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Tatiana Gateva-Ivanova

Publications and source records attributed to Tatiana Gateva-Ivanova.

At least 19 recordsLinked to original sources
arXiv2025

A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equation

We study noninvolutive set-theoretic solutions $(X,r)$ of the Yang-Baxter equations in terms of the properties of the canonically associated algebraic objects-the braided monoid $S(X,r)$, the quadratic Yang-Baxter algebra $A= A(\textbf{k}, X, r)$ over a field $\textbf{k}$ and its Koszul dual, $A^{!}$. More generally, we continue our systematic study of nondegenerate quadratic sets $(X,r)$ and the associated algebraic objects. Next we investigate the class of (noninvolutive) square-free solutions $(X,r)$. It contains the special class of self distributive solutions (quandles). We make a detailed characterization in terms of various algebraic and combinatorial properties each of which shows the contrast between involutive and noninvolutive square-free solutions. We introduce and study a class of finite square-free braided sets $(X,r)$ of order $n\geq 3$ which satisfy "the minimality condition \textbf{M}", that is $\dim_{\textbf{k}} A_2 =2n-1$. Examples are some simple racks of prime order $p$. Finally, we discuss general extensions of solutions and introduce the notion of "a generalized strong twisted union of braided sets". We prove that if $(Z,r)$ is a non-degenerate 2-cancellative braided set splitting as $Z = X\natural^{\ast} Y$, then its braided monoid $S_Z$ is a generalized strong twisted union $S_Z= S_X\natural^{\ast} S_Y$ of the braided monoids $S_X$ and $S_Y$. Moreover, if $(Z,r)$ is injective then its braided group $G_Z=G(Z,r)$ also splits as $G_Z= G_X\natural^{\ast} G_Y$ of the associated braided groups of $X$ and $Y$. We propose a construction of a generalized strong twisted union $Z = X\natural^{\ast} Y$ of braided sets $(X,r_X)$, and $(Y, r_Y)$, where the map $r$ has high, explicitly prescribed order.

math.QA↗
arXiv2024

Quadratic algebras and idempotent braided sets

We study the Yang-Baxter algebras $A(K,X,r)$ associated to finite set-theoretic solutions $(X,r)$ of the braid relations. We introduce an equivalent set of quadratic relations $\Re\subseteq G$, where $G$ is the reduced Gröbner basis of $(\Re)$. We show that if $(X,r)$ is left-nondegenerate and idempotent then $\Re= G$ and the Yang-Baxter algebra is PBW. We use graphical methods to study the global dimension of PBW algebras in the $n$-generated case and apply this to Yang-Baxter algebras in the left-nondegenerate idempotent case. We study the $d$-Veronese subalgebras for a class of quadratic algebras and use this to show that for $(X,r)$ left-nondegenerate idempotent, the $d$-Veronese subalgebra $A(K,X,r)^{(d)}$ can be identified with $A(K,X,r^{(d)})$, where $(X,r^{(d)})$ are all left-nondegenerate idempotent solutions. We determined the Segre product in the left-nondegenerate idempotent setting. Our results apply to a previously studied class of `permutation idempotent' solutions, where we show that all their Yang-Baxter algebras for a given cardinality of $X$ are isomorphic and are isomorphic to their $d$-Veronese subalgebras. In the linearised setting, we construct the Koszul dual of the Yang-Baxter algebra and the Nichols-Woronowicz algebra in the idempotent case, showing that the latter is quadratic. We also construct noncommutative differentials on some of these quadratic algebras.

math.QA↗
arXiv2024

Algebras defined by Lyndon words and Artin-Schelter regularity

Let $X= \{x_1, x_2, \cdots, x_n\}$ be a finite alphabet, and let $K$ be a field. We study classes $\mathfrak{C}(X, W)$ of graded $K$-algebras $A = K\langle X\rangle / I$, generated by $X$ and with a fixed set of obstructions $W$. Initially we do not impose restrictions on $W$ and investigate the case when all algebras in $\mathfrak{C} (X, W)$ have polynomial growth and finite global dimension $d$. Next we consider classes $\mathfrak{C} (X, W)$ of algebras whose sets of obstructions $W$ are antichains of Lyndon words. The central question is "when a class $\mathfrak{C} (X, W)$ contains Artin-Schelter regular algebras?" Each class $\mathfrak{C} (X, W)$ defines a Lyndon pair $(N,W)$ which determines uniquely the global dimension, $gl\dim A$, and the Gelfand-Kirillov dimension, $GK\dim A$, for every $A \in \mathfrak{C}(X, W)$. We find a combinatorial condition in terms of $(N,W)$, so that the class $\mathfrak{C}(X, W)$ contains the enveloping algebra $U\mathfrak{g}$ of a Lie algebra $\mathfrak{g}$. We introduce monomial Lie algebras defined by Lyndon words, and prove results on Groebner-Shirshov bases of Lie ideals generated by Lyndon-Lie monomials. Finally we classify all two-generated Artin-Schelter regular algebras of global dimensions $6$ and $7$ occurring as enveloping $U = U\mathfrak{g}$ of standard monomial Lie algebras. The classification is made in terms of their Lyndon pairs $(N, W)$, each of which determines also the explicit relations of $U$.

math.RA↗
arXiv2023

Quadratic algebras associated to permutation idempotent solutions of the YBE

We study the quadratic algebras $A(K,X,r)$ associated to a class of strictly braided but idempotent set-theoretic solutions $(X,r)$ of the Yang-Baxter or braid relations. In the invertible case, these algebras would be analogues of braided-symmetric algebras or `quantum affine spaces' but due to $r$ being idempotent they have very different properties. We show that all $A(K,X,r)$ for $r$ of a certain permutation idempotent type are isomorphic for a given $n=|X|$, leading to canonical algebras $A(K,n)$. We study the properties of these both via Veronese subalgebras and Segre products and in terms of noncommutative differential geometry. We also obtain new results on general PBW algebras which we apply in the permutation idempotent case.

math.QA↗
arXiv2023

Veronese subalgebras and Veronese morphisms for a class of Yang-Baxter algebras

We study $d$-Veronese subalgebras $A^{(d)}$ of Yang-Baxter algebras $A_X= A(K, X, r)$ related to finite nondegenerate involutive set-theoretic solutions $(X, r)$ of the Yang-Baxter equation, where $K$ is a field and $d\geq 2$ is an integer. We find an explicit presentation of the $d$-Veronese $A^{(d)}$ in terms of one-generators and quadratic relations. We introduce the notion of a $d$-Veronese solution $(Y, r_Y)$, canonically associated to $(X,r)$ and use its Yang-Baxter algebra $A_Y= A(K, Y, r_Y)$ to define a Veronese morphism $v_{n,d}:A_Y \rightarrow A_X $. We prove that the image of $v_{n,d}$ is the $d$-Veronese subalgebra $A^{(d)}$, and find explicitly a minimal set of generators for its kernel. The results agree with their classical analogues in the commutative case. We show that the Yang-Baxter algebra $A(K, X, r)$ is a PBW algebra if and only if $(X,r)$ is a square-free solution. In this case the $d$-Veronese $A^{(d)}$ is also a PBW algebra.

math.QA↗
arXiv2022

Segre products and Segre morphisms in a class of Yang-Baxter algebras

Let $(X,r_X)$ and $(Y,r_Y)$ be finite nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation, and let $A_X = A(\textbf{k}, X, r_X)$ and $A_Y= A(\textbf{k}, Y, r_Y)$ be their quadratic Yang-Baxter algebras over a field $\textbf{k}.$ We find an explicit presentation of the Segre product $A_X\circ A_Y$ in terms of one-generators and quadratic relations. We introduce analogues of Segre maps in the class of Yang-Baxter algebras and find their images and their kernels. The results agree with their classical analogues in the commutative case.

math.QA↗
arXiv2019

The braided group of a square-free solution of the Yang-Baxter equation and its group algebra

Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution $(X,r)$ consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. In this work we study the braided group $G=G(X,r)$ of an involutive square-free solution $(X,r)$ of finite order $n$ and cyclic index $p=p(X,r)$ and the group algebra $\textbf{k} [G]$ over a field $\textbf{k}$. We show that $G$ contains a $G$-invariant normal subgroup $\mathcal{F}_p$ of finite index $p^n$, $\mathcal{F}_p$ is isomorphic to the free abelian group of rank $n$. We describe explicitly the quotient braided group $\widetilde{G}=G/\mathcal{F}_p$ of order $p^n$ and show that $X$ is embedded in $\widetilde{G}$. We prove that the group algebra $\textbf{k} [G]$ is a free left (resp. right) module of finite rank $p^n$ over its commutative subalgebra $\textbf{k}[\mathcal{F}_p]$ and give an explicit free basis. The center of $\textbf{k} [G]$ contains the subalgebra of symmetric polynomials in $\textbf{k} [x_1^p, \cdots, x_n^p]$. Classical results on group rings imply that $\textbf{k}[G]$ is a left (and right) Noetherian domain of finite global dimension.

math.QA↗
arXiv2017

Braces and symmetric groups with special conditions

We study symmetric groups and left braces satisfying special conditions, or identities. We are particularly interested in the impact of conditions like $\textbf{Raut}$ and $\textbf{lri}$ on the properties of the symmetric group and its associated brace. We show that the symmetric group $G=G(X,r)$ associated to a nontrivial solution $(X,r)$ has multipermutation level $2$ if and only if $G$ satisfies $\textbf{lri}$. In the special case of a two-sided brace we express each of the conditions $\textbf{lri}$ and $\textbf{Raut}$ as identities on the associated radical ring $G_*$. We apply these to construct examples of two-sided braces satisfying some prescribed conditions. In particular we construct a finite two-sided brace with condition $\textbf{Raut}$ which does not satisfy $\textbf{lri}$. (It is known that condition $\textbf{lri}$ implies $\textbf{Raut}$). We show that a finitely generated two-sided brace which satisfies \textbf{lri} has a finite multipermutation level which is bounded by the number of its generators.

math.QA↗
arXiv2017

Set-theoretic solutions of the Yang-Baxter equation, Braces, and Symmetric groups

We involve simultaneously the theory of matched pairs of groups and the theory of braces to study set-theoretic solutions of the Yang-Baxter equation (YBE). We show the intimate relation between the notions of a symmetric group (a braided involutive group) and a left brace, and find new results on symmetric groups of finite multipermutation level and the corresponding braces. We introduce a new invariant of a symmetric group $(G,r)$, \emph{the derived chain of ideals of} $G$, which gives a precise information about the recursive process of retraction of $G$. We prove that every symmetric group $(G,r)$ of finite multipermutation level $m$ is a solvable group of solvable length at most $m$. To each set-theoretic solution $(X,r)$ of YBE we associate two invariant sequences of symmetric groups: (i) the sequence of its derived symmetric groups; (ii) the sequence of its derived permutation groups and explore these for explicit descriptions of the recursive process of retraction. We find new criteria necessary and sufficient to claim that $(X, r)$ is a multipermutation solution.

math.QA↗
arXiv2016

On the Yang-Baxter equation and left nilpotent left braces

We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang-Baxter equation, we call them simply solutions. We show that the structure group G(X,r) of a finite non-trivial solution (X,r) cannot be an Engel group. It is known that the structure group G(X,r) of a finite multipermutation solution (X,r) is a poly-Z group, thus our result gives a rich source of examples of braided groups and left braces G(X,r) which are poly-Z groups but not Engel groups. We also show that a finite solution of the Yang-Baxter equation can be embedded in a convenient way into a finite brace and into a finite braided group. For a left brace A, we explore the close relation between the multipermutation level of the solution associated with it and the radical chain $A^{(n+1)}=A^{(n)}* A$ introduced by Rump.

math.GR↗
arXiv2012

Monomial algebras defined by Lyndon words

Assume that $X= {x_1,...,x_g}$ is a finite alphabet and $K$ is a field. We study monomial algebras $A= K /(W)$, where $W$ is an antichain of Lyndon words in $X$ of arbitrary cardinality. We find a Poincaré-Birkhoff-Witt type basis of $A$ in terms of its \emph{Lyndon atoms} $N$, but, in general, $N$ may be infinite. We prove that if $A$ has polynomial growth of degree $d$ then $A$ has global dimension $d$ and is standard finitely presented, with $d-1 \leq |W| \leq d(d-1)/2$. Furthermore, $A$ has polynomial growth iff the set of Lyndon atoms $N$ is finite. In this case $A$ has a $K$-basis $\mathfrak{N} = {l_1^{α_{1}}l_2^{α_{2}}... l_d^{α_{d}} \mid α_{i} \geq 0, 1 \leq i \leq d}$, where $N = {l_1, ...,l_d}$. We give an extremal class of monomial algebras, the Fibonacci-Lyndon algebras, $F_n$, with global dimension $n$ and polynomial growth, and show that the algebra $F_6$ of global dimension 6 cannot be deformed, keeping the multigrading, to an Artin-Schelter regular algebra.

math.RA↗
arXiv2010

Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity

We study quadratic algebras over a field $\textbf{k}$. We show that an $n$-generated PBW algebra $A$ has finite global dimension and polynomial growth \emph{iff} its Hilbert series is $H_A(z)= 1 /(1-z)^n$. Surprising amount can be said when the algebra $A$ has \emph{quantum binomial relations}, that is the defining relations are nondegenerate square-free binomials $xy-c_{xy}zt$ with non-zero coefficients $c_{xy}\in \textbf{k}$. In this case various good algebraic and homological properties are closely related. The main result shows that for an $n$-generated quantum binomial algebra $A$ the following conditions are equivalent: (i) A is a PBW algebra with finite global dimension; (ii) A is PBW and has polynomial growth; (iii) A is an Artin-Schelter regular PBW algebra; (iv) $A$ is a Yang-Baxter algebra; (v) $H_A(z)= 1/(1-z)^n;$ (vi) The dual $A^{!}$ is a quantum Grassman algebra; (vii) A is a binomial skew polynomial ring. So for quantum binomial algebras the problem of classification of Artin-Schelter regular PBW algebras of global dimension $n$ is equivalent to the classification of square-free set-theoretic solutions of the Yang-Baxter equation $(X,r)$, on sets $X$ of order $n$.

math.QA↗
arXiv2009

Binomial skew polynomial rings, Artin-Schelter regularity, and binomial solutions of the Yang-Baxter equation

Let $k$ be a field and $X$ be a set of $n$ elements. We introduce and study a class of quadratic $k$-algebras called \emph{quantum binomial algebras}. Our main result shows that such an algebra $A$ defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual $A^{!}$ is Frobenius of dimension $n,$ with a \emph{regular socle} and for each $x,y \in X $ an equality of the type $xyy=αzzt,$ where $α\in k \setminus\{0\},$ and $z,t \in X$ is satisfied in $A$. We prove the equivalence of the notions \emph{a binomial skew polynomial ring} and \emph{a binomial solution of YBE}. This implies that the Yang-Baxter algebra of such a solution is of Poincaré-Birkhoff-Witt type, and possesses a number of other nice properties such as being Koszul, Noetherian, and an Artin-Schelter regular domain.

math.QA↗
arXiv2009

Garside structure on monoids with quadratic square-free relations

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a "nice" Garside element, certain monoids $S$ with quadratic relations, whose monoidal algebra $A= k[S]$ has a Frobenius Koszul dual $A^{!}$ with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were introduced and studied by the author and in 1995 provided a new class of Noetherian Artin-Schelter regular domains, and the square-free set-theoretic solutions of the Yang-Baxter equation. There is a beautiful symmetry in these objects due to their nice combinatorial and algebraic properties.

math.QA↗
arXiv2009

Multipermutation solutions of the Yang--Baxter equation

Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set $X$ and a function r:X x X --> X x X which satisfies the braid relation. We examine solutions here mainly from the point of view of finite permutation groups: a solution gives rise to a map from $X$ to the symmetric group $Sym(X)$ on $X$ satisfying certain conditions. Our results include many new constructions based on strong twisted union and wreath product, with an investigation of retracts and the multipermutation level and the solvable length of the groups defined by the solutions and new results about decompositions and factorisations of the groups defined by invariant subsets of the solution.

math.QA↗
arXiv2008

Quantum spaces associated to multipermutation solutions of level two

We study finite set-theoretic solutions $(X,r)$ of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over $\C$ with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra $\Acal(\C,X,r)$ having a $q$-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of finite abelian group $\Gcal$ of left actions on $X$. We study the structure of $\Acal(\C,X,r)$ and show that they have a $\bullet$-product form `quantizing' the commutative algebra of polynomials in $|X|$ variables. We obtain the $\bullet$-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite product for a certain crossed $\Gcal$-module (over any field $k$). We provide first steps in the noncommutative differential geometry of $\Acal(k,X,r)$ arising from these results. As a byproduct of our work we find that every such level 2 solution $(X,r)$ factorises as $r=f\circτ\circ f^{-1}$ where $τ$ is the flip map and $(X,f)$ is another solution coming from $X$ as a crossed $\Gcal$-set.

math.QA↗
arXiv2007

Set theoretic solutions of the Yang-Baxter equation, graphs and computations

We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of solutions and use our graphical methods for the computation of solutions of finite order and their automorphisms. Results include a detailed study of solutions of multipermutation level 2.

math.QA↗