arXiv subjects

Natalia Iyudu

Publications and source records attributed to Natalia Iyudu.

At least 19 recordsLinked to original sources

arXiv2022

Golod-Shafarevich-Vinberg type theorems and finiteness conditions for potential algebras

We obtain a lower estimate for the Hilbert series of Jacobi algebras and their completions by providing analogue of the Golog-Shafarevich-Vinberg theorem for potential case. We especially treat non-homogeneous situation. This estimate allows to answer number of questions arising in the work of Wemyss-Donovan-Brown on noncommutative singularities and deformation theory. In particular, we prove that the only case when a potential algebra or its completion could be finite dimensional or of linear growth, is the case of two variables and potential having terms of degree three.

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arXiv2021

Sklyanin algebras and a cubic root of 1

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. This class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry Artin, Tate and Van Den Berg \cite{ATV2} showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates. It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Gröbner basis technique. The authors have previously dispelled this belief. However our previous proof was no less complicated than the one based on algebraic geometry. It used a construcion of a Gröbner basis in a suitable one-sided module over $S$ and had quite a number of cases to consider. In this paper we exhibit a linear substitution after which it becomes possible to determine the leading monomials of a reduced Gröbner basis for the ideal of relations of $S$ itself (without passing to a module). We also find out explicitly (in terms of parameters) which Sklyanin algebras are isomorphic. The only drawback of the new technique is that it fails if the characteristic of the ground field equals 3.

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arXiv2020

Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them $ξδ$-monomials. The Lie structure on this subcomplex is combinatorially described in terms of $ξδ$-monomials. This subcomplex and a basis of $ξδ$-monomials in combination with arguments from Groebner bases theory serves for the cohomology calculations of the higher cyclic Hochschild complex. The language of $ξδ$-monomials in particular allows an interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure.

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arXiv2020

Classification of contraction algebras and pre-Lie algebras associated to braces and trusses

We develop tools for classification of contraction algebras and apply these to solve the problem on classification up to isomorphism of 8 and 9 dimensional algebras corresponding to 3-fold flops. We prove that there is only one up to isomorphism contraction algebra of dimension 8, and two algebras of dimension 9. The formulae for the dimension of algebra, depending on the type of the potential are obtained. In the second part of the paper we show that associated graded structure to brace and truss with appropriate descending ideal filtration is pre-Lie.

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arXiv2020

Homologies of monomial operads and algebras

We consider the bar complex of a monomial non-unital associative algebra $A=k \langle X \rangle / (w_1,...,w_t)$. It splits as a direct sum of complexes $B_w$, defined for any fixed monomial $w=x_1...x_n \in A$. We give a simple argument, showing that the homology of this subcomplex is at most one-dimensional, and describe the place where the nontrivial homology appears. It has a very simple expression in terms of the length of the generalized Dyck path associated to a given monomial in $w \in A$. The operadic analogue of the question about dichotomy in homology is considered. It is shown that dichotomy holds in case when monomial tree-relations form an order. Examples are given showing that in general dichotomy and homological purity does not hold. For quadratic operads, the combinatorial tool for calculating homology in terms of relation graphs is developed. Example of using these methods to compute homology in truncated binary operads is given.

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arXiv2019

Pre-Calabi-Yau algebras and double Poisson brackets

We give an explicit formula showing how the double Poisson algebra introduced in \cite{VdB} appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on $A\oplus A^*$. Specific part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structures. The result holds for any associative algebra $A$ and emphasizes the special role of the fourth component of a pre-Calabi-Yau structure in this respect. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson brackets on representation spaces $({\rm Rep}_n A)^{Gl_n}$ for any associative algebra $A$.

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arXiv2018

Classification of quadratic and cubic PBW algebras on three generators

We give a complete classification of quadratic algebras A, with Hilbert series $H_A=(1-t)^{-3}$, which is the Hilbert series of commutative polynomials on 3 variables. Koszul algebras as well as algebras with quadratic Gröbner basis among them are identified. We also give a complete classification of cubic algebras A with Hilbert series $H_A=(1+t)^{-1}(1-t)^{-3}$. These two classes of algebras contain all Artin-Schelter regular algebras of global dimension 3. As far as the latter are concerned, our results extend well-known results of Artin and Schelter by providing a classification up to an algebra isomorphism.

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arXiv2018

Potential algebras with few generators

We give a complete description of quadratic potential and twisted potential algebras on 3 generators as well as cubic potential and twisted potential algebras on 2 generators up to graded algebra isomorphisms under the assumption that the ground field is algebraically closed and has characteristic different from 2 or 3. We also prove that for two generated potential algebra necessary condition of finite-dimensionality is that potential contains terms of degree three, this answers a question of Agata Smoktunowicz and the first named author, formulated in [AN]. We clarify situation in case of arbitrary number of generators as well.

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arXiv2017

Golod-Shafarevich type theorems and potential algebras

Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than $8$. This answers a question of Wemyss \cite{Wemyss}, related to the geometric argument of Toda \cite{T}. We derive from the improved version of the Golod-Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove, that potential algebra for any homogeneous potential of degree $n\geq 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class ${\cal P}_n$ of potential algebras with homogeneous potential of degree $n+1\geq 4$, the minimal Hilbert series is $H_n=\frac{1}{1-2t+2t^n-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but non-linear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar-Vafa invariants.

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arXiv2017

Quadratic automaton algebras and intermediate growth

We present an example of a quadratic algebra given by three generators and three relations, which is automaton (the set of normal words forms a regular language) and such that its ideal of relations does not possess a finite Gröbner basis with respect to any choice of generators and any choice of a well-ordering of monomials compatible with multiplication. This answers a question of Ufnarovski. Another result is a simple example (4 generators and 7 relations) of a quadratic algebra of intermediate growth.

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arXiv2017

On Koszulity in homology of moduli spaces of stable n-pointed curves of genus zero

We prove Koszulity of the homology of the of moduli spaces of stable n-pointed curves of genus zero $\overline{M}_{0,n}$ for $n=5$, using its presentation due to Keel and the Priddy criterion of Koszulity. For $n=6 $ we establish that $\overline{M}_{0,6}^!$ is potential, find the expression for the potential, and based on that prove that it is Koszul.

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arXiv2016

Three dimensional Sklyanin algebras and Groebner bases

We consider a Sklyanin algebra S with 3 generators, which is the quadratic algebra over a field k with three generators x,y,z given by three relations pxy+qyx+rzz=0, pyz+qzy+rxx=0 and pzx+qxz+ryy=0, where p,q,r are parameters from the fileld k. This class of algebras enjoyed much of attention, in particular, using tools from algebraic geometry, Feigin & Odesskii, and Artin, Tate & Van den Berg, showed that if at least two of the parameters p, q and r are non-zero and at least two of three numbers p^3,q^3 and r^3 are distinct, then S is Koszul and has the same Hilbert series as the algebra of commutative polynomials in three variables. It became commonly accepted, that it is impossible to achieve the same objective by purely algebraic and combinatorial means, like the Groebner basis technique. The main purpose of this paper is to trace combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van den Bergh. Further, we study a wider class of Sklyanin algebras, namely the situation when all parameters of relations could be different. This class called generalized Sklyanin algebras. We classify up to isomorphism genralized Sklyanin algebras with polynomial Hilbert series. We show that generalized Slyanin algebras in general position have the Golod-Shafarevich Hilbert series (with exception of the case of the field with two elements).

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arXiv2016

Two problems from the Polishchuk and Positselski book on Quadratic algebras

In the book 'Quadratic algebras' by Polishchuk and Positselski [23] algebras with a small number of generators (n=2,3) are considered. For some number r of relations possible Hilbert series are listed, and those appearing as series of Koszul algebras are specified. The first case, where it was not possible to do, namely the case of three generators n=3 and six relations r=6 is formulated as an open problem. We give here a complete answer to this question, namely for quadratic algebras with dim A_1=dim A_2=3, we list all possible Hilbert series, and find out which of them can come from Koszul algebras, and which can not. As a consequence of this classification, we found an algebra, which serves as a counterexample to another problem from the same book [23] (Chapter 7, Sec. 1, Conjecture 2), saying that Koszul algebra of finite global homological dimension d has dim A_1 >= d. Namely, the 3-generated algebra A given by relations xx+yx=xz=zy=0 is Koszul and its Koszul dual algebra A^! has Hilbert series of degree 4: H_{A^!}(t)= 1+3t+3t^2+2t^3+t^4, hence A has global homological dimension 4.

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arXiv2016

Asymptotically optimal $k$-step nilpotency of quadratic algebras and the Fibonacci numbers

It follows from the Golod--Shafarevich theorem that if R is an associative algebra given by n generators and $d<\frac{n^2}{4}\cos^{-2}(\fracπ{k+1})$ quadratic relations, then R is not k-step nilpotent. We show that the above estimate is asymptotically optimal, and establish number of related results. For example, we show that for any k this estimate is attained for ifinitely many n.

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arXiv2015

Optimal 5-step nilpotent quadratic algebras

By the Golod--Shafarevich Theorem, an associative algebra R given by n generators and d<n^2/3 homogeneous quadratic relations is not 5-step nilpotent. We prove that this estimate is optimal. Namely, we show that for every positive integer n, there is an algebra R given by n generators and n^2/3 homogeneous quadratic relations such that R is 5-step nilpotent.

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arXiv2015

Complete Characterization of K-Theory for C*-algebras Associated to Locally Finite Unoriented Graphs

In this paper we give a complete description of K-theory groups for Cuntz-Krieger C*-algebras associated to general locally-finite (topologically connected) graphs via Bass-Hashimoto operator. Our result generalizes the one obtained by the second author for the case of graphs with not necessarily finite first Betti numbers. On the basis of purely graph-theoretical method introduced by G. Cornelissen, O. Lorscheid, M. Marcolli and developed further by N.Iyudu, we prove that for the algebra O_E associated to an infinite graph E of the above form holds K_0(O_E)=Z^{β(E)} \oplus Z^{γ(E)} and K_1(O_E) = Z^{γ(E)}, where β(E)=\dim H_1(E) and γ(E) stands for the cardinality of the valency set of E, defined in the paper.

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arXiv2014

The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations

For an arbitrary associative unital ring $R$, let $J_1$ and $J_2$ be the following noncommutative birational partly defined involutions on the set $M_3(R)$ of $3\times 3$ matrices over $R$: $J_1(M)=M^{-1}$ (the usual matrix inverse) and $J_2(M)_{jk}=(M_{kj})^{-1}\,$ (the transpose of the Hadamard inverse). We prove the following surprising conjecture by Kontsevich saying that $(J_2\circ J_1)^3$ is the identity map modulo the ${\rm Diag}_{L} \times \rm{Diag}_R$ action $(D_1,D_2)(M)=D_1^{-1}MD_2$ of pairs of invertible diagonal matrices. That is, we show that for each $M$ in the domain where $(J_2\circ J_1)^3$ is defined, there are invertible diagonal $3\times 3$ matrices $D_1=D_1(M)$ and $D_2=D_2(M)$ such that $(J_2\circ J_1)^3(M)=D_1^{-1}MD_2.$

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arXiv2013

K-theory of locally finite graph $C^*$-algebras

We calculate the K-theory of the Cuntz-Krieger algebra ${\cal O}_E$ associated with an infinite, locally finite graph, via the Bass-Hashimoto operator. The formulae we get express the Grothendieck group and the Whitehead group in purely graph theoretic terms. We consider the category of finite (black-and-white, bi-directed) subgraphs with certain graph homomorphisms and construct a continuous functor to abelian groups. In this category $K_0$ is an inductive limit of $K$-groups of finite graphs, which were calculated in \cite{MM}. In the case of an infinite graph with the finite Betti number we obtain the formula for the Grothendieck group $K_0({\cal O}_E)= {\mathbb Z}^{β(E)+γ(E)},\,$ where $β(E)$ is the first Betti number and $γ(E)$ is the valency number of the graph $E$. We note, that in the infinite case the torsion part of $K_0$, which is present in the case of a finite graph, vanishes. The Whitehead group depends only on the first Betti number: $K_1({\cal O}_E)= {\mathbb Z}^{β(E)}$. These allow us to provide a counterexample to the fact, which holds for finite graphs, that $K_1({\cal O}_E)$ is the torsion free part of $K_0({\cal O}_E)$.

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