Groups, Discrete Magnus Expansion, Pre-Lie and Tridendriform Algebras

We review the discrete evolution problem and the corresponding solution as a discrete Dyson series in order to rigorously derive a generalized discrete version of the Magnus expansion. We also systematically derive the discrete analogue of the pre-Lie Magnus expansion and express the elements of the discrete Dyson series in terms of a tridendriform algebra binary operation. In the generic discrete case, extra significant terms that are absent in the continuous or the linear discrete case appear in both Dyson and Magnus expansions. Based on the rigorous discrete derivation key links between quantum algebras, tridendriform and pre-Lie algebras are then established. This is achieved by examining tensor realizations of quantum groups, such as the Yangian. We show that these realizations can be expressed in terms of tridendriform and pre-Lie algebras. The continuous limit as expected provides the corresponding non-local charges of the Yangian as members of the pre-Lie Magnus expansion.

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arXiv2025

Combinatorial twists in gl_n Yangians

We introduce the special set-theoretic Yang-Baxter algebra and show that it is a Hopf algebra subject to certain conditions. The associated universal R-matrix is also obtained via an admissible Drinfel'd twist. The structure of braces emerges naturally in this context by requiring the special set-theoretic Yang-Baxter algebra to be a Hopf algebra and a quasi-triangular bialgebra after twisting. The fundamental representation of the universal R-matrix yields the familiar set-theoretic (combinatorial) solutions of the Yang-Baxter equation. We then apply the same Drinfel'd twist to the gl_n Yangian after introducing the augmented Yangian. We show that the augmented Yangian is also a Hopf algebra and we also obtain its twisted version.

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arXiv2025

Self-distributive structures, braces & the Yang-Baxter equation

The theory of the set-theoretic Yang-Baxter equation is reviewed from a purely algebraic point of view. We recall certain algebraic structures called shelves, racks and quandles. These objects satisfy a self-distributivity condition and lead to solutions of the Yang-Baxter equation. The quantum algebra as well as the integrability associated to Baxterized involutive set-theoretic solutions is briefly discussed. We then present the theory of the universal algebras associated to rack and general set-theoretic solutions. We show that these are quasi-triangular Hopf algebras and we derive the universal set-theoretic Drinfel'd twist. It is shown that this is an admissible twist allowing the derivation of the universal set-theoretic R-matrix.

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arXiv2024

Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices

We present connections between left non-degenerate solutions of the set-theoretic braid equation and left shelves using Drinfel'd homomorphisms. We generalize the notion of affine quandle, by using heap endomorphisms and metahomomorphisms, and identify the underlying Yang-Baxter algebra for solutions of the braid equation associated to a given quandle. We introduce the notion of the pre-Lie skew brace and identify certain affine quandles that give rise to pre-Lie skew braces. Generalisations of the braiding of a group, associated to set-theoretic solutions of the braid equation are also presented. These generalized structures encode part of the underlying Hopf algebra. We then introduce the quasi-triangular (quasi) Hopf algebras and universal R-matrices for rack and set-theoretic algebras. Generic set-theoretic solutions coming from heap endomorphisms are also identified.

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arXiv2024

Novel non-involutive solutions of the Yang-Baxter equation from (skew) braces

We produce novel non-involutive solutions of the Yang-Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from braces and skew braces, and surprisingly in the case of braces they are not necessarily involutive. In the case of two-sided (skew) braces one can assign such solutions to every element of the set. Novel bijective maps associated to the inverse solutions are also introduced. Moreover, we show that the recently derived Drinfeld twists of the involutive case are still admissible in the non-involutive frame and we identify the twisted $r$-matrices and twisted coproducts. We observe, as in the involutive case that the underlying quantum algebra is not a quasi-triangular bialgebra, as one would expect, but a quasi-triangular quasi-bialgebra. The same applies to the quantum algebra of the twisted $r$-matrices, contrary to the involutive case.

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arXiv2024

Algebraic structures in set-theoretic Yang-Baxter & reflection equations

We present resent results regarding invertible, non-degenerate solutions of the set-theoretic Yang-Baxter and reflection equations. We recall the notion of braces and we present and prove various fundamental properties required for the solutions of the set theoretic Yang-Baxter equation. We then restrict our attention on involutive solutions and consider lambda parametric set-theoretic solutions of the Yang-Baxter equation and we extract the associated quantum algebra. We also discuss the notion of the Drinfeld twist for involutive solutions and their relation to the Yangian. We next focus on reflections and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the symmetric group. We show that there exists a ``reflection'' finite sub-algebra for some special choice of reflection maps.

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arXiv2023

Near braces and p-deformed braided groups

Motivated by recent findings on the derivation of parametric non-involutive solutions of the Yang-Baxter equation we reconstruct the underlying algebraic structures, called near braces. Using the notion of the near braces we produce new multi-parametric, non-degenerate, non-involutive solutions of the set-theoretic Yang-Baxter equation. These solutions are generalisations of the known ones coming from braces and skew braces. Bijective maps associated to the inverse solutions are also constructed. Furthermore, we introduce the generalized notion of p-deformed braided groups and p-braidings and we show that every p-braiding is a solution of the braid equation. We also show that certain multi-parametric maps within the near braces provide special cases of p-braidings.

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arXiv2023

Applications of Grassmannian flows to coagulation systems

We demonstrate how many classes of Smoluchowski-type coagulation models can be realised as multiplicative Grassmannian flows and are therefore linearisable, and thus integrable in this sense. First, we prove that a general Smoluchowski-type equation with a constant frequency kernel, that encompasses a large class of such models, is realisable as a multiplicative Grassmannian flow. Second, we establish that several other related constant kernel models can also be realised as such. These include: the Gallay--Mielke coarsening model; the Derrida--Retaux depinning transition model and a general mutliple merger coagulation model. Third, we show how the additive and multiplicative frequency kernel cases can be realised as rank-one analytic Grassmannian flows.

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arXiv2022

Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation

We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-deformed case of set-theoretic solutions we also construct admissible Drinfeld twists similar to the set-theoretic ones, subject to certain extra constraints dictated by the q-deformation. These findings greatly generalise recent relevant results on set theoretic solutions and their q-deformed analogues.

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arXiv2022

From Braces to Hecke algebras & Quantum Groups

We examine links between the theory of braces and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify new quantum groups associated to set-theoretic solutions coming from braces. We also construct a novel class of quantum discrete integrable systems and we derive symmetries for the corresponding periodic transfer matrices.

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arXiv2022

Applications of Grassmannian flows to integrable systems

We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable in the sense that they are realisable as Fredholm Grassmannian flows. In other words, time-evolutionary solutions to such systems can be constructed from solutions to the corresponding underlying linear partial differential system, by solving a linear Fredholm equation. For example, it is well-known that solutions to classical integrable partial differential systems can be generated by solving a corresponding linear partial differential system for the scattering data and then solving the linear Fredholm (or Volterra) integral equation known as the Gel'fand-Levitan-Marchenko equation. In this paper and in a companion paper, Doikou et al. [DMSW:graphflows], we both, survey the classes of nonlinear systems that are realisable as Fredholm Grassmannian flows, and present new example applications of such flows. We also demonstrate the usefulness of such a representation. Herein we extend the work of Poppe and demonstrate how solution flows of the non-commutative potential Korteweg de Vries and nonlinear Schrodinger systems are examples of such Grassmannian flows. In the companion paper we use this Grassmannian flow approach as well as an extension to nonlinear graph flows, to solve Smoluchowski coagulation and related equations.

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arXiv2021

Set theoretic Yang-Baxter equation, braces and Drinfeld twists

We consider involutive, non-degenerate, finite set theoretic solutions of the Yang-Baxter equation. Such solutions can be always obtained using certain algebraic structures that generalize nil potent rings called braces. Our main aim here is to express such solutions in terms of admissible Drinfeld twists substantially extending recent preliminary results. We first identify the generic form of the twists associated to set theoretic solutions and we show that these twists are admissible, i.e. they satisfy a certain co-cycle condition. These findings are also valid for Baxterized solutions of the Yang-Baxter equation constructed from the set theoretical ones.

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arXiv2021

Set theoretic Yang-Baxter & reflection equations and quantum group symmetries

Connections between set-theoretic Yang-Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic $R$-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for $R$-matrices being Baxterized solutions of the $A$-type Hecke algebra ${\cal H}_N(q=1)$. We show in the case of the reflection algebra that there exists a ``boundary'' finite sub-algebra for some special choice of ``boundary'' elements of the $B$-type Hecke algebra ${\cal B}_N(q=1, Q)$. We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the $B$-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the $B$-type Hecke algebra. These are universal statements that largely generalize previous relevant findings, and also allow the investigation of the symmetries of the double row transfer matrix.

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arXiv2020

An algebraic approach to discrete time integrability

We propose the systematic construction of classical and quantum two dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrodinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz-Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two dimensional quantum lattice is thus also examined.

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