2021 Online Seminars
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Date: 30 November 2021
Speaker: Arnaldo Mandel (USP)
Title: Automorphisms of the cone of quasi semimetrics
Abstract:
A quasi semimetrics on a finite set V is a nonnegative real map on the edges of a complete directed graph with vertices V satisfying all possible triangle inequalities; alternatively, it can be thought of as a square matrix with null diagonal and some inequalities involving its elements. Integral quasi semimetrics are known as exponent matrices, and form an additive monoid, whose automorphisms were described by Dokuchaev, Kirichenko, Kudryavtseva and Plakhotnyk. The quasi semimetrics comprise a polyhedral cone whose symmetry groups were studied for small V by Deza, Dutour and Panteleeva. We determine the combinatorial symmetry group of the cone, thus obtaining both the preceding results in one shot.
It turns out that the only automorphisms are the "obvious" ones. We conclude with a digression on what automorphisms should be considered obvious, in a categorical setting.
Joint work with Misha Dokuchaev and Makar Plakhotnyk.
Date: 23 November 2021
Speaker: Nicola Sambonet (UFBA)
Title: Partial actions and category theory
Abstract:
In this talk we will review some elementary facts about partial actions, from the point of view of category theory. There are several ways to extend R.Exel's original definition of partial action of groups, most remarkably, C.Hollings defines a partial action of a monoid as a partial homomorphism into an inverse semigroup of partial symmetries, thus, as a particular premorphism into a restriction monoids. Some of the basic results about partial actions have categorical flavour, most evidently, the correspondence between the partial actions of a group to the ordinary action of the Exel inverse monoid or the M.Szendrei expansion. On the other hand, inverse categories and restriction categories are known structures with applications in computer science. We will extend Holling's definitions to generic categories, to obtain the above correspondence as a consequence of Yoneda's lemma on universal objects and adjoint functors. Thus, we will discuss the relation between the categorical versions of the Exel universal and the Szendrei expansion. Finally, we will describe some aspects about the globalization property of F.Abadie, J.Kellendonk and M.V.Lawson.
Joint work with Mikhailo Dokuchaev.
Date: 16 November 2021
Speaker: Érica Z. Fornaroli (UEM)
Title: Lie automorphisms of incidence algebras
Abstract:
Let $X$ be a finite partially ordered set and let $K$ be a field. The incidence algebra $I(X,K)$ of $X$ over $K$ is the set
$I(X,K)={f : X \times X \to K : f(x,y)=0 if x\nleq y}$
endowed with the usual structure of a vector space over $K$ and the product defined by $(fg)(x,y)=\sum_{x\leq t\leq y} f(x,t)g(t,y)$ for any $f, g\in I(X,K)$. We will present a full description of the Lie automorphisms of the incidence algebra I(X,K) in the case where $X$ is connected.
Joint work with Ednei A. Santulo Jr and Mykola Khrypchenko
Date: 26 October 2021
Speaker: Mykola Khrypchenko (UFSC)
Title: Crossed module extensions of inverse semigroups and the third inverse semigroup cohomology group
Abstract:
We introduce the notion of a crossed module over an inverse semigroup which generalizes the notion of a module over an inverse semigroup in the sense of Lausch [1], as well as the notion of a crossed module over a group in the sense of Whitehead [2] and Maclane [3]. With any crossed S-module A we associate a 4-term exact sequence of inverse semigroups A \xrightarrow{i} N \xrightarrow{\beta} S \xrightarrow{\pi} T, which we call a crossed module extension of A by T. We then introduce the so-called admissible crossed module extensions and show that equivalence classes of admissible crossed module extensions of A by T are in a one-to-one correspondence with the elements of the cohomology group H^3_\le(T^1,A^1), whenever T is an F-inverse monoid.
This is a joint work [4] with Mikhailo Dokuchaev (Universidade de Sao Paulo) and Mayumi Makuta (Universidade de Sao Paulo).
References
[1] Lausch, H. Cohomology of inverse semigroups. J. Algebra 35 (1975), 273-303.
[2] Whitehead, J. H. C. Combinatorial homotopy. II. Bull. Am. Math. Soc. 55 (1949), 453-496.
[3] MacLane, S. Cohomology theory in abstract groups. III. Operator homomorphisms of kernels. Ann. Math. (2) 50 (1949), 736-761.
[4] Dokuchaev, M., Khrypchenko, M., and Makuta, M. Inverse semigroup cohomology and crossed module extensions of semilattices of groups by inverse semigroups. (arXiv:2104.13481) (2021).
Date: 19 October 2021
Speaker: Renato Fehlberg Júnior (DMAT-UFES)
Title: On free subalgebras of varieties
Abstract:
We discuss about Makar-Limanov conjecture (about existence of free subalgebras) in the context of varieties of algebras. More precisely, we show that some results of L. Makar-Limanov, P. Malcolmson and Z. Reichstein on the existence of free associative algebras are valid in the more general context of varieties of algebras. This is a joint work with Javier Sánchez Serdà.
Date: 05 October 2021
Speaker: Javier Sánchez (IME-USP)
Title: On graded division rings
Abstract:
Given an associative ring with unit, P. M. Cohn characterized the homomorphisms from R to a division ring by means of a structure defined over the set of square matrices over R. P. Malcolmson described alternative ways of determining such homomorphisms using functions induced from the notions of rank of a matrix and of dimension over a division ring. In this work, we show that these characterizations can be implemented in the context of graded rings. More precisely, given a ring R graded by a group G we adapt the theory of Cohn and Malcolmson to determine the different graded homomorphisms from R to G-graded division rings. This is a joint work with Daniel E. N. Kawai.
Date: 28 September 2021
Speaker: Misha Dokuchaev (IME-USP)
Title: Homology and cohomology via the partial group algebra
Abstract:
Edson Ribeiro Alvares, Marcelo Muniz Alves and Maria Julia Redondo introduced and studied group cohomology based on modules over the partial group algebra. In a joint paper with Marcelo Muniz Alves (UFPR) and Dessislava Kochloukova (Unicamp) we link such partial homology and cohomology of a group G with coefficients in an irreducible (resp. indecomposable) module over the partial group algebra of G with the ordinary homology and cohomology of G with in general non-trivial coefficients. Furthermore, we compare the standard cohomological dimension of G with the partial cohomological dimension.
Date: 21 September 2021
Speaker: Alexander Lichtman (University of Wisconsin, Parkside, USA)
Title: Valuations on division rings
Abstract:
Let R be a ring with a valuation function v whose group of values is an arbitrary ordered group G. Assume that the graded ring gr(R) associated to this valuation is a left Ore domain. Then there exists a division ring D(R) which contains a subring isomorphic to R and is generated by this subring, and the valuation v extends to D(R) (Theorem I). This theorem generalizes Cohn's Theorem which considered the case when the valuation v is discrete. We derive Theorem I from more general Theorems II-IV. We study the division rings which are constructed in these theorems and obtain results which are new also in the discrete valuation case, and in particular for division rings generated by enveloping algebras or group rings.
Date: 14 September 2021
Speaker: Jairo Goncalves (IME-USP)
Title: Free symmetric and unitary pairs in normal subgroups of division rings with involution
Abstract:
Let D be a division ring with center of char \neq 2, with an involution *. Let D^{\dagger} be the multiplicative group of D, and let N be a normal subgroup of D^{\dagger}. An element u in D^{\dagger} is said to symmetric (resp. unitary) if u*=u (resp. u*= u^{-1}). With some mild restrictions, we show that if D is a symbol algebra of index p, and if there exists a non central symmetric element u in N, then N contains a free symmetric pair. We also present some general results for the presence of free unitary pairs in N and free symmetric pairs in the field of fractions of group algebras of torsion free nilpotent groups.