Session on Rings and Algebras:
=Monday, August 5
14.30-15.00
Olivier Mathieu (Institut Camille Jordan, Lyon, France)
On self similar groups
A self similar group is a group with a self similar action on a tree. Here we provide an exact condition for a finitely nilpotent group to be self similar.
15.10-15.30
Rodrigo Lucas Rodrigues (Universidade Federal do Ceara, Brasil)
The algebraic classification of commutative power-associative algebras of low dimension
This talk is concerned with small dimension commutative power-associative algebras over an algebraically closed field $k$ with characteristic relative prime with 30. We prove that an algebra of this variety is Jordan if its dimension as vector space over $k$ is not greater than 3. Albeit in dimension 4, non-Jordan commutative power-associative algebras exist, we can show that such algebras yet admits Wedderburn decomposition. Finally, we classify algebraically all commutative power-associative algebras of dimension less or equal than 4.
This is a joint work with Angelo Papa Neto and Elkin Quintero Vanegas.
15.40-16.10
Sergio Roberto López-Permouth (Ohio University, USA)
On the extent of amenability of bases of infinite dimensional algebras
Let $F$ be a field and $A$ an $F$-algebra . For $r \in A$, let $l_r$ denote the left multiplication map $l_r: A \rightarrow A$. A basis $B$ for $A$ is amenable if the matrix representation with respect to $B$ of every $l_r$ is row finite. An algebra that has an amenable basis is said to be an amenable algebra; countable dimension algebras are amenable. For every countable basis $B= \{ b_i\}_{i=1}^{\infty}$ of $A$, there exists a topology $\tau_{B}$ on $F^{(B)}$ such that $B$ is amenable if and only if, for all $r \in A$, the sequence $\{[rb_i]_{B}\}_{i=1}^{\infty}$ of representations of $\{rb_i\}_{i=1}^{\infty}$ with respect to $B$ converges to $0$ in $\tau_{B}$.
Given an algebra $A$, and a basis $B$ for $A$, the amenability domain of $B$ is the subalgebra of $A$ consisting of all elements $r \in A$ with row finite matrix representation with respect to $B$. A basis having amenability domain $F$ is said to be contrarian; we show that, under some mild additional hypotheses, amenable algebras always have contrarian bases. The collection of amenability domains of bases of an algebra $A$ is called the amenability profile of $A$. The profile is a measurement of the diversity of the bases of $A$ and serves to sort them according to the extent of their amenability.
We consider when profiles are minimal (consisting of only $F$ and $A$) and when they are maximal (consisting of all subalgebras of $A$); the former algebras are said to lack discernment and the latter to be full rank. We provide an example of a graph magma algebra without discernment. We show that $F[x]$ does not lack discernment and that graph magma algebras are never full rank.
This talk relates to ongoing collaborations with many coauthors including Al-Essa, Aydogdu, Díaz Boils, Muhammad, Muthana, and Stanley.
16.20-16.40
Martha Lizbeth Shaid Sandoval Miranda (UAM-Iztapalapa, México)
On the weak-injectivity profile of a ring
In order to study rings in which every essential cyclic module is embedded in a projective module, a concept of weak injectivity was introduced by S.K. Jain and S. López-Permouth. Given modules $N, M \in Mod(R),$ $M$ is said to be weakly $N$-injective if whenever $\varphi \in Hom_R(N, E(M)),$ there exists $X\leq E(M)$ satisfying that $X\cong M$ and $\varphi(N)\subseteq X.$ A module $M$ is said to be weakly injective if it is weakly $N-$injective for every $N\in fgmod(R),$ where $fgmod(R)$ denotes the full subcategory of finitely generated $R-$modules (see S.K Jain and S.R López-Permouth, A survey of theory of weakly injective modules, Computational Algebra. Lecture Notes in Pure and Applied Mathematics 151 (1994) 205 - 232). Weakly injective modules have served to offer characterizations of qfd rings and to shed light about subtleties of open conjectures regarding QI rings.
Now, given a module $M,$ $$\displaystyle {\mathcal{W}\mathfrak{In}^{-1}(M)}:=\{N \in Mod(R) \mid M \mbox{ is weakly }\displaystyle N-\mbox{injective} \},$$ denotes the domain of weak injectivity of M. Domains of weak-injectivity may be used to gauge at least two different properties, namely, injectivity and weak injectivity. The first one gauged via the complete domains of weak injectivity, and the second one by looking only at only the finitely generated modules in them.
In this work, we will explore properties of weak injectivity domains. In particular, we study $\displaystyle\bigcap \{ \mathcal{W}\mathfrak{In}^{-1}(M)\mid M\in Mod(R)\}$ looking for some analogous to the situation of poverty but now for weak injectivity case.
This is a joint work with Pınar Aydoğdu (Hacettepe University), and Sergio López-Permouth (Ohio University).
17.10-17.40
Misha Dokuchaev (Universidade de São Paulo, Brasil)
Group cohomology related to the partial group algebra
With respect to theory of partial actions and partial representations, two group cohomologies were introduced. The first one was given in [DKh], in which actions of a group $G$ on abelian groups ($G$-modules) were replaced by unital partial actions of $G$ on commutative semigroups (partial $G$-modules). Another cohomology was defined in [AAR], where $KG$-modules (or $G$-modules) were substituted by modules over the partial group algebra $K _{par} G.$ The latter algebra governs the partial representations of $G$ in a similar fashion as $KG$ does for the representations of $G.$ Despite the relations between partial actions and partial representations, the two cohomologies have little in common.
Given a unital partial action $\alpha $ of $G$ on an algebra ${\mathcal A}$ we consider ${\mathcal A}$ as a $K_{par} G$-module in a natural way and study the globalization problem for the cohomology in the sense of [AAR] with values in ${\mathcal A}.$ Assuming that ${\mathcal A}$ is a product of blocks, we prove that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous. As a consequence we obtain that the Alvares-Alves-Redondo cohomology group $H_{par}^n(G,{\mathcal A})$ is isomorphic to the usual cohomology group $H^n(G,{\mathcal M}({\mathcal B})),$ where ${\mathcal M}({\mathcal B})$ is the multiplier algebra of ${\mathcal B}$ and ${\mathcal B}$ is the algebra under the enveloping action of $\alpha .$ The results are obtained using the ideais of our earlier preprint on the globalization problem for partial cohomologies in the sense of [DKh].
This is a joint work with Mykola Khrypchenko and Juan Jacobo Simón.
Bibliography
[AAR] E.R.Alvares, M.M.S.Alves, M.J.Redondo, Cohomology of partial smash products, J. Algebra, 482 (2017), 204-223.
[DKh] M.Dokuchaev, M.Khrypchenko, Partial cohomology of groups, J. Algebra, 427 (2015), 142-182.
17.50-18.10
Elkin Oveimar Quintero Vanegas (Universidade Federal do Amazonas, Brasil)
Train algebras of rank and dimension 4 that are power-associative
Train algebras of rank 4 which are power-associative was classified in three families in which two of them are Jordan algebras. Commutative power-associative algebras which are non Jordan belong to the third one. Recently, we classified all commutative power-associative algebras of dimension 4 over an algebraically closed field. The aim of this work is to show that all commutative power-associative and non Jordan algebras of dimension 4 are train algebras of rank 4. Furthermore, we show that class, the power-associative and train algebras of rank and dimension 4, has a unique non trivial irreducible representation.
This is a joint work with R. Lucas Rodrigues.
Tuesday, August 6
14.30-15.00
Ángel del Río (Universidad de Murcia, España)
Torsion units of integral group rings: Positive and negative results
Let $G$ be a finite group and let $\mathbb{Z}G$ denote the integral group ring of $G$. Hans Zassenhaus conjectured in the 1970's that every torsion unit of $\mathbb{Z}G$ is conjugate in the rational group algebra to an element of $\pm G$. This conjecture was proved for many families of groups but recently Eisele and Margolis have shown a counterexample, or rather a family of counterexamples. They comes from a strategy to produce candidates to minimal counterexamples proposed by Margolis and del Río. This leads to the question of classifying the finite groups for which the Zassenhaus Conjecture holds. We will present the strategy leading to the counterexample and some families of cyclic-by-nilpotent groups for which the Zassenhaus Conjecture holds.
This is a joint work with Mauricio Caicedo.
15.10-15.30
Héctor Suárez (UPTC, Tunja, Colombia)
$\sigma$-Filtered skew PBW extensions and their homogenization
In this work we provide a special filtration on the skew PBW extensions. We define $\sigma$-filtered skew PBW extensions and present some properties of these algebras. We show that the homogenization of a $\sigma$-filtered skew PBW extension $A$ of $R$ is a graded skew PBW extension of the homogenization of $R$. Using this fact we prove that if the homogenization of $R$ is Auslander-regular, then the homogenization of $A$ is a domain noetherian, Artin-Schelter regular and $A$ is noetherian, Zariski and (ungraded) skew Calabi-Yau.
This is a joint work with Armando Reyes.
15.40-16.10
Plamen Koshlukov (State University of Campinas, Brazil)
Trace ideals of almost polynomial growth
In this talk we consider algebras with trace and their trace polynomial identities over a field of characteristic 0. We study the algebras of $2\times 2$ diagonal matrices and describe all possible traces on these algebras. We prove that in the non-degenerate cases the corresponding trace codimensions have exponential growth. Moreover we prove that a variety generated by a finite dimensional algebra with trace has a polynomially bounded (trace) codimension sequence if and only if it does not contain any of the algebras of two concrete series of trace algebras defined on the $2\times 2$ diagonal matrices. In fact we were able to exhibit finite generating sets for the trace identities of each algebra in these two series, and to compute the codimension sequences in an explicit form. As a consequence of our methods of proof we prove that given a variety of trace algebras its codimension growth is either polynomial or exponential.
This is a joint work with D. La Mattina (Palermo, Italy) and A. Ioppolo (Palermo, Italy, and UNICAMP, Brazil).
16.20-16.40
Mauricio Medina-Barcenas (Northern Illinois University, USA)
A generalization of right hereditary rings
In this talk we will study the concept of hereditary ring in the module-theoretic context. We will define $\Sigma$-Rickart modules which generalize right hereditary rings. We will define these modules in terms of Rickart modules and we will see some of their properties which extend classical results of hereditary rings. Also, we will see when the endomorphism ring of a $\Sigma$-Rickart module is a right hereditary ring.
This is a joint work with Gangyong Lee
17.10-17.40
Victor Petrogradsky (University of Brasilia, Brazil)
Nil restricted Lie algebras and superalgebras analogous to Grigorchuk and Gupta-Sidki groups
The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2, Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic.
There are also analogues of the Grigorchuk and Gupta-Sidki groups in the world of Lie superalgebras of an arbitrary characteristic constructed by the author. In these examples, $\mathrm{ad}(a)$ is nilpotent, $a$ being even or odd with respect to ${\mathbb{Z}}_2$-grading as Lie superalgebras. This property is an analogue of the periodicity of the Grigorchuk and Gupta-Sidki groups. So, we get an example of a nil finely-graded Lie superalgebra of slow polynomial growth, which shows that an extension of a theorem due to Martinez and Zelmanov for the Lie superalgebras of characteristic zero is not valid.
We have a family of 2-generated restricted Lie algebras of slow polynomial growth with a nil $p$-mapping, a field of positive characteristic being arbitrary. In particular, we obtain a continuum subfamily of nil restricted Lie algebras having Gelfand-Kirillov dimension one but the growth is not linear.
17.50-18.10
Helbert Venegas (Universidad Nacional de Colombia)
Cancellation problem for AS-Regular algebras of dimension three
A noncommutative version of the Zariski cancellation problem (ZCP) asks whether $A[x]\cong B[x]$ implies $A\cong B$ when $A$ and $B$ are noncommutative algebras. In this talk we study the ZCP for noncommutative noetherian connected graded Artin-Schelter (abbreviated as AS) regular algebras of global dimension three. In particular, we prove that if $A$ is generated in degree 1 and $A$ is not polynomial identity, then it is cancellative.
This is a joint work with Xin Tang, Fayetteville state university and James Zhang, University of Washington, USA.