Papers by Dmitri Piontkovski
Journal of Symbolic Computation, Nov 1, 2020
In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose... more In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous contextfree grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series.
Journal of Algebra, Apr 1, 2008
A well-known conjecture says that every one-relator group is coherent. We state and partly prove ... more A well-known conjecture says that every one-relator group is coherent. We state and partly prove a similar statement for graded associative algebras. In particular, we show that every Gorenstein algebra A of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line P 1 as a noncommutative scheme based on the coherent noncommutative spectrum qgr A of such an algebra A, that is, the category of coherent A-modules modulo the torsion ones. This category is always abelian Ext-finite hereditary with Serre duality, like the category of coherent sheaves on P 1. In this way, we obtain a sequence P 1 n (n ≥ 2) of pairwise non-isomorphic noncommutative schemes which generalize the scheme P 1 = P 1 2. Theorem 1.2 (Theorem 4.1). Every graded algebra defined by a single homogeneous quadratic relation is graded coherent. Note that there are non-coherent quadratic algebras with two relations, for example, the algebras k x, y, z, t|tz − zy, zx [Pi2, Prop. 10] or even k x, y, z|yz − zy, zx [Po, Example 2].
arXiv (Cornell University), Aug 29, 2005
We develop the cohomology theory of color Lie superalgebras due to Scheunert-Zhang in a framework... more We develop the cohomology theory of color Lie superalgebras due to Scheunert-Zhang in a framework of nonhomogeneous quadratic Koszul algebras. In this approach, the Chevalley-Eilenberg complex of a color Lie algebra becomes a standard Koszul complex for its universal enveloping algebra. As an application, we calculate cohomologies with trivial coefficients of Z n 2graded 3-dimensional color Lie superalgebras.
arXiv (Cornell University), May 19, 2004
Quadratic algebras associated to pseudo-roots of noncommutative polynomials have been introduced ... more Quadratic algebras associated to pseudo-roots of noncommutative polynomials have been introduced by I. Gelfand, Retakh, and Wilson in connection with studying the decompositions of noncommutative polynomials. Later they (with S. Gelfand and Serconek) shown that the Hilbert series of these algebras and their quadratic duals satisfy the necessary condition for Koszulity. It is proved in this note that these algebras are Koszul.
arXiv (Cornell University), Jun 20, 2017
Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra... more Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then the rationality of the Hilbert series of the algebra follows from the affirmative answer to Ufnarovski's question. Assuming that the ground field has a positive characteristic, we show that the answer to Ufnarovskii's question is positive if and only if the basic field is an algebraic extension of its prime subfield. Moreover, in the "only if" part we show that there exists a finitely presented graded algebra of linear growth with irrational Hilbert series. In addition, over an arbitrary infinite basic field, the set of Hilbert series of the quadratic algebras of linear growth with 5 generators is infinite. Our approach is based on a connection with the dynamical Mordell-Lang conjecture. This conjecture describes the intersection of an orbit of an algebraic variety endomorphism with a subvariety. We show that the positive answer to Ufnarovski's question implies some known cases of the dynamical Mordell-Lang conjecture. In particular, the positive answer for a class of algebras is equivalent to the Skolem-Mahler-Lech theorem which says that the set of the zero elements of any linear recurrent sequence over a zero characteristic field is the finite union of several arithmetic progressions. In particular, the counterexamples to this theorem in the finite characteristic case give examples of algebras with irrational Hilbert series.
Linear Programming
Springer eBooks, 2011
Advances in Mathematics, Feb 1, 2019
Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra... more Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then the rationality of the Hilbert series of the algebra follows from the affirmative answer to Ufnarovski's question. Assuming that the ground field has a positive characteristic, we show that the answer to Ufnarovskii's question is positive if and only if the basic field is an algebraic extension of its prime subfield. Moreover, in the "only if" part we show that there exists a finitely presented graded algebra of linear growth with irrational Hilbert series. In addition, over an arbitrary infinite basic field, the set of Hilbert series of the quadratic algebras of linear growth with 5 generators is infinite. Our approach is based on a connection with the dynamical Mordell-Lang conjecture. This conjecture describes the intersection of an orbit of an algebraic variety endomorphism with a subvariety. We show that the positive answer to Ufnarovski's question implies some known cases of the dynamical Mordell-Lang conjecture. In particular, the positive answer for a class of algebras is equivalent to the Skolem-Mahler-Lech theorem which says that the set of the zero elements of any linear recurrent sequence over a zero characteristic field is the finite union of several arithmetic progressions. In particular, the counterexamples to this theorem in the finite characteristic case give examples of algebras with irrational Hilbert series.
We consider varieties of linear multioperator algebras, that is, classes of algebras with several... more We consider varieties of linear multioperator algebras, that is, classes of algebras with several multilinear operations satisfying certain identities. To each such a variety one can assign a numerical sequence called a sequence of codimensions. e n-th codimen-* e article was prepared within the framework of the Academic Fund Program at the
arXiv (Cornell University), Mar 5, 2021
Let A be a finitely presented associative monomial algebra. We study the category qgr(A) which is... more Let A be a finitely presented associative monomial algebra. We study the category qgr(A) which is a quotient of the category of graded finitely presented A-modules by the finite-dimensional ones. As this category plays a role of the category of coherent sheaves on the corresponding noncommutative variety, we consider its bounded derived category D b (qgr(A)). We calculate the categorical entropy of the Serre twist functor on D b (qgr(A)) and show that it is equal to the (natural) logarithm of the entropy of the algebra A itself. Moreover, we relate these two kinds of entropy with the topological entropy of the Ufnarovski graph of A and the entropy of the path algebra of the graph. If A is a path algebra of some quiver, the categorical entropy is equal to the logarithm of the spectral radius of the quiver's adjacency matrix.
Journal of Algebra, Dec 1, 2005
We call a graded connected algebra R effectively coherent, if for every linear equation over R wi... more We call a graded connected algebra R effectively coherent, if for every linear equation over R with homogeneous coefficients of degrees at most d, the degrees of generators of its module of solutions are bounded by some function D(d). For commutative polynomial rings, this property has been established by Hermann in 1926. We establish the same property for several classes of noncommutative algebras, including the most common class of rings in noncommutative projective geometry, that is, strongly Noetherian rings, which includes Noetherian PI algebras and Sklyanin algebras. We extensively study so-called universally coherent algebras, that is, such that the function D(d) is bounded by 2d for d ≫ 0. For example, finitely presented monomial algebras belong to this class, as well as many algebras with finite Groebner basis of relations.
Linear Transformations
Springer texts in business and economics, 2011
We will begin with a general definition of transformations and then study linear transformations.... more We will begin with a general definition of transformations and then study linear transformations. A mappingF from a set S to another set S ′ is a relation which, to every element x of S, associates an element y of S ′ . This mapping is denoted by F : S → S ′ .
International Mathematics Research Notices, 2006
Quadratic algebras associated to graphs have been introduced by I. Gelfand, S. Gelfand, and Retak... more Quadratic algebras associated to graphs have been introduced by I. Gelfand, S. Gelfand, and Retakh in connection with decompositions of noncommutative polynomials. Here we show that, for each graph with rare triangular subgraphs, the corresponding quadratic algebra is a Koszul domain with global dimension equal to the number of vertices of the graph.
arXiv (Cornell University), Jul 13, 2018
In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose... more In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous contextfree grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series.
Journal of Algebra, Oct 1, 2007
We develop the cohomology theory of color Lie superalgebras due to Scheunert-Zhang in a framework... more We develop the cohomology theory of color Lie superalgebras due to Scheunert-Zhang in a framework of nonhomogeneous quadratic Koszul algebras. In this approach, the Chevalley-Eilenberg complex of a color Lie algebra becomes a standard Koszul complex for its universal enveloping algebra. As an application, we calculate cohomologies with trivial coefficients of Z n 2graded 3-dimensional color Lie superalgebras.
Two operads are said to belong to the same Wilf class if they have the same generating series. We... more Two operads are said to belong to the same Wilf class if they have the same generating series. We discuss possible Wilf classifications of non-symmetric operads with monomial relations. As a corollary, this would give the same classification for the operads with a finite Groebner basis. Generally, there is no algorithm to decide whether two finitely presented operads belong to the same Wilf class. Still, we show that if an operad has a finite Groebner basis, then the monomial basis of the operad forms an unambiguous context-free language. Moreover, we discuss the deterministic grammar which defines the language. The generating series of the operad can be obtained as a result of an algorithmic elimination of variables from the algebraic system of equations defined by the Chomsky-Schützenberger enumeration theorem. We then focus on the case of binary operads with a single relation. The approach is based on the results by Rowland on pattern avoidance in binary trees. We improve and refine Rowland's calculations and empirically confirm his conjecture. Here we use both the algebraic elimination and the direct calculation of formal power series from algebraic systems of equations. Finally, we discuss the connection of Wilf classes with algorithms for calculation of the Quillen homology of operads. CCS CONCEPTS • Mathematics of computing → Generating functions; • Computing methodologies → Algebraic algorithms.
On the Growth of Noetherian Filtered Rings
Communications in Algebra, Jan 4, 2003
Abstract The goal of this note is to show that associated graded ring of a Noetherian ring with a... more Abstract The goal of this note is to show that associated graded ring of a Noetherian ring with a descending filtration, grows subexponentially. The same is true for completed group algebras of Noetherian pro-p groups and for group algebras of Noetherian groups which are residually a finite p-group.
Vectors and Matrices
Springer texts in business and economics, 2011
Ordered n-tuple of objects is called a vector $$\mathbf{y} = ({y}_{1},{y}_{2},\ldots, {y}_{n}).$$... more Ordered n-tuple of objects is called a vector $$\mathbf{y} = ({y}_{1},{y}_{2},\ldots, {y}_{n}).$$ Throughout the text we confine ourselves to vectors the elements y i of which are real numbers.
Journal of Mathematical Sciences, Nov 18, 2009
We consider a couple of versions of classical Kurosh problem (whether there is an infinite-dimens... more We consider a couple of versions of classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra?) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a variety of algebras of this signature such that the free algebra of the variety contains multilinear elements of arbitrary large degree, while the clone of every such element satisfies some nontrivial identity. If, in addition, the number of binary operations is at least 2, then one can guarantee that each such clone is finitely-dimensional. Our approach is the following: we translate the problem to the language of operads and then apply usual homological constructions in order to adopt Golod's solution of the original Kurosh problem. The paper is expository, so that some proofs are omitted. At the same time, the general relations of operads, algebras, and varieties are widely discussed.
International Journal of Algebra and Computation, Aug 1, 2005
Quadratic algebras associated to pseudo-roots of noncommutative polynomials have been introduced ... more Quadratic algebras associated to pseudo-roots of noncommutative polynomials have been introduced by I. Gelfand, Retakh, and Wilson in connection with studying the decompositions of noncommutative polynomials. Later they (with S. Gelfand and Serconek) shown that the Hilbert series of these algebras and their quadratic duals satisfy the necessary condition for Koszulity. It is proved in this note that these algebras are Koszul.
arXiv (Cornell University), Dec 16, 2017
If a finitely generated monoid M is defined by a finite number of degree-preserving relations, th... more If a finitely generated monoid M is defined by a finite number of degree-preserving relations, then it has linear growth if and only if it can be decomposed into a finite disjoint union of subsets (which we call "sandwiches") of the form a w b where a, b, w ∈ M and w denotes the monogenic semigroup generated by w. Moreover, the decomposition can be chosen in such a way that the sandwiches are either singletons or "free" ones (meaning that all elements aw n b in each sandwich are pairwise different). So, the minimal number of free sandwiches in such a decompositions becomes a new numerical invariant of a homogeneous (and conjecturally, non-homogeneous) finitely presented monoid of linear growth.
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Papers by Dmitri Piontkovski