## Schedule

(Thursday's, 14:00, Room: A-259)#### Past seminars

- 22 Aug, 2024.
*Daniela Martinez Correa (IME-USP)*

**Métodos para calcular cocaracteres de álgebras associativas.**Seja \(F\) um corpo de característica zero e considere \(E\) a álgebra de Grassman de dimensão infinita sobre \(F\). Neste seminário apresentaremos dois algoritmos para calcular as multiplicidades da sequência de cocaracteres da álgebra das matrizes triangulares superiores de tamanho \(n\times n\) com entradas em \(E\). Este é um trabalho em conjunto com Lucio Centrone (Uniba, Itália) e Vesselin Drensky (IMI-BAS, Bulgária).

Show abstract - 01 Jul, 2024.
*(room A-136) Arturo Pianzola (University of Alberta, Canada)*

**Fiberwise criteria for forms of algebras.**Many interesting objects in infinite dimensional Lie theory become "geometric" when viewed as objects over suitable rings and not the base field. For example the derived algebra of an affine Kac-Moody Lie algebras, when we factored out the centre, is a Lie algebra over the ring of complex Laurent polynomials. This allows us to look at the fibres of such geometric object with the hope of gaining information about our original algebraic object. We provide a fiberwise criterion that yields information, say about sheaves of OX-algebras for example, based on their behaviour at closed fibres. This criterion permits to answer a question about affine Kac-Moody Lie algebras that I. Burban had asked the authors. Joint work with P. Gille.

Show abstract - 13 Jun, 2024.
*João Fernando Schwarz (UFABC)*

**Noether's Problems.**Noether's problem is a classical and very important problem in algebra. First we dissipate the confusion regarding the statement of this problem: we consider five versions that are common in the literature. It is an intrinsecally interesting problem in invariant theory, but with far reaching applications in the sutdy of moduli spaces, PI-algebras, and the Inverse problem of Galois theory, among others. To obtain a noncommutative analogue of Noether's problem, one would need a significant skew field that shares a role similar to the field of rational functions. Given the importance of the Weyl fields due to Gelfand-Kirillov's Conjecture, in 2006 J. Alev and F. Dumas introduced what is nowdays called the noncommutative Noether's problem. Their problem was stated in zero characteristic and linear actions. More general versions of noncommutative Noether's problem are introduced, and we consider its analogue in prime characteristic.

Show abstract - 16 May, 2024.
*Vladimir Sokolov (Russian Academy of Sciences, Russia)*

**Decomposition of a Lie algebra into sum of two subalgebras.**Any vector space decomposition of a Lie algebra into a direct sum of two subalgebras generate an integrable ODE system. We discuss algebraic properties to this system. Some reductions of the system are related to special types of non-associative algebras.

Show abstract - 25 Apr, 2024.
*(15:00) Henrique de Oliveira Rocha (IME-USP)*

**Representações de álgebras de Lie de campos vetoriais em variedades e supervariedades algébricas.**Apresentaremos resultados obtidos sobre as representações de álgebras de Lie de campos vetoriais em variedades algébricas que admitem uma ação compatível do anel de coordenadas da variedade e são finitamente geradas como módulos sobre essa álgebra comutativa. Também mostraremos que alguns desses resultados podem ser generalizados para supervariedades.

Show abstract - 25 Apr, 2024.
*Alexandre Grichkov (IME-USP)*

**Álgebras de Malcev com identidades \(J(xy,z,t)=J(x,y,z)t=0\).**Descrevemos estrutura das álgebras de Malcev com identidades \(J(xy,z,t)=J(x,y,z)t=0\). Para isso introduzimos várias cohomologias para estas álgebras e construímos os exemplos onde estas cohomologias não são triviais.

Show abstract - 21 Mar, 2024.
*Gilson Reis dos Santos Filho*

**Coálgebras não-associativas, o radical localmente nilpotente e a construção de coálgebras de Jordan não-comutativas.**Nesta apresentação descrevemos resultados associados ao estudo da seguinte conjectura formulada por I. Shestakov: Uma variedade de álgebras (não necessariamente associativa) admite o radical localmente nilpotente se, e somente se, as coálgebras associadas à respectiva variedade são localmente finitas. Vamos dar especial enfoque no resultado obtido por I. Shestakov, L. Murakami, GRS, que mostra que, para toda variedade em uma certa classe de variedades de álgebras, se existe radical localmente nilpotente, então as coálgebras da respectiva variedade são localmente finitas. Em seguida, abordaremos as recentes tentativas de estudar a conjectura no âmbito de subvariedades de álgebras de Jordan não-comutativas, especificamente a existência de exemplos de coálgebras não localmente finitas nestas variedades.

Show abstract - 14 Mar, 2024.
*Alexandre Grichkov*

**Variedades nilpotentes de classe dois os loops diassociativos.**No trabalho [Grishkov A., Rasskazova D., Rasskazova M., Stulh I., Nilpotent Steiner loops of class 2, COMMUNICATIONS IN ALGEBRA v. 46(2019), n.12,5480-5486] descrevemos o loop de Steiner livre nilpotente de classe dois. Relembramos que o loop é de Steiner se este loop é diassociativo de expoente dois. Nesta apresentação vamos generalizar este resultado para loops diassociativos nilpotentes de classe dois de expoente quatro.

Show abstract - 24 Aug, 2023.
*(15:00) Faber Gómez (Universidad de Antioquia, Colombia)*

**Finite dimensional Jordan superalgebras and the Wedderburn principal theorem.**In this talk, we will show some results about the Wedderburn principal theorem for finite dimensional Jordan superalgebras. Also, we would like to talk about other classical problems associated with it, such as the second cohomology group. Some examples are shown.

Show abstract - 24 Aug, 2023.
*Frank Michael Forger (IME-USP)*

**Polinômios hiperbólicos a valores em álgebras C*.**O conceito de polinômio hiperbólico foi introduzido nos anos 40/50 do século passado por Lars Garding no âmbito do estudo de equações diferenciais parciais (lineares) hiperbólicas para captar propriedades essenciais do símbolo principal de um operador diferencial (linear) hiperbólico, e desde então se alastrou para outras e bem diferentes áreas da Matemática, tais como otimização e programação. Mas na sua aplicação original a equações diferenciais parciais, mostrou-se insuficiente para garantir a validade do teorema fundamental sobre boa postura do problema de Cauchy, de modo que é natural procurar por uma generalização que se refira diretamente ao símbolo principal do operador e não apenas ao seu determinante. Ao longo dos últimos anos, o palestrante e o seu aluno de doutorado Tiago Radins desenvolveram "ab initio" uma nova teoria de polinômios hiperbólicos a valores em uma álgebra de matrizes, e mais geralmente em uma álgebra C*, que promete ter o potencial para superar essa lacuna e assim chegar a uma definição do conceito de um operador diferencial hiperbólico do mesmo nível que a já tradicional definição de um operador diferencial elíptico.

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Nesta palestra, pretende-se apresentar alguns aspectos básicos desta nova teoria que, apesar de ainda se encontrar num estágio embrional, já mostra aspectos muito interessantes e inéditos que podem impulsionar novas direções de pesquisa e abrir caminhos absolutamente inéditos, além de novas aplicações, hoje ainda completamente imprevisíveis. - 03 Aug, 2023.
*Bruno L. M. Ferreira (UTFPR)*

**Problema de Herstein para álgebras alternativas.**Nesta palestra abordaremos o famoso Teorema de Herstein, a saber:

Show abstract

**Teorema:**Se \(\phi\) é um homomorfismo de Jordan de um anel \(R\) em um outro anel \(R'\) de característica diferente de \(2\) e \(3\) então \(\phi\) é um homomorfismo ou um anti-homomofismo.

Contaremos um pouco sobre a história de outros resultados anteriores que motivaram Herstein, bem como resultados posteriores, provados em estruturas associativas, que tem sido influenciados pelo seu Teorema. Já em estruturas não associativas, como por exemplo os anéis alternativos, falaremos sobre a extensão de um resultado devido a Hua:

**Teorema:**Seja \(\mathcal{K}\) um anel de divisão. Todo semi-automorfismo de \(\mathcal{K},\) isto é, uma aplicação \(\sigma :\mathcal{K} \longrightarrow \mathcal{K}\) satisfazendo \[\sigma (a+b)=\sigma (a)+\sigma (b),\] \[\sigma (aba)=\sigma (a)\sigma(b) \sigma (a)\] e \(\sigma (1)=1,\) é um automorfismo ou um anti-automorfismo.

Por fim, concluiremos mostrando alguns resultados que tem sido feitos na direção a solucionar o famoso Teorema de Herstein para o caso de anéis alternativos. - 29 Jun, 2023.
*Elena Aladova (IME-USP)*

**Automorphisms of the category of free finitely generated non-associative algebras with unit.**Let \(\Theta\) be an arbitrary variety of algebras and \(\Theta^0\) the category of all free finitely generated algebras in \(\Theta\). The group \(Aut(\Theta^0)\) of automorphisms of the category \(\Theta^0\) plays an important role in universal algebraic geometry. It turns out that for a wide class of varieties, the group \(Aut(\Theta^0)\) can be decomposed into a product of the subgroup \(Inn(\Theta^0)\) of inner automorphisms and the subgroup \(St(\Theta^0)\) of strongly stable automorphisms.

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In this talk we would like to give some clarifying remarks describing the place of \(\Theta\) and \(Aut(\Theta^0)\) in the general set up of the universal algebraic geometry. Then we present the method of verbal operations which provides a machinery to calculate the group \(St(\Theta^0)\) and discuss some new results concerning the group of strongly stable automorphisms for the variety of non-associative algebras with unit.

This research is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, No. 101829/20221. - 22 Jun, 2023.
*S.V.Pchelintsev (Financial Academy of the Russian Government, Moscow City Pedagogical University)*

**Binary (-1, 1)-bimodules over semisimple algebras.**The birepresentations of semisimple binary (-1,1)-algebras are studied. It is proved that any irreducible binary (-1,1)-bimodule over a simple algebra is alternative. Examples of indecomposable not irreducible (-1,1)-bimodules are constructed.

Show abstract - 15 Jun, 2023.
*V. Bovdi (UAE University, United Arab Emirates)*

**Modular group algebras whose group of units is locally nilpotent.**Let \(p\) be a prime integer. Let \(F\) be a field of characteristic \(p\), \(G\) a group, \(FG\) the group algebra, and \(V(FG)\) the group of the units of \(FG\) with augmentation 1. The anti-automorphism \(g\mapsto g^{-1}\) of \(G\) extends linearly to the algebra \(FG\). This extension leaves \(V(FG)\) setwise invariant, and its restriction to \(V(FG)\) followed by \(u\mapsto u^{-1}\) gives an automorphism of the group \(V(FG)\). The elements of \(V(FG)\) fixed by this automorphism are said unitary. They form a subgroup, which we denote by \(V_*(FG)\). The group \(V_*(FG)\) has a complex structure and has several applications. The investigation of the structure of \(V_*(FG)\) was initiated by A. Bovdi.

Show abstract

In my talk I will present characterization those modular group algebras \(FG\) whose:

1. the group \(V(FG)\) is locally nilpotent,

2. the group \(V_*(FG)\) is locally nilpotent or nilpotent under the classical involution of \(FG\). - 01 Jun, 2023.
*Thiago Castilho de Mello (UNIFESP)*

**Gradings and images of multilinear polynomials on matrix algebras.**Let \(f=f(x_1, \dots, x_m)\) be a noncommutative polynomial over a field \(K\). If \(A\) is a \(K\)-algebra, such a polynomial defines a map \(f:A^m\longrightarrow A\) by evaluation of variables by elements of \(A\).

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A famous problem, known as "Lvov-Kaplansky's Conjecture" asks whether the image of a multilinear polynomial \(f\) on \(M_n(K)\) is a vector subspace, or, equivalently, the image of \(f\) is one of the following subspaces: \(\{0\}\), \(K\) (the set of scalar matrices), \(sl_n(K)\) (the set of traceless matrices) or \(M_n(K)\). When \(K\) is an algebraically closed field, such problem has a positive answer only for \(n=2\) or \(m=2\). There are some parcial answers for \(n=3\) and \(m=3\), but no complete solution is known.

Recently, many variations of this conjecture have been studied. For instance, images of multilinear polynomials on upper (and strictly upper) triangular matrices, images of polynomials on quaternion algebras and on Jordan algebras of a symmetric bilinear form have been completely described.

In this talk we intend to explore how gradings on \(M_n(K)\) can be used to obtain results on images of multilinear polynomials and also polynomial identities on \(M_n(K)\). Also, we intend to present some new results and conjectures about images of graded polynomials on \(M_n(K)\), when it is endowed with some special kind of gradings. - 25 May, 2023.
*Claudemir Fideles Bezerra Júnior (IMECC-Unicamp)*

**Ferramentas Aritméticas aplicadas a PI-teoria.**Pesquisas em PI-teoria utilizam métodos e técnicas provenientes de outras áreas da Álgebra (estrutura de anéis, representações de álgebras, álgebras graduadas, ações de grupos, para citar algumas), da Combinatória, da teoria de representações de grupos (especialmente dos grupos simétrico e geral linear), da álgebra linear, da teoria de invariantes, e outras áreas da Matemática. O objetivo principal desta palestra é apresentar a aplicação de Ferramentas Aritméticas básica para determinar uma base para as identidades graduadas da álgebra de Grassmann \(E\) de dimensão infinita munida de estruturas graduadas cujo o suporte coincide com um subgrupo do grupo cíclico infinito \(\mathbb{Z}\). Para chegar a tal objetivo daremos uma rápida introdução as identidades polinomiais e álgebras graduadas. Apresentaremos, em mais detalhes, as chamadas \(\mathbb{Z}\)-graduações \(n\)-induzidas sobre \(E\), com \(n\leq 2\). E no final, descreveremos uma base para as identidades graduadas em cada um dos casos anteriores. Nestes trabalhos notam-se um forte uso da Aritmética como ferramenta, providenciando uma interessante conexão entre esta área e a PI-Teoria.

Show abstract - 18 May, 2023.
*Dmitry Logachev (UFAM)*

**Drinfeld modules and Anderson t-motives - characteristic p analogs of abelian varieties.** - 20 Apr, 2023.
*Misha Verbitsky (IMPA)*

**Hyperbolic groups are not Ulam stable.**Let \(G\) be a Lie group equipped with a left-invariant Riemannian metric \(d\), and \(\Gamma\) any group. An \(\epsilon\)-homomorphism is a map \(\rho: \Gamma \rightarrow G\) which is "not far" from a homomorphism. More formally, an \(\epsilon\)-homomorphism is a map \(\rho: \Gamma \rightarrow G\) satisfying \(d(\rho(xy), \rho(x)\rho(y)) <\epsilon\) for all \(x, y\in \Gamma\). A group \(\Gamma\) is called Ulam stable if any \(\epsilon\)-homomorphism \(\Gamma\rightarrow U(n)\) can be approximated by homomorphisms. Ulam stability was originally treated by D. Kazhdan (1982), following a question of V. Milman. Kazhdan has proven that all amenable groups are Ulam stable. Then he constructed an \(\epsilon\)-homomorphism \(\rho: \Gamma \rightarrow U(2),\) for any given \(\epsilon >0,\) which cannot be 1/10-approximated by a homomorphism, where \(\Gamma\) is the fundamental group of a genus 2 Riemann surface. I would give a geometric version of his construction, and construct an \(\epsilon\)-homomorphism \(\rho: \Gamma \rightarrow G\) which cannot be 1/10-approximated for any Lie group \(G\), where \(\Gamma\) is the fundamental group of a compact Riemannian manifold of strictly negative sectional curvature. The proof is based on a quantitative version of Ambrose-Singer theorem, which claims that the holonomy of a connection can be expressed through its curvature, and Gromov's comparison inequality used to bound an area of a geodesic triangle in a CAT(-\(\epsilon\))-space. This is a joint work with Misha Brandenbursky.

Show abstract - 28 Nov, 2019.
*Zerui Zhang (IME-USP)*

**On Novikov superalgebras, Novikov-Poisson algebras, Poisson algebras and metabelian Poisson algebras.** - 21 Nov, 2019.
*André Zaidan (IME-USP)*

**Reductions in representation theory of Lie algebras of Vector Fields.**For a smooth irreducible affine algebraic variety, we studied two classes of modules: gauge modules and Rudakov modules, admitting compatible actions of both the algebra of functions and the Lie algebra of vector fields on the variety. We proved that gauge modules and Rudakov modules corresponding to simple \({gl}_n\)-module remain irreducible as a module over the Lie algebra of vector fields unless it appears in the de Rhan complex. We also studied irreducibility of tensor product of Rudakov modules.

Show abstract - 10 Oct, 2019.
*Yuly Billig (Univ Carleton, Canada)*

**Representations of Lie algebras of vector fields on algebraic varieties.**We study a category of representations of the Lie algebras of vector fields on affine algebraic variety X that admit a compatible action of the algebra of polynomial functions on X. We investigate two classes of simple modules in this category: gauge modules and Rudakov modules, and establish a covariant pairing between modules of these two types. We state a conjecture that gauge modules exhaust modules in this category that are finitely generated over the algebra of functions. We give a proof of this conjecture when X is the affine space. This is a joint work with Slava Futorny, Jonathan Nilsson, Andre Zaidan, Colin Ingalls and Amir Nasr.

Show abstract - 26 Sep, 2019.
*A. Tsurkov (UFRN)*

**IBN-varieties of algebras.** - 19 Sep, 2019.
*João Fernando Schwarz (IME-USP)*

**An algorithm in noncommutative invariant theory of division algebras.**In this talk we discuss the problem of finding a nice set of generators - satisfying the Weyl relations - in the invariant division subring of the Weyl Fields, which are the division ring of fractions of the Weyl Algebra. We show that despite being a combinatorially complex object, having for instance the free associative algebra as a subalgebra, such an algorithm can be found with very small computational cost, in case we consider invariants under the action of finite pseudo-reflection groups. We finish discussing some general questions about computability and open problems. This is a joint work with V. Futorny.

Show abstract - 22 Aug, 2019.
*Marina Rasskazova (Omsk State Technologic University, Russia)*

**Cubic hipersufaces and commutative Moufang loops.**Yu.Manin noted that for any non-degenerated cubic surface \(V(k)\) over a field \(k\) there exists universal equivalence \(\sim\) on \(V(k)\) such that the set \(V(k)/\sim\) admits natural structure of commutative Moufang loop of exponent \(6\). Yu.Manin asked the order of free commutative Moufang loop with exponent \(6\) and \(n\) generators. In this talk we give answer on Manin's question for \(n<9\).

Show abstract - 15 Aug, 2019.
*(15:00) Mikhail Zaicev (Lomonosov Moscow University)*

**Central polynomials of associative algebras and their growth.**A central polynomial for an algebra A is a polynomials in noncommutative variables taking central values in A. If an algebra has central polynomials, for example, the algebra of kxk matrices, can one measure how many are there? We study the growth of central polynomials for any algebra satisfying a polynomial identity over a field of characteristic zero. We prove the existence of two limits called the central exponent and the proper central exponent of A. They give a measure of the exponential growth of the central polynomials and the proper central polynomials of A. They are comparable with exp the PI-exponent of the algebra.

Show abstract - 15 Aug, 2019.
*Misha Neklyudov (UFAM, Manaus)*

**Ergodicity of infinite particle systems with locally conserved quantities.**We analyse certain degenerate infinite dimensional sub-elliptic generators, and obtain estimates on the long-time behaviour of the corresponding Markov semigroups that describe a certain model of heat conduction. In particular, we establish ergodicity of the system for a family of invariant measures, and show that the optimal rate of convergence to equilibrium is polynomial. Consequently, there is no spectral gap, but a Liggett-Nash type inequality is shown to hold. The proof is based on application of Lie algebra structure of family of vector fields corresponding to our generator.

Show abstract - 13 Jun, 2019.
*Bakhrom Omirov (National University of Uzbekistan)*

**Some cohomologically properties of solvable Leibniz algebras with maximal rank nilradicals.**This talk is devoted to the description of solvable Leibniz algebras with maximal rank nilradicals. It will be shown that such algebras are complete (in sense that they are centerless and their the first cohomology group is trivial). The quantity of such Leibniz algebras in terms of the properties of nilradical will be discussed. Comparisons with some existed results for Lie and Leibniz algebras will be given, as well. Moreover, among of such solvable Leibniz algebras we shall indicate a subclass of Lie algebras whose cohomology group is trivial.

Show abstract - 06 Jun, 2019.
*Alexei Krasilnikov (UnB)*

**The central polynomials of a strongly Lie nilpotent associative algebra.** - 23 May, 2019.
*Alexander V. Mikhailov (University of Leeds, UK)*

**Polynomial integrable Hamiltonian systems and symmetric powers of \(\mathbb{C}^2\).** - 02 May, 2019.
*Vladimir Grebenev (Institute of Computational Technologies, SD RAS, Russia and IME-USP)*

**Conformal invariance of the zero-vorticity isolines in 2D turbulence.**It was clearly validated experimentally in [ Bernard D., Boffetta G., Celani A. and Falkovich G. Conformal invariance in two-dimensional turbulence. Nature Physics. 2006. 2(2). P. 124-128.] that the zero-vorticity isolines in 2D turbulence belongs to the class of conformal invariant SLE(k) (Schram-Lowner evolution) curves with k= 6. The diffusion coefficient k classifies the conformally invariant random curves. With this motivation, we performed a Lie group analysis in [Grebenev V.N., Wac lawczyk M. and Oberlack M. Conformal invariance of the Lungren-Monin-Novikov equations for vorticity fields in 2D turbulence. J. Phys. A: Math. Theor. 2017. 50(43) P. 435502-44.] of the first equation (i.e. for the evolution of the 1-point probability density function (PDF) \(f_1(x_{(1)},w_{(1)},t))\) of the inviscid Lundgren-Monin-Novikov (LMN) equations for 2D vorticity fields. We proved that the conformal group (CG) is broken for the 1-point PDF but the CG is recovered for the equation restricted on the characteristics with zero-vorticity. As for the zero-vorticity isolines, it implicitly leads to their CG invariance. The main focus of the present work is directed to a Lie group analysis of the characteristic equations of the inviscid LMN hierarchy truncated to the first equation. With this, the CG invariance of the characteristics with zero-vorticity is explicitly derived. Actually, this chain describes the motion of Lagrangian fluid particles that are moving within the conditionally averaged velocity fields. We also show the CG invariance of the separation and coincidence properties of the PDFs. Besides the derivation of the CG invariance of the zero-vorticity isolines, we demonstrate that the infinitesimal operator admitted by the characteristic equations forms a Lie algebra which is the Witt algebra, whose central extension represents exactly the Virasoro algebra. The numerical value of the central charge c occurring here could not be calculated exactly without additional impact into the mathematical tools. But from the previous DNS results performed by Bernard et al the value c = 0 is given and corresponds to k = 6 for the SLE(k).

Show abstract - 28 Mar, 2019.
*Alexander Pozhidaev (Sobolev Institute of Mathematics, Novosibirsk, Russia)*

**The universal conservative superalgebra.**In 1972, I.Kantor introduced the class of conservative algebras, which includes some well-known classes of algebras, such as associative, Jordan, Lie and Leibniz algebras. We introduce the class of conservative superalgebras, in particular, the superalgebra \(U(n,m)\) of bilinear operations of an \((n+m)\)-dimensional vector superspace. Moreover, we show that each conservative superalgebra modulo its maximal Jacobian ideal is embedded into \(U(n,m)\) for certain values of \(n,m\). This is a joint work with I.Kaigorodov (UFABC) and Yu.Popov (UNICAMP)

Show abstract - 14 Mar, 2019.
*Alexander Pozhidaev (Sobolev Institute of Mathematics, Novosibirsk, Russia)*

**On simple finite-dimensional right-symmetric (super)algebras.**We describe the simple right-symmetric superalgebras with an idempotent and the superidentity \((x,y,z) + (-1)^{z(x+y)}(z,x,y)+(-1)^{x(y+z)}(y,z,x)=0\) over a field of characteristic prime to 6. We classify all right-symmetric algebras \(A=W\oplus M_2(F)\) such that \(W\) is an irreducible unital right module over \(M_2(F)\). We show that for every natural \(n\) there exists a simple nonassociative right-symmetric algebra of dimension \(n(n+1)\), which possesses the ``unital'' matrix subalgebra \(M_n(F)\).

Show abstract - 10 Jan, 2019.
*Alfredo Najera Chavez (UNAM, Mexico)*

**An illustrated introduction to cluster algebras.**In this talk I will give an introduction to the theory of cluster algebras, focusing on distinguished families of examples. In particular, I will elaborate on the strong relation between cluster algebras and representation theory. If time permits I will explain some of my results relating cluster algebras and root systems of Kac-Moody Lie algebras.

Show abstract - 29 Nov, 2018.
*S. R. Sverchkov (RANEPA, Siberian Institute of Management, Novosibirsk, Russia)*

**The Lie algebra of skew-symmetric elements and its application in the theory of Jordan algebras.**It is proved that the Lie algebra of skew-symmetric elements of a free associative algebra of rank 2 with respect to the standard involution is spanned by elements of the form \([a, b], [a, b]^3\), where \(a\), \(b\) are Jordan polynomials. Using this result, it is proved that the Lie algebra of Jordan derivations of a free Jordan algebra of rank 2 is generated, as a \(T\)-space, by two derivations. It is shown that all commutator Jordan \(s\)-identities are consequences of the Glennie-Shestakov \(s\)-identity.

Show abstract - 11 Oct, 2018.
*Liudmila Sabinina (Universidade Estadual de Morelos, Mexico)*

**Binary-Lie algebras and correspondent loops.**By definition an algebra L is binary-Lie iff any two elements of this algebra generate a Lie subalgebra.In general, for given finite dimensional binary-Lie algebra L we have not classical theorems such Levi's theorem, Weyl's theorem, Malcev's theorem, etc... We study some subvariety of binary-Lie algebras given by the identity J(x,y,zt)=0, where we prove all classical theorems. This variety appear as example of non-Malcev variety where we have the following analog of Moufang theorem: if J(x,y,z)=0, then x,y,z generate a Lie subalgebra. Finally, we discuss the isotops of commutative Moufang loops and its applications.

Show abstract - 04 Oct, 2018.
*Plamen Koshlukov (IMECC-UNICAMP)*

**Trace-preserving embeddings of Jordan algebras.**We discuss embeddings of PI algebras into "good" behaved algebras. It is well known that there exist associative PI algebras satisfying all identities of the \(n\times n\) matrices that cannot be embedded into \(n\times n\) matrices over any commutative ring. On the other hand, Procesi proved that if an associative algebra satisfies all trace identities of the \(n\times n\) matrices then it can be embedded into \(n\times n\) matrices. Let \(K\) be a field of characteristic 0, and assume \(A\) is a central simple finite dimensional \(K\)-algebra with a trace. We establish a sufficient condition for an algebra with trace to be embeddable into \(A\). Then we show that the Jordan algebra of a nondegenerate symmetric bilinear form on a finite dimensional vector space satisfies the condition. Hence for the latter algebra the embedding theorem holds. We also show that the associative algebra of the \(n\times n\) matrices satisfies our condition. Thus we obtain a "new" proof of Procesi's Theorem. Another consequence of our result is a result of Berele's concerning the embedding of trace algebras with involution. As a corollary we obtain the Specht property for algebras satisfying our condition. In the case of \(n\times n\) matrices with the usual trace this is a result of Razmyslov's. For the matrix algebras with trace and involution the result is new as well as for the Jordan algebra of a bilinear form. These are joint results with Diogo Diniz and Claudemir Fideles.

Show abstract - 20 Sep, 2018.
*Dmitry Logachev (UFAM)*

**Relações entre os t-motivos de Anderson e seus reticulados.**Os t-motivos de Anderson são análogos das variedades abelianas na característica \(p\). Particularmente, para um t-motivo \(M\) é possível definir seu reticulado \(L(M)\) e o reticulado dual \(L(M)^*\). Em diferença do caso das variedades abelianas, não é conhecido se há ou não há correspondência 1 - 1 entre os t-motivos e seus reticulados, exceto em alguns casos simples, onde há. Ainda mais, não para todos os t-motivos seu reticulado é "completo". É conhecido que se o reticulado é completo, então \(L(M)^*\) é dual a \(L(M)\) no sentido de álgebra linear. Na conferência serão apresentadas todas as definições e alguns resultados novos, por exemplo, o fato que na vizinhança dum ponto inicial a aplicação do reticulado é 1 - 1, e que se \(L(M)\) não é completo, então \(L(M)^*\) pode não ser dual a \(L(M)\) no sentido de álgebra linear (ele pode ter mesmo outra dimensão).

Show abstract - 13 Sep, 2018.
*Eduardo do Nascimento Marcos (IME-USP)*

**Hochschild-Mitchell (co)-Homology of \(G-k\)-categories over a ring, Galois Coverings and Skew Categories.** - 07 Jun, 2018.
*Felipe Albino dos Santos (IME-USP)*

**On the n-point algebras.**We will recall some advances reached in the last 20 years since the definition of the n-point rings and algebras by Bremner in 1995. We will discuss about how it motivated us to study further generalizations of loop algebras and show a procedure to find a finite basis for the center of the universal central extension in one of this examples. Finally we show that \(\mathbb{C}[t^{\pm1},u]/\langle u^m-t^n-1\rangle\) is not a \(n\)-point ring.

Show abstract - 24 May, 2018.
*(15:00) Victor Hugo López Solís (Universidad Nacional Santiago Antúnez de Mayolo, Peru)*

**O problema de Nathan Jacobson.**Temos resolvido um problema de longa data que foi anunciado por Nathan Jacobson sobre a descrição das álgebras alternativas que contém a álgebra de matrizes de ordem 2, com o mesmo elemento identidade.

Show abstract - 24 May, 2018.
*Marcelo Lanzilotta (Udelar, Uruguay)*

**Funções e álgebras de Igusa-Todorov.** - 17 May, 2018.
*Oleg Shashkov (Financial University under the Government of the Russian Federation, Russia and IME-USP)*

**Simple right alternative superlagebras with some conditions on even parts.**We have studied simple finite-dimensional right alternative superalgebras with some conditions on the even part. For strongly alternative superalgebras with a semisimple even part, a complete description of their structure is given.

Show abstract - 10 May, 2018.
*Ashish Mishra (UFPA, Brazil)*

**Decomposition of \(G \wr S_n\)-action on generalized Boolean algebras.** - 03 May, 2018.
*João Fernando Schwarz (IME-USP)*

**Generalized Weyl Algebras as Galois Orders.**This talk is a natural continuation of the preceeding ones this year. We expose the final results concerning the representation theory of Weyl Algebras invariants, and on the Noncommutative Noether's Problem. Then we consider quantum versions of what we have done before: we discuss the q-difference Noether Problem, we show that many invariant rings of many quantum groups have natural Galois Orders structures. We unify all known results with the conceptual tool of Quantum Linear Galois Algebras - a quantum analogue of one introduced before by Eshmatov, Futorny and Schwarz. Finally, we exploit the galois algebra structure of Generalized Weyl Algebras to study a problem posed by Kirkman, Kuzmanovich and Zhang.

Show abstract - 26 Apr, 2018.
*Nurlan Ismailov (Suleyman Demirel University, Kazakhstan and IME-USP)*

**Bicommutative algebra under (anti-)commutators.**An algebra with identities \(a(bc) = b(ac), (ab)c = (ac)b\) is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free bicommutative algebra to be Lie or Jordan.

Show abstract - 05 Apr, 2018.
*Alexandre Grichkov (IME-USP)*

**Moufang theorem and its generalization.**The classical Moufang theorem says that for three elements \(x,y,z\) from any Moufang loop \(M\) such that \((xy)z=x(yz)\), a subloop generated by \(x,y,z\) is a group. In 2003 A.Rajah proposed a question (open untill now): does there exist a non-Moufang variety \(V\) of diassociative loops such that for every loop from variety \(V\) the Moufangs theorem holds? In this talk we give some result in the direction of Rajah's question and its generalizations.

Show abstract - 22 Mar, 2018.
*João Fernando Schwarz (IME-USP)*

**Invariants of the Weyl Algebra II - Structure and Representation.**We continuous the discussion of the previous talk. We also discuss the representation theory of invariants of the Weyl Algebra - and certain other rings of differential operators. Despite the rigidity mentioned before, it closely resemble the original one in the sense that they have a very natural structure of Galois Orders - with good induction/restriction properties in relation to certain maximal commutative subalgebras. We mention quantum analogues of the previous discussions - put in short, the same phenomena happens. Finally, we have some words to say on arbitrary Generalized Weyl Algebras.

Show abstract - 15 Mar, 2018.
*João Fernando Schwarz (IME-USP)*

**Rigidity of Rings and Invariants of the Weyl Algebra I.**In this first talk we are going to discuss in depth the phenomena described by the celebrated Chevalley-Shephard-Tod Theorem: when \(A^G \cong A\) for \(A\) an algebra and \(G\) a finite group of algebra automorphisms, for many classes of algebras. We shall see that it is rather unusual for many known noncommutative rings that the above can happen at all. If, properly stated, we change the point of view from isomorphism to birational equivalence, the situation changes dramatically. The Classical Noether's Problem and the Gelfand-Kirillov Conjecture serve as a point of departure of our discussion, focused on invariants of the Weyl Algebra.

Show abstract slides - 08 Mar, 2018.
*Andrzej Zuk (University Paris 7, France)*

**Groups associated to box-ball systems.** - 23 Nov, 2017.
*Yury Volkov (Saint Petersburg State University, Russia)*

**s-homogeneous algebras and s-homogeneous triples.** - 09 Nov, 2017.
*(15:00) Edson Ribeiro Alvares (UFPR, Brazil)*

**(m, n)-Quasitilted and (m, n)-Almost hereditary algebras.** - 09 Nov, 2017.
*Patrick Le Meur (Paris 7, France)*

**Duality for non commutative algebras.**Homological duality in noncommutative algebra occurs in various classes of algebras like, for intance, Artin-Schelter regular algebras, algebras with Van den Bergh duality and (skew-)Calabi-Yau algebras. In the noncommutative geometry of Artin and Zhang, Artin-Schelter regular algebras are used to produce noncommutative analogs of smooth projective schemes in which the counterpart of the Grothendieck-Verdier duality is enhanced by the homological duality of that algebra. In representation theory, Calabi-Yau (dg) algebras are used to construct additive categorifications of cluster algebras. From the viewpoint of differential calculus, all these algebras have in common a Poincaré duality on their Hochschild (co)homology spaces. The talk will present these classes of algebras and examples of them. It will discuss various contexts of noncommutative geometry, representation theory and differential calculus in which such an algebra shows up, and it will explain some implications of the existence of a homological duality.

Show abstract - 19 Oct, 2017.
*(16:00) Malihe Yousofzadeh (University of Isfahan, Iran)*

**Affine Root Supersystems and their Generalizations.**Following the interest of physicists in the study of supersymmetries, in 1997, V. Kac introduced Lie superalgebras. He studied the super version of contragredient Lie algebras and classified the finite dimensional simple ones. Then, in 1986, J. Van de Lour classified affine Lie superalgebras, i.e., those contragredinet Lie superalgebras which are not finite dimensional but of finite growth. Here, we first recall affine Lie superalgebras and elaborate their root systems. We call these root systems affine root supersystems. Affine root supersystems are examples of extended affine root supersystems which appear naturally in the study of generalizations of affine Lie superalgebras. We give a complete description of extended affine root supersystems.

Show abstract - 19 Oct, 2017.
*Saeid Azam (University of Isfahan and IPM, Iran)*

**Extended affine Weyl groups, structure and presentations.**Extended affine Weyl groups are certain reflection groups which arise naturally in the theory of extended affine Lie algebras. We recall briefly the definition of extended affine Lie algebras and extended affine root systems and review the basic structural results concerning extended affine Weyl groups. In contrast to finite and affine cases, extended affine Weyl groups are not in general Coxeter groups. We discuss certain presentations of these reflection groups.

Show abstract - 01 Jun, 2017.
*Vladimir Sokolov (USP)*

**Compatible Lie and associative algebras.**Two Lie brackets on the same vector space are called compatible if any their linear combination is a Lie bracket. In the integrability theory there exist many motivations for investigation and classification of compatible brackets. The most interesting problem is a description of all brackets compatible with given simple (in the Lie algebra sense) bracket. More rigid problem is a classification of associative algebras compatible with a given semi-simple associative algebra A. In my talk A is the matrix algebra. A new algebraic structure related to the problem is presented. Unexpectedly, a class of associative algebras compatible with A is related to Dynkin diagrams of the Kac-Moody algebras of A,D,E-type.

Show abstract - 18 May, 2017.
*(15:30, room 259A) Libor Krizka (USP)*

**Verma modules, singular vectors and equivariant differential operators.**For a semisimple Lie group G with a parabolic subgroup P we consider the generalized flag manifold G/P. By the classical duality theorem, G-equivariant differential operators between sections of associated vector bundles over the generalized flag manifold G/P correspond to homomorphisms of generalized Verma modules. As homomorphisms of generalized Verma modules are uniquely determined by the so called singular vectors, we use a geometric realization of these generalized Verma modules to explicitly describe singular vectors. This gives us also an explicit description of the correspoding G-equivariant differential operators.

Show abstract - 18 May, 2017.
*Victor Petrogradsky (UnB)*

**Fibonacci Lie algebra and its properties.**We discuss properties of Fibonacci (restricted) Lie algebra. This algebra is a natural analogue of Grigorchuk and Gupta-Sidki groups. It is defined over a field of characteristic 2, has 2 generators, fractal (or self-similar), has a slow polynomial growth and a nil \(p\)-mapping. All these properties will be proven.

Show abstract - 11 May, 2017.
*Jonathan Nilsson (Carleton, Canada)*

**Tensor Modules for the Lie Algebra of Vector Fields on a Sphere.**For an affine manifold \(X\), let \(A\) be the algebra of polynomial functions on \(X\) and let \(D=\) Der \((A)\) be the Lie algebra of polynomial vector fields on \(X\). Although \(D\) is a very classical object, much is still unknow about its representations. One natural class of representation are the \(AD\)-modules - vector spaces equipped with compatible \(A\)- and \(D\)-module structures. \(AD\)-modules can be constructed from gl\(_s\)-modules where \(s=\) dim \(X\), and modules that embed into such \(AD\)-modules in a natural way are called tensor modules. In this talk I'll focus on the case where \(X\) is the sphere \(S^2\). In particular I will prove a decomposition theorem for tensor products of such modules. This talk is based on joint work with Yuly Billig.

Show abstract - 27 Apr, 2017.
*(15:30, room 259A) Lucas Calixto (UFMG)*

**Módulos de Weyl para superálgebras de Lie.** - 27 Apr, 2017.
*Pablo Zadunaisky (University of Buenos Aires/USP)*

**How to build 1-singular Gelfand-Tsetlin modules.**A Gelfand-Tsetlin tableau is a triangular array of complex numbers, corresponding to each tableau of height n there is a family of gl(n,C) modules. Such modules were built explicitly in many cases: Gelfand and Tsetlin's classical work implies that finite dimensional modules are associated to certain tableaux with integer entries. When the entries of the tableau are generic [i.e. the difference between two entries in the same row are noninteger] the construction is due to Drozd, Futorny and Ovsienko. Finally, in an article published in 2016, Futorny, Grantcharov and Ramirez built a Gelfand-Tsetlin module associated to a 1-singular tableau [where two entries in the same row are equal]. However there is at the moment no recipe to build a module associated to an arbitrary tableau. We will review some of this theory and give a construction which simplifies and unifies the construction of generic and 1-singular GT-modules, and points to the construction of modules associated to arbitrary tableaux.

Show abstract - 20 Apr, 2017.
*(15:30, room 259A) John MacQuarrie (UFMG)*

**The path algebra as a left adjoint functor.**Given a finite quiver, the path algebra constuction generates an algebra as freely as possible given the constraints imposed by the quiver. This construction is functorial. Given a finite dimensional algebra, there is a construction going in the other direction: the Gabriel quiver. As the path algebra is a free construction, it seems natural to suppose that it should be a left adjoint, with Gabriel quiver as an excellent candidate for the corresponding right adjoint. There are technical reasons why life isn't quite so easy, but we will see how to make this intuitive idea a theorem.

Show abstract - 20 Apr, 2017.
*A. Borovik (University of Manchester, UK)*

**Black box groups.**Let \(x_1, ..., x_m\) be matrices of size \(n\) by \(n\) over the finite field of order \(q\). What can we say about the group \(X\) that they generate? This group \(X\) could be of astronomic size. For example, can we *explicitely* find in \(X\) a nontrivial unipotent element or prove that it does not exist? This problem appears, at the first glance, to be deceptively simple -- but it remained open since 1999. I will explain its solution in case of odd \(q\): a probabilistic algorithm which finds the desired element (if it exists) in probabilistic time polynomial in \(n\) and log \(q\). I will also explain some basics of the probabilistic set up for solving problems of that kind: the so-called black box group theory. It is a surprisingly beautiful theory on the boundary between finite and infinite in mathematics: one can compute in very big finite groups as if they were compact Lie groups. Our algorithm is implemented as a computer problem and has been successfully tested for \(q\) being 30 decimal digits long. In this context, its most surprising feature is that it imitates the historic development of geometry from Euclid to Lobachevky to Minkowski, and even more striking are parallels with quantum mechanics. This is a joint work with Sukru Yalcinkaya.

Show abstract - 30 Mar, 2017.
*V. V. Bavula (University of Sheffield, UK)*

**Classical left regular left quotient ring of a ring and its semisimplicity criteria.** - 21 Feb, 2017.
*(Tuesday, 14:00, room 241A) Dessislava Kochloukova (IMECC-UNICAMP)*

**Propriedades homológicas de produto fibra.**Na primeira parte da palestra vamos discutir propriedades gerais de propriedade homologica FPm e de propriedade homotópica Fm (a última generaliza o conceito de grupos finitamente apresentáveis). Na segunda parte da palestra vamos focar no conceito de produto fibra de grupos. Vamos discutir um criterio conhecido como Teorema 1-2-3 de Bridson-Baumslag-Howie-Miller-Short e uma conjectura nesta direção, conhecida como Conjectura \((n-1)-n-(n+1)\) de Kuckuck (o tema principal do seu doutorado em Oxford, 2013), que abordam quando o produto fibra é finitamente apresentável e quando tem tipo homotópico Fn. Finalmente vamos enunciar alguns resultados novos obtidos com o aluno de doutorado Francismar Lima (defesa 2016, UNICAMP) sobre propriedade homológica FPn do produto fibra de grupos. Se o tempo permitir vamos discutir como a teoria de produto fibra tem applicações para teoria de produto subdreto e grupos residualmente livres.

Show abstract - 08 Dec, 2016.
*Elizaveta Vishnyakova (IME-USP)*

**Lie supergroups and the splitting problem for complex homogeneous supermanifolds.**Our talk is devoted to Lie supergroups and homogeneous supermanifolds.The main structure theorem in the theory of real supergroups was proven by B.Kostant. It is an equivalence of the category of real Lie supergroups and the category of real Harish-Chandra pairs. From this theorem it follows that a Lie supergroup depends only on the underlying Lie group and its Lie superalgebra with certain compatibility conditions. Further the structure sheaf of a Lie supergroup and the supergroup morphisms can be explicitly described in terms of the corresponding Lie superalgebra and underlying Lie supergroup. We will discuss another proof of this result that works also in the complex-analytic and algebraic cases.Further, we will consider a very important class of complex supermanifolds, non-split supermanifolds. A supermanifold is called split if it is isomorphic to a vector bundle with a purely even base and purely odd fiber. In the smooth category, all supermanifolds are known to be split (although non-canonically). Another example of split supermanifolds is any Lie supergroup. The second question is how to find out whether a given complex homogeneous supermanifold is split or non-split.

Show abstract - 24 Nov, 2016.
*Claudio Gorodski (IME-USP)*

**Polar symplectic representations.** - 17 Nov, 2016.
*Germán Alonso Benitez Monsalve (IME-USP)*

**Variedade de Gelfand-Tsetlin.**Serge Ovsienko provou que a variedade de Gelfand-Tsetlin para gl(n) é equidimensional (i.e., todas suas componentes irredutíveis tem a mesma dimensão) com dimensão n(n-1)/2. Este resultado é conhecido como "Teorema de Ovsienko" e tem importantes consequências na Teoria de Representações de Álgebras. Neste seminário, será apresentado uma versão fraca do Teorema de Ovsienko e estendemos tal versão fraca para uma estrutura que tem como caso particular gl(3), esse é o caso do grupo quântico Yangian Yp(gl(3)) de nível p. Além disso, o Teorema de Ovsienko também tem como consequências na Geometria Simplética, especificamente na equidimensionalidade das fibras de uma projeção da aplicação de Kostant-Wallach. Também será apresentada uma generalização de este resultado.

Show abstract - 27 Oct, 2016.
*Cristian Ortiz (IME-USP)*

**The adjoint representation up to homotopy of a Lie groupoid.**The notion of Lie groupoid generalizes that of a Lie group, but also includes smooth manifolds and actions of Lie groups as particular instances. There exists a quite natural extension of representation theory of Lie groups to the framework of Lie groupoids, which in spite of being natural, it is very restrictive and it does not provide a good notion of adjoint representation. One can deal with the lack of examples by introducing a more flexible notion of representation, that of representation up to homotopy. The talk aims at introducing representations up to homotopy, having as main examples the adjoint and coadjoint representations.The goal of the talk consists on showing a necessary and sufficient condition for a Lie groupoid to have equivalent adjoint and coadjoint representations up to homotopy. In the search of examples, we will see that symplectic geometry offers a good scenario to find classes of Lie groupoids with the property described above.

Show abstract - 29 Sep, 2016.
*15:00 Pavel Shumyatsky (UnB)*

**On finiteness of verbal subgroups in residually finite groups.**A group-word \(w\) is called concise if whenever the set of \(w\)-values in a group \(G\) is finite it always follows that the verbal subgroup \(w(G)\) is finite. More generally, a word \(w\) is said to be concise in a class of groups \(X\) if whenever the set of \(w\)-values is finite for a group \(G\in X\), it always follows that \(w(G)\) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. It is still an open problem whether every word is concise in the class of residually finite groups. Our talk will be about recent developments with respect to that problem.

Show abstract - 29 Sep, 2016.
*14:00 Evgeny Khukhro (University of Lincoln, U.K.)*

**Engel-type subgroups and length parameters of finite groups.** - 15 Sep, 2016.
*João Fernando Schwarz (IME-USP)*

**Folcloric and Edgy results of Invariants of the Weyl Algebra.**In this talk we have a mixture of past and new results on invariants of the Weyl Algebra. Some results well known to especialists, but whose proof are hard to find in the literature, are discussed. Then we discuss some facts about invariants of the Weyl Algebra: such as an analogue of Jung-van der Kulk theorem for the Weyl Algebra of rank 1, and effective computational methods for calculating a generating set of invariants. Finally, we discuss some new results on Noncommutative Noether's Problem and applications to the structure and representation theory of Rational Cherednik Algebras.

Show abstract - 25 Aug, 2016.
*Alessandra Frabetti (Universidade de Lyon, France)*

**Loop of formal diffeomorphisms.**Formal diffeomorphisms in one variable form a proalgebraic group represented by the so-called Faà di Bruno Hopf algebra, which is commutative and not cocommutative. This algebra, and some variations generated by trees and Feynman graphs, encode very efficiently the renormalization of Green's functions in perturbative quantum field theory. A model in quantum electrodynamics (QED) makes use of a non-commutative version of the Faà di Bruno Hopf algebra, which of course can not represent a proalgebraic group. However, formal series with matrix (non-commutative) coefficients make plainly sense in the physical context, and do form a loop. In a work in progress with Ivan P. Shestakov, we show how this loop can be represented by the non-commutative Faà di Bruno Hopf algebra and study the non-commutative version of loop's infinitesimal structure, given by Sabinin algebras.

Show abstract - 18 Aug, 2016.
*Olivier Mathieu (Université de Lyon)*

**On the chromatic polynomials.**A coloring of a graph \(\Gamma\) means a coloring of the vertices in a way that two connected vertices have different colors. Let \(\Gamma\) be a graph with \(n\) vertices and let \(P_\Gamma(q)\) be the number of coloring of \(\Gamma\) with \(q\)-colors. It turns out that \(P_\Gamma(q)=q^n -a_1 q^{n-1} + a_2 q^{n-2}\dots\) is a polynomial in \(q\). As it is well known, there is a combinatorics to describe the coefficients \(a_i\) which are are non negative integers and also to describe the value of \((-1)^n P_\Gamma(-1)\). At first these combinatorial results look very surprising. In this talk we will explain why they become very natural in the context of algebraic geometry.

Show abstract - 09 Jun, 2016.
*Nick Early (PennState University, USA)*

**The Hidden Symmetry and Geometric Structure of the Worpitzky Identity.**The classical Worpitzky identity for the symmetric group \(S_n\) decomposes a cubical lattice into \(n!\) simplices of different sizes, each with a multiplicity counted by the Eulerian number of permutations of \(n\) variables with a fixed number of descents. It is well-known in combinatorics that the Eulerian numbers can be represented as volumes of suitably normalized hypersimplices. Using vector spaces generated by certain permutohedral cones called plates, due to Ocneanu in his forthcoming work, we replace the relative volume of a hypersimplex with linear degrees of freedom in a highly constrained analog of categorification. Our main result is to introduce for the first time and prove an \(S_n\)-equivariant generalization of the Worpitzky identity in which the classical Worpitzky is a relation between linear dimensions. Our generalization is an isomorphism between two new graded, symmetric group modules, in which the character values coincide with the scaled volumes of a certain generalized hypersimplices. The proof combines simplicial geometry with basic properties of a new \(q\)-deformed commutative algebra which we introduce.

Show abstract - 02 Jun, 2016.
*Pedram Hekmati (IMPA)*

**Group actions on abelian categories.**Representations of compact Lie groups on vector spaces are well-understood and their classification dates back to the fundamental work of Cartan and Weyl. In this talk, I will discuss a categorified notion of group actions and provide an explicit example of how such a categorical representation can be constructed.

Show abstract - 12 May, 2016.
*Alexei Kotov (UFPR, Curitiba)*

**Integration of differential graded Lie algebras via Harish-Chandra pairs.**We construct an explicit formula for the integration of differential graded Lie algebras into differential graded Lie groups. We also explain the notion of a dg Lie group and show some applications of it.

Show abstract - 05 May, 2016.
*Victor Petrogradsky (UnB)*

**Álgebras de Lie de crescimento lento.**Discutimos construções bastante antigas e recentes de álgebras de Lie de crescimento lento. Em particular, temos exemplos de álgebras de Lie (vinculadas) auto-similares finitamente geradas de crescimento polinomial lento, com nil p-mapping. Por suas propriedades, estas álgebras de Lie restritas se assemelham a grupos Grigorchuk e Gupta-Sidki. Discutimos diferentes propriedades dessas álgebras e suas envolventes associativas.

Show abstract - 28 Apr, 2016.
*Yuly Billig (Carleton University, Canada)*

**Homogeneous symmetric functions and Sturm-Liouville problem on a Fock space.**We compute the eigenvalues of the differential operator \(\sum_{a+b=c+d} x_a x_b \frac{d}{dx_c} \frac{d}{dx_d}\) acting on the Fock space \(C[x_1, x_2, ...]\). It turns out that this operator is diagonalizable with integer non-negative eigenvalues. Not surprisingly, the eigenvectors and the eigenvalues are indexed by Young diagrams.

Show abstract - 07 Apr, 2016.
*Kaiming Zhao (Wilfrid Lourier University, Canada)*

**Simple modules over conformal Galilei algebras.**Conformal Galilei algebras is a class of objects in theoretical physics and attract a lot of attention in related mathematical areas. Some irreducible representations of two classes of conformal Galilei algebras in 1-spatial dimension will be constructed. In particular, a classification of simple weight modules with finite dimensional weight spaces over conformal Galilei algebras will be given.

Show abstract - 31 Mar, 2016.
*E. Vishnyakova (IME-USP)*

**Graded manifolds and \(n\)-fold vector bundles.** - 17 Mar, 2016.
*Wilson Fernando Mutis Cantero (IME-USP)*

**Mishchenko-Fomenko subalgebras in \(S(gl_n)\) and regular sequences.** - 11 Feb, 2016.
*(14:30) Kinvi Kangni (Université Félix Houphouet Boigny, Ivory Coast)*

**On some recent developpement of Fourier analysis.**We'll start to introduce classical Fourier analysis, spherical Fourier transform on locally compact group and after, according to an unitary dual of some compact subgroups, and applications to connected and reductive Lie group.

Show abstract - 10 Dec, 2015.
*Dimitar Grantcharov (University of Texas, Arlington, USA)*

**Twisted localization of weight modules.**The twisted localization functor is a localization-type functor for non-commutative rings. This functor plays crucial role in the classification of the simple objects of various categories of weight modules, i.e. modules that decompose as direct sums of weight spaces. In this talk we will discuss various applications of the twisted localization for finite-dimensional Lie algebras and algebras of differential operators. Most of the talk will be based on a joint work with Vera Serganova.

Show abstract - 19 Nov, 2015.
*Tomoyuki Arakawa (Kyoto University, Japan)*

**Associated varieties of vertex algebras.**For a vertex algebra \(V\) one associates an affine Poisson scheme \(X_V\) called an associated variety, which plays an important role on the representation theory of \(V\). In the case that \(V\) is a simple affine vertex algebra its associated variety \(X_V\) is a \(G\)-invariant, conic subscheme of \(g^*\) that is not necessarily contained in the nilpotent cone. I will talk about the Feigin-Frenkel conjecture on the associated variety of affine vertex algebras, and its application to the representation theory of \(W\)-algebras.

Show abstract - 12 Nov, 2015.
*Ivan Struchiner (IME-USP)*

**Deformations of Lie Groupoids: Cohomological Aspects.**Lie groupoids are natural algebraic/geometric structures which arise in the study of manifolds with symmetries. In this talk I will explain what a Lie groupoid is and describe its deformation theory by means of its deformation cohomology. At the end of the talk I will relate this to a recent talk given by Cristian Ortiz, and pose an open question about the underlying algebraic structure of the deformation cohomology. I will also try to explain why this is a relevant problem. This talk is based on joint work with Marius Crainic and João Nuno Mestre available at arXiv: 1510.02530.

Show abstract - 01 Oct, 2015.
*Daniel Bravo (Universidad Austral de Chile)*

**Finiteness conditions and cotorsion pairs.**Given a ring \(R\) and a nonnegative integer \(n\), we say that an \(R\)-module \(M\) is finitely \(n\)-presented if it has a partial projective resolution \(P_n \to P_{n-1} \to \cdots \to P_1 \to P_0 \to M \to 0\), with each \(P_i\) finitely generated. Denote by \(FP_{n}\) the class of finitely \(n\)-presented modules. In this talk, we will focus on some module theoretical properties of \(FP_n\), how they allow us to generalize coherent rings, and what happens to \(FP_{n}\) under this generalization of \(R\). Furthermore, we investigate the relative injective modules and relative flat modules, with respect to the class \(FP_{n}\), and study associated cotorsion pairs to these classes. If time allows, we will show how the class of finitely \(n\)-presented modules for any \(n \geq 0\), allow us to define a generalization of the stable module category.

Show abstract - 24 Sep, 2015.
*Yury Yolkov (IME-USP)*

**Derived Picard groups of selfinjective Nakayama algebras.** - 17 Sep, 2015.
*Cristian Ortiz (USP)*

**Lie 2-algebra of vector fields of a Lie groupoid.**A Lie 2-algebra is a categorified version of a Lie algebra. The corresponding global counterpart is that of a Lie 2-group. It is known that Lie 2-algebras (resp. Lie 2-groups) are in one-to-one correspondence with crossed modules of Lie algebras (resp. Lie groups). In this talk I will show an example of a crossed module of Lie algebras, hence a Lie 2-algebra, with origin in a geometric problem: describing the algebraic structure of vector fields on stacks. For that, we use Lie groupoids and vector fields on Lie groupoids. This is work in progress joint with J. Waldron.

Show abstract - 06 Aug, 2015.
*(15:00) Marina Rasskazova (Omsk, Russia)*

**Deformations of group algebra of dihedral group.**We define some notion of deformation of associative algebra and prove that deformation of group algebra of finite Coxeter group has dimension less then this group. In particular case of Coxeter groups-dihedral groups we recieved more exact results in the case of deformations of maximal dimensions. The talk is based on join work with A.Grishkov (IME-USP) and S.Sidki (UnB-DF).

Show abstract - 18 Jun, 2015.
*Willian Franca (IME-USP)*

**Commuting maps.**Let \(R\) be a simple unital ring. Under a mild technical restriction on \(R\), we will characterize biadditve mappings \(G: R^2\to R\) satisfying \(G(u,u)u=uG(u,u)\), and \(G(1,r)=G(r,1)=r\) for all unit \(u\in R\) and \(r\in R\) respectively. As an application we describe bijective linear maps \(\theta: R\to R\) satisfying \(\theta(xyx^{-1}y^{-1})=\theta(x)\theta(y)\theta(x)^{-1}\theta(y)^{-1}\) for all invertible \(x,y\in R\).

Show abstract - 11 Jun, 2015.
*Fernando Borges (IME-USP)*

**A new class of cluster algebra.**In this talk we present a new class of cluster algebra with coefficients of Dynkin type A, which we call c-cluster algebra. In order to obtain the cluster variables of a c-cluster algebra, we give a generalization of the Caldero-Chapoton map.

Show abstract - 28 May, 2015.
*Dmitri Vassilevich (UFABC)*

**Non-associative deformation quantization.**Deformation quantization is a formal deformation of the algebra of smooth functions on some manifold. In the classical setting, the Poisson bracket serves as an initial conditions, while the associativity allows to proceed to higher orders. Some applications to string theory require deformation in the direction of a quasi-Poisson bracket (that does not satisfy the Jacobi identity). This initial conditions is incompatible with associativity, it is quite unclear which restrictions can be imposed on the deformation. We show that for any quasi-Poisson bracket the deformation quantization exists and is essentially unique if one requires hermiticity and the Weyl condition.(joint work with Vladislav Kupriyanov).

Show abstract - 23 Apr, 2015.
*Jacob Mostovoy (Centro de Investigación y de Estudios Avanzados del IPN, México)*

**The Campbell-Baker-Hausdorff formula in non-associative algebras.**In the non-associative context there is more than one candidate for the exponential power series. As a consequence, there are several versions of the Campbell-Baker-Hausdorff formula. I will talk about a geometrically motivated exponential series and CBH formula which can be used to construct a version of the Lie theory for non-associative product.

Show abstract - 16 Apr, 2015.
*Ilya Gorshkov (Sobolev Institute of Mathematics)*

**Some arithmetic characterizations of finite groups.**Arithmetic characteristics of a finite group such as order, the set of element orders, the set of indices, indices of subgroups play an important role in group theory.In my talk I will say about characterization finite group by spectrum and by set of conjugacy classes size.

Show abstract - 09 Apr, 2015.
*Ivan Shestakov*

**The Freiheitssatz for generic Poisson algebras.**A generic Poisson algebra is a linear space with two operations: associative and commutative product \(x \cdot y = xy\) and anti-commutative bracket \(\{x, y\}\), satisfying the Leibniz identity \(\{x, yz\} = \{x, y\}z + \{x, z\}y\). These algebras were introduced by the author in the study of speciality and deformations of Malcev–Poisson algebras. When the bracket \(\{x,y\}\) is a Lie one, one get a usual Poisson algebra.

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The Freiheitssatz problem for a variety \(M\) is to determine whether every nontrivial equation over the free algebra \(M(X), X = \{x_1, x_2,\dots \}\), is solvable over \(M(X)\). It was first proved by W.Magnus for the variety of groups, and then was considered by various authors in various varieties. In particular, the Freiheitssatz was proved for ordinary Poisson algebras by L.Makar-Limanov and U.Umirbaev. We prove it for generic Poisson algebras.

It is a joint work with P.Kolesnikov and L.Makar-Limanov. - 26 Mar, 2015.
*Alexandr Pozhidaev (Sobolev Institute of Mathematics, Novosibirsk, Russia)*

**On the classification of simple finite-dimensional noncommutative Jordan superalgebras.**In this talk we will discuss the classification of simple finite-dimensional noncommutative Jordan superalgebras of arbitrary characteristic.

Show abstract - 19 Mar, 2015.
*Yu. L. Ershov (Sobolev Institute of Mathematics, Novosibirsk, Russia)*

**Separant of arbitrary polynomial.**A separant of a polynomial over a valued field was defined by G.D. Brink only for separable polynomials. Applications of this notion will be discussed for generalizations of Henzel's lemma and a root continuty theorem. Means of finding (computing) of a separant of a polyntomial will be discussed, using constructive models and classical theory of resultants (polyresultants, polydiscriminant...)

Show abstract - 12 Mar, 2015.
*V. Sokolov (Landau Institute for Theoretical Physics, Moscou, Russia)*

**Integrable differential equations and non-associative algebraic structures.**We discuss the symmetry approach to integrability of partial differential equations. It turns out that some special classes of integrable polynomial multi-component models are closely related to Jordan and left-symmetric algebras and to Jordan triple systems. Rational integrable models involve inverse elements in Jordan triple systems.

Show abstract - 05 Mar, 2015.
*Alexandr Pozhidaev (Sobolev Institute of Mathematics)*

**Poisson-Farkas algebras and Filippov algebras.**In this talk we present a connection of the Poisson-Farkas algebras and the Filippov algebras (n-Lie algebras). This construction gives all known examples of simple Filippov algebras. Also we find some new examples of Poisson-Farkas algebras which give some simple Filippov algebras. In particular, we obtain the first example of simple nontrivial Filippov superalgebra of characteristic 2.

Show abstract - 31 Dec, 1969.
*(postponed) Oscar Armando Hernández Morales.*

**Módulos sobre Álgebras de Vertex Afim I.** - 31 Dec, 1969.
*(postponed) Oscar Armando Hernández Morales.*

**Módulos sobre Álgebras de Vertex Afim II.**