arXiv subjects

Stanislav Shkarin

Publications and source records attributed to Stanislav Shkarin.

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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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

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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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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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

On numerically hypercyclic operators

According to Kim, Peris and Song, a continuous linear operator $T$ on a complex Banach space $X$ is called {\it numerically hypercyclic} if the numerical orbit $\{f(T^nx):n\in\N\}$ is dense in $\C$ for some $x\in X$ and $f\in X^*$ satisfying $\|x\|=\|f\|=f(x)=1$. They have characterized numerically hypercyclic weighted shifts and provided an example of a numerically hypercyclic operator on $\C^2$. We answer two questions of Kim, Peris and Song. Namely, we construct a numerically hypercyclic operator, whose square is not numerically hypercyclic as well as an operator which is not numerically hypercyclic but has two numerical orbits whose union is dense in $\C$. We characterize numerically hypercyclic operators on $\C^2$ as well as the operators similar to a numerically hypercyclic one and those operators whose conjugacy class consists entirely of numerically hypercyclic operators. We describe in spectral terms the operator norm closure of the set of numerically hypercyclic operators on a reflexive Banach space. Finally, we provide criteria for numeric hypercyclicity and decide upon the numerical hypercyclicity of operators from various classes.

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arXiv2013

On Pauli Pairs

The state of a system in classical mechanics can be uniquely reconstructed if we know the positions and the momenta of all its parts. In 1958 Pauli has conjectured that the same holds for quantum mechanical systems. The conjecture turned out to be wrong. In this paper we provide a new set of examples of Pauli pairs, being the pairs of quantum states indistinguishable by measuring the spatial location and momentum. In particular, we construct a new set of spatially localized Pauli pairs.

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arXiv2012

Orbits of coanalytic Toeplitz operators and weak hypercyclicity

We prove a new criterion of weak hypercyclicity of a bounded linear operator on a Banach space. Applying this criterion, we solve few open questions. Namely, we show that if $G$ is a region of $\C$ bounded by a smooth Jordan curve $Γ$ such that $G$ does not meet the unit ball but $Γ$ intersects the unit circle in a non-trivial arc, then $M^*$ is a weakly hypercyclic operator on $H^2(G)$, where $M$ is the multiplication by the argument operator $Mf(z)=zf(z)$. We also prove that if $g$ is a non-constant function from the Hardy space $H^\infty(\D)$ on the unit disk $\D$ such that $g(\D)\cap\D=\varnothing$ and the set $\{z\in\C:|z|=1,\ |g(z)|=1\}$ is a subset of the unit circle $\T$ of positive Lebesgue measure, then the coanalytic Toeplitz operator $T^*_g$ on the Hardy space $H^2(\D)$ is weakly hypercyclic. On the contrary, if $g(\D)\cap\D=\varnothing$, $|g|>1$ almost everywhere on $\T$ and $\log(|g|-1)\in L^1(\T)$, then $T^*_g$ is not 1-weakly hypercyclic and hence is not weakly hypercyclic (a bounded linear operator $T$ on a complex Banach space $X$ is called $n$-weakly hypercyclic if there is $x\in X$ such that for every surjective continuous linear operator $S:X\to \C^n$, the set $\{S(T^mx):m\in\N\}$ is dense in $\C^n$). The last result is based upon lower estimates of the norms of the members of orbits of a coanalytic Toeplitz operator. Finally, we show that there is a 1-weakly hypercyclic operator on a Hilbert space, whose square is non-cyclic and prove that a Banach space operator is weakly hypercyclic if and only if it is $n$-weakly hypercyclic for every $n\in\N$.

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arXiv2012

The Kitai Criterion and backward shifts

It is proved that for any separable infinite dimensional Banach space $X$, there is a bounded linear operator $T$ on $X$ such that $T$ satisfies the Kitai Criterion. The proof is based on quasisimilarity argument and on showing that $I+T$ satisfies the Kitai Criterion for certain backward weighted shifts $T$.

math.FA↗

arXiv2012

A weighted bilateral shift with cyclic square is supercyclic

It is shown that for a bounded weighted bilateral shift $T$ acting on $\ell_p(\Z)$ for $1\leq p\leq 2$ supercyclicity of $T$, weak supercyclicity of $T$, cyclicity of $T\oplus T$ and cyclicity of $T^2$ are equivalent. A new sufficient condition for cyclicity of a weighted bilateral shift is proved, which implies, in particular, that any compact weighted bilateral shift is cyclic.

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arXiv2012

Compact operators without extended eigenvalues

A complex number $λ$ is called an extended eigenvalue of a bounded linear operator $T$ on a Banach space $\B$ if there exists a non-zero bounded linear operator $X$ acting on $\B$ such that $XT=λTX$. We show that there are compact quasinilpotent operators on a separable Hilbert space, for which the set of extended eigenvalues is the one-point set ${1}$.

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arXiv2012

On similarity of quasinilpotent operators

Bounded linear operators on separable Banach spaces algebraically similar to the classical Volterra operator $V$ acting on $C[0,1]$ are characterized. From this characterization it follows that $V$ does not determine the topology of $C[0,1]$, which answers a question raised by Armando Villena. A sufficient condition for an injective bounded linear operator on a Banach space to determine its topology is obtained. From this condition it follows, for instance, that the Volterra operator acting on the Hardy space $\H^p$ of the unit disk determines the topology of $\H^p$ for any $p\in[1,\infty]$.

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arXiv2012

Non-sequential weak supercyclicity and hypercyclicity

A bounded linear operator $T$ acting on a Banach space $\B$ is called weakly hypercyclic if there exists $x\in \B$ such that the orbit ${T^n x: n=0,1,...}$ is weakly dense in $\B$ and $T$ is called weakly supercyclic if there is $x\in \B$ for which the projective orbit ${λT^n x: λ\in \C, n=0,1,...}$ is weakly dense in $\B$. If weak density is replaced by weak sequential density, then $T$ is said to be weakly sequentially hypercyclic or supercyclic respectively. It is shown that on a separable Hilbert space there are weakly supercyclic operators which are not weakly sequentially supercyclic. This is achieved by constructing a Borel probability measure $μ$ on the unit circle for which the Fourier coefficients vanish at infinity and the multiplication operator $Mf(z)=zf(z)$ acting on $L_2(μ)$ is weakly supercyclic. It is not weakly sequentially supercyclic, since the projective orbit under $M$ of each element in $L_2(μ)$ is weakly sequentially closed. This answers a question posed by Bayart and Matheron. It is proved that the bilateral shift on $\ell_p(\Z)$, $1\leq p <\infty$, is weakly supercyclic if and only if $2<p<\infty$ and that any weakly supercyclic weighted bilateral shift on $\ell_p(\Z)$ for $1\leq p\leq 2$ is norm supercyclic. It is also shown that any weakly hypercyclic weighted bilateral shift on $\ell_p(\Z)$ for $1\leq p<2$ is norm hypercyclic, which answers a question of Chan and Sanders.

math.FA↗