Koszul algebras and Donaldson-Thomas invariants

For a given symmetric quiver $Q$, we define a supercommutative quadratic algebra $\mathcal{A}_Q$ whose Poincar\'e series is related to the motivic generating function of $Q$ by a simple change of variables. The Koszul duality between supercommutative algebras and Lie superalgebras assigns to the algebra $\mathcal{A}_Q$ its Koszul dual Lie superalgebra $\mathfrak{g}_Q$. We prove that the motivic Donaldson-Thomas invariants of the quiver $Q$ may be computed using the Poincar\'e series of a certain Lie subalgebra of $\mathfrak{g}_Q$ that can be described, using an action of the first Weyl algebra on $\mathfrak{g}_Q$, as the kernel of the operator $\partial_t$. This gives a new proof of positivity for motivic Donaldson--Thomas invariants. In addition, we prove that the algebra $\mathcal{A}_Q$ is numerically Koszul for every symmetric quiver $Q$ and conjecture that it is in fact Koszul; we also prove this conjecture for quivers of a certain class.Comment: 25 pages, the main result on DT invariants of symmetric quivers is now not conditional on Koszulnes

Gröbner–Shirshov basis of the Adyan extension of the Novikov group

AbstractThe goal of this paper is to give a comparatively short and simple analysis of the Adyan origional group constraction (S.I. Adyan, Unsolvability of some algorithmic problems in the theory of groups, Trudy MMO 6 (1957) 231–298)

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

This project concerns the classification and study of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain ladder-like structures). We determine a linear system of parameters for each ring and explicitly determine the Hilbert series for the resulting Artinian reduction. As corollaries, we obtain the multiplicity and regularity of the original rings. This work extends results known for a subfamily coming from a two-sided ladder and includes constructive proofs which may be useful in future study of these rings and others. We also develop explicit elements in the Priddy complex which correspond via known isomorphisms to Tate variables in the acyclic closure of the residue field over the localization of our rings at their homogeneous maximal ideals

Higman-Neumann-Neumann extension and embedding theorems for Leibniz algebras

In this work we introduce the Higman-Neumann-Neumann (HNN)- extensions and the appropriate embedding theorems for dialgebras and Leibniz algebras. Due to the importance of the connection between the dialgebras and Leibniz algebras and the relationship between associative algebras and Lie algebras, we recall the theory of Groebner-Shirshov bases, and the Composition-Diamond Lemma in associative algebras and Lie algebras, as well as the theory of Groebner-Shirshov bases for dialgebras. As an application of the HNN-extensions of dialgebras and Leibniz algebras, we provide embedding theorems for dialgebras and Leibniz algebras, respectively: every dialgebra embeds inside its any HNN-extension and every Leibniz algebra embeds inside its any HNN-extension

Signature Gr\"obner bases in free algebras over rings

We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal domain. We give an algorithm for computing them, combining elements from the theory of commutative and noncommutative (signature) Gr\"obner bases, and prove its correctness. Applications include extensions of the free algebra with commutative variables, e.g., for homogenization purposes or for performing ideal theoretic operations such as intersections, and computations over $\mathbb{Z}$ as universal proofs over fields of arbitrary characteristic. By extending the signature cover criterion to our setting, our algorithm also lifts some technical restrictions from previous noncommutative signature-based algorithms, now allowing, e.g., elimination orderings. We provide a prototype implementation for the case when $R$ is a field, and show that our algorithm for the mixed algebra is more efficient than classical approaches using existing algorithms.Comment: 10 page

Veronese and Segre morphisms between non-commutative projective spaces

Number theory, Algebra and Geometr

Propriedades homológicas de finitude

Orientador: Dessislava Hristova KochloukovaTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Consideramos problemas nas teorias de grupos discretos, álgebras de Lie e grupos pro-p. Apresentamos resultados relacionados sobretudo a propriedades homológicas de finitude de tais estruturas algébricas. Primeiramente, discutimos Sigma-invariantes de produtos entrelaçados de grupos discretos. Descrevemos completamente o invariante Sigma1, relacionado à herança por subgrupos da propriedade de ser finitamente gerado, e descrevemos parcialmente o invariante Sigma2, relacionado à herança por subgrupos da propriedade de admitir uma apresentação finita. Aplicamos tais resultados ao estudo de números de Reidemeister de isomorfismos de certos produtos entrelaçados. Na sequência definimos e estudamos uma versão da construção de comutatividade fraca de Sidki na categoria de álgebras de Lie sobre um corpo de característica diferente de dois. Tal construção pode ser vista como um funtor que recebe uma álgebra de Lie g e retorna um certo quociente chi(g) da soma livre de duas cópias isomorfas de g. Demonstramos resultados sobre a preservação de certas propriedades algébricas por tal funtor e mostramos que o multiplicador de Schur de g é um subquociente de chi(g). Mostramos em particular que, para uma álgebra de Lie livre g de posto ao menos três, chi(g) é finitamente apresentável mas não é de tipo FP3 , e tem dimensão cohomológica infinita. Por fim, consideramos também uma versão da construção de comutatividade fraca na categoria de grupos pro-p para um número primo fixado p. Mostramos que tal construção também preserva diversas propriedades algébricas, como ocorre nos casos de grupos discretos e álgebras de Lie. Para tanto estudamos também produtos subdiretos de grupos pro-p; em particular demonstramos uma versão do Teorema (n ? 1) ? n ? (n + 1)Abstract: We consider problems in the theories of discrete groups, Lie algebras, and pro-p groups. We present results related mainly to homological finiteness properties of such algebraic structures. First, we discuss Sigma-invariants of wreath products of discrete groups. We give a complete description of the Sigma1-invariant, which is related to the inheritance of the property of being finitely generated by subgroups. We also describe partially the invariant Sigma2, which is related to the inheritance of finite presentability by subgroups. We apply such results in the study of Reidemeister numbers of isomorphisms of certain wreath products. Then we define and study a version of Sidki¿s weak commutativity construction in the category of Lie algebras over a field whose characteristic is not two. Such construction can be seen as a functor that receives a Lie algebra g and returns a certain quotient chi(g) of the free sum of two isomorphic copies of g. We prove some results on the preservation of certain algebraic properties by this functor, and we show that the Schur multiplier of g is a subquotient of chi(g). We show in particular that, for a free Lie algebra g with at least three free generators, chi(g) is finitely presentable but not of type FP3 , and has infinite cohomological dimension. Finally, we also consider a version of the weak commutativity construction in the category of pro-p groups for a fixed prime number p. We show that such construction also preserves several algebraic properties, as occurs in the cases of discrete groups and Lie algebras. To this end, we also study subdirect products of pro-p groups. In particular we prove a version of the (n ? 1) ? n ? (n + 1) TheoremDoutoradoMatematicaDoutor em Matemática2015/22064-6; 2016/24778-9FAPES