Schedule
Future seminars
Past seminars
- 01.06.2017. The Fischer decomposition in representation theory ( Libor Krizka).
The pair of polynomials \(P(x) \in \mathbb{C}[x_1,\dots,x_n]\) and \(Q(\partial_x) \in \mathbb{C}[\partial_{x_1},\dots,\partial_{x_n}]\) is called a Fischer pair, if \(\mathbb{C}[x]=\mathbb{C}[P(x)] \otimes \ker Q(\partial_x)\). The corresponding decomposition is the so called Fischer decomposition. In the talk, I will construct some examples of Fischer pairs and apply this decomposition to particular examples in representation theory.
Show abstract - 25.05.2017. Center of Brauer Configuration Algebras ( Alex Sierra Cardenas).
Biserial algebras are a slightly generalization of what is called a serial algebra. Serial algebras were introduced by Nakayama in 1941 and they were the first non-semisimple algebras with only finitely many indecomposable modules. It was until 1961 when Tachikawa gave the defining property of biserial algebras, but was Fuller who coined the name of “biserial algebra” in 1979. Whitin the vast class of biserial algebras there is a particular one that has the feature that its representation theory is largely controlled by the uniserial modules. This class is that of the symmetric special biserial algebras. In 2015 Schroll showed that the class of symmetric special biserial algebras coincides with the class of Brauer graph algebras. One of the principal features of a Brauer graph algebra is that the associated Brauer graph encodes its representation theory. Brauer configuration algebras are a generalization of Brauer graph algebras, in the sense that every Brauer graph is a Brauer configuration and every Brauer graph algebra is a Brauer configuration algebra. As the Brauer graph algebras, the Brauer configuration algebras have additional structure arising from combinatorial data, called a Brauer configuration. Brauer configuration algebras were introduced by Green and Schroll in 2015 and it is expected that the Brauer configuration also will encode the representation theory of Brauer configuration algebras. In this talk we will define a Brauer configuration and a Brauer configuration algebra, and also we will compute the dimension of one of the invariants of a Brauer configuration algebra: the dimension of its center.
Show abstract - 18.05.2017. Polinomios invariantes para álgebras de funções truncadas ( Tiago Macedo (UNIFESP)).
Considere uma álgebra de Lie \(a\) e a álgebra de funções polinomiais em \( a\) que são \( a\)-invariantes, \(S( a)^{ a}\). Existe uma relação estreita entre \(S( a)^{ a}\) e o centro \(Z( a)\) da álgebra universal envelopante de \( a\). Esse centro e um objeto bastante estudado por fornecer informações sobre as representações de \( a\). Por exemplo, quando \( a\) e uma álgebra de Lie simples de dimensão finita, \(Z( a)\) é isomorfa a uma álgebra de polinomios finitamente gerada, e ajuda a descrever a decomposição em blocos de uma certa categoria de representações de \( a\). Considere \( g\) uma álgebra de Lie simples de dimensão finita e \(A\) uma álgebra associativa, comutativa e de dimensão finita. Nesse seminário, vamos descrever a álgebra \(S( g \otimes A)^{ g \otimes A}\) e, como consequencia, vamos mostrar como \(Z( g \otimes A)\) age em representações irredutiveis de dimensão finita de \( g \otimes A\). Estes resultados foram obtidos em colaboração com Alistair Savage (da Universidade de Ottawa)
Show abstract Handwritten Draft - 06.04.2017. Combinatorial construction of Gelfand-Tsetlin modules for \(gl(n)\) ( Jian Zhang).
We propose a new effective method of constructing explicitly Gelfand-Tsetlin modules for gl(n). We obtain a large family of irreducible modules (conjecturally all) that have a basis consisting of Gelfand-Tsetlin tableaux, the action of the Lie algebra is given by the Gelfand-Tsetlin formulas and with all Gelfand-Tsetlin multiplicities equal 1. As an application of our construction we prove necessary and sufficient condition for the Gelfand and Graev's continuation construction to define a module which was conjectured by Lemire and Patera.
Show abstract - 29.11.2016. Algebras de incidencia hereditarias por partes ( Marcelo Moreira da Silva).
Apresentamos um estudo das algebras de incidencia que sao hereditarias por partes, as quais denominamos Phias, piecewise hereditary incidence algebras. Atraves da aljava com relacoes, descrevemos as Phias de tipo Dynkin e introduzimos uma nova familia de Phias de tipo Dynkin extendido chamada familia ANS, em referencia a Assem, Nehring e Skowro\'nski. Nessa descricao, o importante metodo foi o dos cortes admissiveis em extensoes triviais, os quais inspiraram a elaboracao de um programa que concebe exatamente os cortes admissiveis na extensao trivial dada que resultam em algebras de incidencia.
Show abstract - 22.11.2016. Vertices of Gelfand-Tsetlin polytopes ( Luiz Enrique Ramirez).
In this talk we study the polyhedral geometry of Gelfand-Tsetlin tableaux arising in the theory of finite dimensional representations of the Lie algebra gl_n. Is presented a method to calculate the dimension of the lowest-dimension face containing a given Gelfand-Tsetlin tableau. In particular is disproved a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand-Tsetlin polytopes. This talk is a review of the paper: 'Vertices of Gelfand-Tsetlin polytopes' (https://arxiv.org/abs/math/0309329)
Show abstract - 18.11.2016. Variadades de Gelfand-Tsetlin ( German Alonso).
- 08.11.2016. Groethendieck Theorem ( Marcos Orseli).
The goal of this talk is to present a proof of classical Groethendieck theorem about classification of vector bundles over projective line \(\mathbb P^1\). The proof will be based on Birkhoff factorization of an invertible matrix with coefficients that are Laurent polynomials.
Show abstract - 25.10.2016. \(n\)-point Lie algebras II ( Felipe Albino dos Santos).
The goal of this presentation is to introduce the \(n\)-point Lie algebras. We will discuss the motivation, give some concrete examples following previous work, and talk about the Heisenberg Lie algebras of these structures. Finally, we will present an overview of \(\varphi\)-Verma modules, an irreducibility criterion for them, and explain how these ideas could be applied to classify irreducible modules for Heisenberg subalgebras of \(4\)-point Lie algebras.
Show abstract - 18.10.2016. \(n\)-point Lie algebras ( Felipe Albino dos Santos).
The goal of this presentation is to introduce the \(n\)-point Lie algebras. We will discuss the motivation, give some concrete examples following previous work, and talk about the Heisenberg Lie algebras of these structures. Finally, we will present an overview of \(\varphi\)-Verma modules, an irreducibility criterion for them, and explain how these ideas could be applied to classify irreducible modules for Heisenberg subalgebras of \(4\)-point Lie algebras.
Show abstract - 04.10.2016. Singular Gelfand-Tsetlin Modules for \(gl(n)\) ( Carlos A. Gomes).
In this lecture we will present a large class of non-generic Gelfand-Tsetlin modules - the class of 1-singular Gelfand-Tsetlin modules. Our explicit construction of 1-singular-modules provides a large family of new irreducible modules for \(gl(n)\).
Show abstract - 27.09.2016. Algebras, quivers and adjoint functors. III ( Kostiantyn Iusenko).
This mini-course is a short introduction to the basic concepts of category theory and representation theory of finite-dimensional algebras. We will learn the concept of adjoint functors and will show that the construction ''quiver'' \(\leftrightarrow\) ''algebra'' can be interpreted as a pair of adjoint functors between certain categories. Lectures almost do not contain the proofs, theoretical part will be accompanied with examples, and sometimes introduced in the form of exercises.
Show abstract Outlines - 20.09.2016. Algebras, quivers and adjoint functors. II ( Kostiantyn Iusenko).
This mini-course is a short introduction to the basic concepts of category theory and representation theory of finite-dimensional algebras. We will learn the concept of adjoint functors and will show that the construction ''quiver'' \(\leftrightarrow\) ''algebra'' can be interpreted as a pair of adjoint functors between certain categories. Lectures almost do not contain the proofs, theoretical part will be accompanied with examples, and sometimes introduced in the form of exercises.
Show abstract Outlines - 30.08.2016. Algebras, quivers and adjoint functors. I ( Kostiantyn Iusenko).
This mini-course is a short introduction to the basic concepts of category theory and representation theory of finite-dimensional algebras. We will learn the concept of adjoint functors and will show that the construction ''quiver'' \(\leftrightarrow\) ''algebra'' can be interpreted as a pair of adjoint functors between certain categories. Lectures almost do not contain the proofs, theoretical part will be accompanied with examples, and sometimes introduced in the form of exercises.
Show abstract Outlines - 20.05.2016. BV-differential on Hochschild cohomology of Frobenius algebra. ( Y. Volkov).
Let \(R\) be be a Frobenius algebra over a field \(k\) with Nakayama automorphism \(v: R \rightarrow R\). We define the Gerstenhaber algebra \(HH^{*}(R)^v\) that is isomorphic to \(HH^{*}(R)\) if the order of \(v\) is a natural number, which is not divisible by the characteristic of \(k\). We construct the BV-differential on the algebra \(HH^{*}(R)^v\).
Show abstract - 13.05.2016. Optimal control on \(SO(3)\) ( Yuly Billig).
Euler proved in 1776 that every rotation of a 3-dimensional body can be realized as a sequence of three rotations around two given axes. If we allow sequences of an arbitrary length, such a decomposition will not be unique. It is then natural to ask a question about decompositions that minimize the total angle of rotation. In the talk we present a solution to this problem. Orientation of Kepler space telescope is controlled with reaction wheels. In 2013, two of these wheels failed, and as a result Kepler may now be rotated only around two axes. Our theorem provides optimal algorithms for Kepler's attitude control. Other possible applications arise in quantum information theory, where transformations on a single orbit are described by the group \(SU(2)\), which is closely related to \(SO(3)\).
Show abstract - 06.05.2016. Double vector bundles and graded manifolds of degree 2 ( Elizaveta Vishnyakova).
Our talk is devoted to:
Show abstract
1) elements of supergeometry: graded manifolds, tangent spaces, vector fields and so on.
2) the notion of a Lie algebroid.
3) a prove of the equivalence of the category of double vector bundles and the category of graded manifolds of degree 2. - 15.04.2016. Module Depth and Representation Theory (Christopher James Young).
- 13.04.2016. On the base problem for the polynomial identities of matrix algebras. (Axelei Kuzmin).
- 08.04.2016. On Duflo's Theorem and Gelfand-Tsetlin modules. (Luiz Enrique Ramirez).
Duflo's Theorem gives a description of annihilators of Verma modules. The known proofs of this result are rather technical. In this seminar I will discuss a proof of this theorem for Verma modules with tableaux realization. We will also use this arguments to compute annihilators of some Gelfand-Tsetlin modules. This talk is based on the paper: O. Khomenko. Some applications of Gelfand-Zetlin modules, Representations of algebras and related topics, Fields Inst. Commun., vol. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 205
Show abstract - 18.03.2016. Quantum determinants and quantum Pfaffians (Jian Zhang).
We use quantum exterior algebras to give a new and elementary formulation of quantum Pfaffians and generalized quantum Pfaffians based on quantum Plucker relations. In this approach, the quantum Pfaffians are for any square matrix satisfying a simple quadratic relation. In particular, we prove the fundamental identity expressing any quantum determinant as a quantum Pfaffian.
Show abstract - 11.03.2016. Metodo de deslocamento de argumento (Wilson).
Seja \(S_n\) a algebra simetrica da algebra de Lie \(gl_n\) das matrizes de tamanho \(n \times n\) sobre o corpo dos numeros complexos. Para \(N\) em no espaco dual de \(gl_n\) seja \(A_N\) a subalgebra de Mishchenko-Fomenko de \(S_n\) construida pelo famoso metodo de deslocamento de argumento associada ao parametro \(N\). E conhecido que se \(N\) e um elemento semisimples regular ou nilpotente regular entao a subalgebra \(A_N\) e gerada por uma sequencia regular em \(S_n\). Neste seminario vou apresentar uma descricao geral da prova do seguinte teorema. Teorema: Se \(N\) e uma matriz nilpotente em \(gl_4\) entao a subalgebras de Mishchenko-Fomenko \(A_N\) e gerada por uma sequencia regular em \(S_4\).
Show abstract - 04.03.2016. Algebras hereditarias por partes (Marcelo).
Nesse seminario, introduziremos as chamadas algebras hereditarias por partes e seus tipos. Isso envolve o conceito de equivalencia de categorias derivada. Aproveitaremos para expor algumas propriedades algebricas invariantes sobre essa equivalencia.
Show abstract - 18.02.2016. Modular representations of profinite groups ( John MacQuarrie (UFMG)).
The modular representation theory of a finite group \(G\) concerns the study of finitely generated modules over the group algebra \(kG\), where \(k\) is a field of positive characteristic. Such modules can be organised using the concepts of relative projectivity, vertex and source. Basic results of Green and Higman along these lines constitute fundamentals upon which the subject is built. A profinite group is the inverse limit of an inverse system of finite groups. In this talk we will see how many of the foundational results of modular representation theory of finite groups pass through this inverse system, to the limit.
Show abstract - 17.11.2015. McKay correspondence. VI ( Kostiantyn).
We will continue with geometric McKay correspondence.
Show abstract - 10.11.2015. McKay correspondence. V ( Iryna).
We will discuss the blow-up of singularities on algebric varieties.
Show abstract - 03.11.2015. Algebras Multiseriais, Algebras Multiseriais Especiais y Algebras de Configuracao de Brauer. III ( Alex Sierra).
Os conceitos de algebra multiserial e algebra multiserial especial sao generalizacoes naturais de conceitos muito bem conhecidos de classes de algebras, as quais sao: Algebras biseriais e as Algebras biseriais especiais. Vamos chamar de modulo multiserial a um modulo satisfazendo que o quociente de seu radical pelo seu socalo e uma soma direta de modulos uniseriais. Vamos mostrar que, qualquer modulo finitamente gerado sobre uma algebra multiserial especial e multiserial. As algebras de configuracao de Brauer sao um tipo particular de algebras de algebras multiseriais. Vamos definir esse conceito algebrico e vamos ver que, e uma generalizacao natural das algebras de grafos de Brauer.
Show abstract - 27.10.2015. Representacoes Irredutiveis De Grau Dois Da Primeira Algebra De Weyl ( Cesar Augusto Rodriguez. ).
- 06.10.2015. Algebras Multiseriais, Algebras Multiseriais Especiais y Algebras de Configuracao de Brauer. II ( Alex Sierra).
Os conceitos de algebra multiserial e algebra multiserial especial sao generalizacoes naturais de conceitos muito bem conhecidos de classes de algebras, as quais sao: Algebras biseriais e as Algebras biseriais especiais. Vamos chamar de modulo multiserial a um modulo satisfazendo que o quociente de seu radical pelo seu socalo e uma soma direta de modulos uniseriais. Vamos mostrar que, qualquer modulo finitamente gerado sobre uma algebra multiserial especial e multiserial. As algebras de configuracao de Brauer sao um tipo particular de algebras de algebras multiseriais. Vamos definir esse conceito algebrico e vamos ver que, e uma generalizacao natural das algebras de grafos de Brauer.
Show abstract - 29.09.2015. Algebras Multiseriais, Algebras Multiseriais Especiais y Algebras de Configuracao de Brauer. ( Alex Sierra).
Os conceitos de algebra multiserial e algebra multiserial especial sao generalizacoes naturais de conceitos muito bem conhecidos de classes de algebras, as quais sao: Algebras biseriais e as Algebras biseriais especiais. Vamos chamar de modulo multiserial a um modulo satisfazendo que o quociente de seu radical pelo seu socalo e uma soma direta de modulos uniseriais. Vamos mostrar que, qualquer modulo finitamente gerado sobre uma algebra multiserial especial e multiserial. As algebras de configuracao de Brauer sao um tipo particular de algebras de algebras multiseriais. Vamos definir esse conceito algebrico e vamos ver que, e uma generalizacao natural das algebras de grafos de Brauer.
Show abstract - 22.09.2015. Introducao a extensao trivial sob o ponto de vista do seu quiver com relacoes ( Marcelo).
- 15.09.2015. McKay correspondence. IV ( Kostiantyn).
We aim to understand geometric quotients of group action on algebraic varieties. Then we will describe quotients \(\mathbb C^2/G\), in which \(G \subset SL(2,\mathbb{C})\) is finite subgroup, and will study the singularities of these varieties.
Show abstract - 01.09.2015. McKay correspondence. III ( Kostiantyn).
We construct the McKay graphs of finite subgroups of \(SL(2,\mathbb{C})\).
Show abstract - 18.08.2015. McKay correspondence. II ( Kostiantyn).
We aim to classify finite subgroups of \(SL(2,\mathbb{C})\).
Show abstract - 30.06.2015. McKay correspondence. I ( Kostiantyn).
We consider McKay construction of a graph of finite group (using its representation theory). We will construct the graph of two groups \(C_n\) (cyclic group) and \(D_n\) and will formulate the statement of McKay correspondence.
Show abstract - 23.06.2015. Representations of finite groups. II ( Felipe).
- 16.06.2015. Representations of finite groups. I ( Felipe).
- 26.05.2015. Constructions of Ext functor IV ( German).
We consider two construction of Ext functors: as derived functor and via Yoneda approach. Then we aim to compare these two approach.
Show abstract - 19.05.2015. Constructions of Ext functor III ( German).
We consider two construction of Ext functors: as derived functor and via Yoneda approach. Then we aim to compare these two approach.
Show abstract - 12.05.2015. Constructions of Ext functor II ( German).
We consider two construction of Ext functors: as derived functor and via Yoneda approach. Then we aim to compare these two approach.
Show abstract - 05.05.2015. Constructions of Ext functor I ( German).
We consider two construction of Ext functors: as derived functor and via Yoneda approach. Then we aim to compare these two approach.
Show abstract - 28.04.2015. Quantization of the shift of argument subalgebras in type A. III (Wilson Fernando Mutis Cantero).
Given a simple Lie algebra \(g\) and an element \(\mu \in g^*\), the corresponding shift of argument subalgebra of \(S(g)\) is Poisson commutative. In the case where \(\mu\) is regular, this subalgebra is known to admit a quantization, that is, it can be lifted to a commutative subalgebra of the universal enveloping algebra \(U(g)\). We will present the main result of Futorny and Molev in the article “Quantization of the shift of argument subalgebras in type A” which states that if \(g\) is a Lie algebra of type \(A\), then this property extends to arbitrary \(\mu\), thus proving a conjecture of Feigin, Frenkel and Toledano Laredo.
Show abstract - 21.04.2015. Quantization of the shift of argument subalgebras in type A. II (Wilson Fernando Mutis Cantero).
Given a simple Lie algebra \(g\) and an element \(\mu \in g^*\), the corresponding shift of argument subalgebra of \(S(g)\) is Poisson commutative. In the case where \(\mu\) is regular, this subalgebra is known to admit a quantization, that is, it can be lifted to a commutative subalgebra of the universal enveloping algebra \(U(g)\). We will present the main result of Futorny and Molev in the article “Quantization of the shift of argument subalgebras in type A” which states that if \(g\) is a Lie algebra of type \(A\), then this property extends to arbitrary \(\mu\), thus proving a conjecture of Feigin, Frenkel and Toledano Laredo.
Show abstract - 14.04.2015. Quantization of the shift of argument subalgebras in type A (Wilson Fernando Mutis Cantero).
Given a simple Lie algebra \(g\) and an element \(\mu \in g^*\), the corresponding shift of argument subalgebra of \(S(g)\) is Poisson commutative. In the case where \(\mu\) is regular, this subalgebra is known to admit a quantization, that is, it can be lifted to a commutative subalgebra of the universal enveloping algebra \(U(g)\). We will present the main result of Futorny and Molev in the article “Quantization of the shift of argument subalgebras in type A” which states that if \(g\) is a Lie algebra of type \(A\), then this property extends to arbitrary \(\mu\), thus proving a conjecture of Feigin, Frenkel and Toledano Laredo.
Show abstract - 07.04.2015. Extension Fullness of the Category of Gelfand-Zeitlin Modules III (Luis Enrique).
We discuss the categories of Gelfand-Zeitlin modules of \(g = gl_n\) and Whittaker modules associated with a semi-simple complex finite-dimensional algebra \(g\). There will be proven that such categories are extension full in the category of all \(g\)-modules (based on the paper by Kevin Coulembier and Volodymyr Mazorchuk).
Show abstract - 31.03.2015. Extension Fullness of the Category of Gelfand-Zeitlin Modules II (Luis Enrique).
We discuss the categories of Gelfand-Zeitlin modules of \(g = gl_n\) and Whittaker modules associated with a semi-simple complex finite-dimensional algebra \(g\). There will be proven that such categories are extension full in the category of all \(g\)-modules (based on the paper by Kevin Coulembier and Volodymyr Mazorchuk).
Show abstract - 24.03.2015. Extension Fullness of the Category of Gelfand-Zeitlin Modules (Luis Enrique).
We discuss the categories of Gelfand-Zeitlin modules of \(g = gl_n\) and Whittaker modules associated with a semi-simple complex finite-dimensional algebra \(g\). There will be proven that such categories are extension full in the category of all \(g\)-modules (based on the paper by Kevin Coulembier and Volodymyr Mazorchuk).
Show abstract - 17.03.2015. Variedades de Gelfand-Tsetlin para Yangians (German).
Sergei Ovsienko provou que a variedade de Gelfand-Tsetlin para \(gl(n)\) é equidimensional de dimensão \(\frac{n(n-1)}{2}\). Como uma das consequências tem-se que a álgebra envolvente universal de \(gl(n)\) é livre como módulo à esquerda (ou à direita) sobre sua subálgebra Gelfand-Tsetlin. Motivados sobre a importância desse fato na Teoria de Representações de álgebras, se pretende dar continuidade a este problema sobre o grupo quântico Yangian \(Y_p(n)\).
Show abstract - 20.10.2014. Duflo isomorphism. Kontsevich Formality Theorem III (Kostiantyn).
We plan to sketch the prrof of formality theorem. We discuss admissible graphs and polydifferential operator associated to them, as well as Configuration spaces \(C_{n,m}^+\) and their compactifications as they form the main ingredients in Kontsevich's proof.
Show abstract Handwritten drafts - 13.10.2014. Duflo isomorphism. Kontsevich Formality Theorem II (Kostiantyn).
Given symplectic manifold \(X\) we define two differential graded Lie algebra \(T_{poly}(X)\) and \(D_{poly}(X)\). Formality theorem says that such algebras are quasi-isomorphic, which gives formal deformations of Poisson manifold \(X\). We plan to discuss how this theorem is related with classical Duflo's theorem.
Show abstract Handwritten drafts - 06.10.2014. Duflo isomorphism. Kontsevich Formality Theorem I (Kostiantyn).
Firstly we recall the statement of Duflo's theorem about the isomorphism between the center of universal enveloping algebra of finite dimensional Lie algebra and ring of invariants of corresponding symmetric algebra. Then we plan to discuss its relation with Kashiwara-Vergne conjecture and Bernoulli numbers. Finally we will discuss Poisson algebras and formal deformations of associative algebras.
Show abstract Handwritten drafts - 15.09.2014. Recent developments of representation theory of nonrelativistic conformal algebras. (Naruhiko Aizawa, Prefecture University of Osaka, Japan.).
Nonrelativistic conformal algebra is a particular class of non-semisimple Lie algebras. The member of the class is a finite or an infinite dimensional Lie algebra. The semisimple part of the finite dimensional algebras is the direct sum of sl(2) and so(d), while the Virasoro algebra is the semisimple part of the infinite-dimensional algebras. This class of Lie algebra appears in various kind of problems in theoretical and mathematical physics. For instance, one can find them in connection with fluid dynamics, gravity theory, AdS/CFT correspondence and vertex operator algebras. This motivate us to study representations of the nonrelativistic conformal algebras. In the beginning of this talk, I introduce various members of the nonrelativistic conformal algebra. Then I pick up some members of physical interest and study them in some more detail. Our first problem is the central extensions of the algebras. The list of possible central extensions is given. Our second problem is the irreducible representations of highest (lowest) weight type. We start with the Verma module and study its irreducibility. This is done by calculating the Kac determinant and explicit construction of singular vectors. If the Verma module is reducible, then it will be shown that how to obtain the irreducible modules.
Show abstract - 29.05.2014. Duflo isomorphism III. Introduction to cohomology of Lie and associative algebras III. (Pasha Zusmanovich).
I will give a brief introduction into the subject, keeping in mind its utility in topics related to Duflo isomorphism, and mainly following the first two chapters of Calaque and Rossi book "Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry".
Show abstract - 22.05.2014. Duflo isomorphism III. Introduction to cohomology of Lie and associative algebras II. (Pasha Zusmanovich).
I will give a brief introduction into the subject, keeping in mind its utility in topics related to Duflo isomorphism, and mainly following the first two chapters of Calaque and Rossi book "Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry".
Show abstract - 15.05.2014. Duflo isomorphism III. Introduction to cohomology of Lie and associative algebras. (Pasha Zusmanovich).
I will give a brief introduction into the subject, keeping in mind its utility in topics related to Duflo isomorphism, and mainly following the first two chapters of Calaque and Rossi book "Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry".
Show abstract - 08.05.2014. Duflo isomorphism II. Overview of Duflo's result (Evan).
We give an overview of original paper of Michel Duflo.
Show abstract - 29.04.2014. Duflo isomorphism I. Isomorphism Theorem of Harish-Chandra (Kostiantyn).
We discuss the isomorphism theorem of Harish-Chandra which describes the center of universal enveloping algebra of a semi-simple Lie algebra. We will see how it works in the cases \(sl_2\) and \(sl_3\) and will prove it for an arbitrary semi-simple Lie algebra.
Show abstract Handwritten drafts - 04.04.2014. Dixmier groups and Calogero-Moser spaces. (Farkhod Eshmatov).
The talk will consist of two parts. In the first part I will discuss some results of Y.Berest and G.Wilson on the ideal structure of \(A_1=C[t,d/dt]\), the first Weyl algebra of ordinary differential operators with polynomial coefficients. Let \(G=Aut(A_1)\) be the group of automorphisms of \(A_1\), and let \(R\) be the set of isomorphism classes of (right) ideals of \(A_1\). There is a description of G given by J.Dixmier. Let \(C\) be the disjoint union of the spaces \(C_n\), the space of equivalence classes (modulo simultaneous conjugation) of pairs \((X,Y)\) of nxn complex matrices such that \([X,Y]+1\) has rank one. The space \(C\) is called the Calogero-Moser space. First, they showed that the natural action of \(G\) on each \(C_n\) is transitive. Second, there is a \(G\)-equivariant bijective map from \(C\) to \(R\). Since the \(G\)-orbits on \(R\) classify algebras Morita equivalent to \(A_1\), Berest-Wilson's theorem gives a complete classification of such algebras. In the second part I will discuss our joint work with Y.Berest and A.Eshmatov. The starting point of this work is an observation that the stabilizers \(G_n\) of \(G\) action on \(C_n\) are exactly automorphism groups of algebras Morita equivalent to \(A_1\). Using the Bass-Serre theory we give a geometric presentation for these groups, thus answering question posed by T. Stafford. We also discuss isomorphism problem for these groups. For this end, following Borel, Tits, Steinberg we define an abstract Borel subgroup for \(G_n\). Then we show that up to conjugation there are exactly \(p(n)\), the number of partitions of \(n\), Borel subgroups of \(G_n\). If time permits I will discuss a natural ind-group structure on these groups.
Show abstract - 20.03.2014. Hall algebras of bound quivers and quantumn groups (Evan Wilson).
We describe the twisted Ringel-Hall algebra of bound quiver with small homological dimension. The description is given in the terms of quadratic form associated with the corresponding bound quiver.
Show abstract - 30.01.2014. Counterexample to Lusztig's conjecture (Geordie Williamson, Max-Planck-Institute, Bonn).
- 24.10.2013. On the classification of irreducible Gelfand-Tsetlin modules of \(sl(3)\) (part 3) (Enrique).
In this talk we wil introduce derivative Gelfand-Tsetlin tableaux. Which allow us to generalize Gelfand-Tsetlin formulas for \(gl(3)\). Then we will see how every Irreducible Gelfand-Tsetlin \(sl(3)\)-module can be realized by tableaux with action given by generalized Gelfand-Tsetlin formulas.
Show abstract - 17.10.2013. On the classification of irreducible Gelfand-Tsetlin modules of \(sl(3)\) (part 2) (Enrique).
In this talk we wil define generic modules and the concept of Gelfand-Tsetlin modules. Some results about existence of irreducible Gelfand-Tsetlin modules for \(gl(n)\) are presented. Then we will present these results for \(sl(3)\).
Show abstract - 10.10.2013. On the classification of irreducible Gelfand-Tsetlin modules of \(sl(3)\) (part 1) (Enrique).
In this seminar we will discuss the concept of Gelfand-Tsetlin modules for \(gl(n)\) and the main examples (modules with tableau realization). In next talks we will focus in \(sl(3)\) case and give the main ideas behind the classification of irreducible Gelfand-Tsetlin modules in this case.
Show abstract - 03.10.2013. Realizations of DJKM algebras (part 2/2) (Renato).
In this talk we give two realizations of DJKM algebras in terms of sums of partial diferential operators.
Show abstract - 26.09.2013. Realizations of DJKM algebras (part 1/2) (Renato).
In this seminar we will introduce first definitions and discuss some results about DJKM algebras. In next talks we will study the construction of free field realizations for this algebras.
Show abstract - 19.09.2013. Root Multiplicities Part 3: The Witt formula and Kang's multiplicity formula (Evan).
In this talk we describe the Witt formula for giving the dimensions of the graded components of a finitely generated free Lie algebra. Kang extended this result to root multiplicities of infinite dimensional Kac-Moody algebras using Kac's construction. We describe this formula, and explain its application to low-level multiplicity computations.
Show abstract - 20.06.2013. Nakajima quiver varieties 3: categorical quotients (Kostiantyn).
An algebro-geometric analogue of symplectic quotient is categorical quotient. In this talk we will give a very brief introduction into Mumford's Geometric Invariant Theory concentrating on the examples rather than on formal definitions. Also we plan to discuss Kirwan-Ness theorem which connects symplectics and categorical quotients.
Show abstract Handwritten drafts - 13.06.2013. Realizations of the three point Lie algebra (Ben Cox).
We describe the universal central extension of the three point current algebra \(sl(2,R)\) where \(R=\mathbb C[t,t^-1,u | u^2=t^2+4t]\) and construct realizations of it in terms of sums of partial differential operators. This is joint work with Elizabeth Jurisich.
Show abstract - 16.05.2013. Irreducible finite dimensional representations for \(gl(n)\) (Enrique).
In this talk we present a classical result given by I.Gelfand and M.Tsetlin (1950), which give as explicit bases and formulas for each irreducible finite dimensional representation of gl(n). We give some examples of this construction for \(n=2\) and \(n=3\).
Show abstract - 09.05.2013. Nakajima quiver varieties 2: The moment map and Hamiltonian reduction (continuation) (Evan).
The moment map is an important step in the construction of Nakajima quiver varieties, with important applications in physics and geometry. In this talk, we will give a brief introduction to the foundations of the subject, including an introduction to the symplectic geometry that is needed.
Show abstract - 02.05.2013. Nakajima quiver varieties 2: The moment map and Hamiltonian reduction (Evan).
The moment map is an important step in the construction of Nakajima quiver varieties, with important applications in physics and geometry. In this talk, we will give a brief introduction to the foundations of the subject, including an introduction to the symplectic geometry that is needed.
Show abstract - 25.04.2013. Nakajima's Quiver varieties. I. (Kostiantyn).
Ringel produced a construction of \(U_q({n})\), the positive part of the quantized enveloping algebra of a Kac-Moody Lie algebra \(g\), in terms of a Hall algebra associated with an appropriated quiver. Quivers varieties give a geometric construction of the whole \(U(g)\) rather than its positive part. In this introductory talk we will present a brief look at Nakajima's quiver varieties without going into the deep details.
Show abstract Handwritten drafts - 18.04.2013. Vertex Algebras (conclusion) (Yuly Bilig).
In these talks we will present an introduction to vertex algebras, focusing on examples, rather on formal theory. We will discuss in detail the vertex algebra structure on the basic module for affine \(sl(2)\), as well as the vertex algebra interpretation of the Sugawara construction.
Show abstract - 11.04.2013. A geometric form of Drozd's theorem (Vladimir Sergeichuk).
G.R. Belitskii (1983) gave an algorithm for reducing a pair of matrices to canonical form under similarity. His algorithm is extended to the problem of classifying representations of a finite dimensional algebra. Belitskii's canonical matrices represent in a one-to-one manner the sets of equivalent representations. If the algebra is of tame type, then for each n the set of indecomposable canonical n-by-n matrices consists of a finite number of straight lines and points in the affine space of n-by-n matrices. If the algebra is of wild type, then there exists n such that the set of indecomposable canonical n-by-n matrices contains a 2-dimensional plane.
Show abstract Paper related to the talk - 04.04.2013. Vertex Algebras (continuation) (Yuly Bilig).
In these talks we will present an introduction to vertex algebras, focusing on examples, rather on formal theory. We will discuss in detail the vertex algebra structure on the basic module for affine \(sl(2)\), as well as the vertex algebra interpretation of the Sugawara construction.
Show abstract Handwritten drafts - 21.03.2013. Vertex Algebras (Yuly Bilig).
In these talks we will present an introduction to vertex algebras, focusing on examples, rather on formal theory. We will discuss in detail the vertex algebra structure on the basic module for affine \(sl(2)\), as well as the vertex algebra interpretation of the Sugawara construction.
Show abstract Handwritten drafts - 14.03.2013. Verma modules 2 (continuation of the previous talk) (Renato).
Neste seminário definiremos o conceito de módulos do tipo Verma a partir de uma subálgebra parabólica de \(\mathbb{g}\) contendo uma fixada subálgebra de Borel \(\mathbb{b}\). Então obteremos a definição de módulos de Verma clássico e imaginário.
Show abstract Handwritten drafts - 07.03.2013. Verma modules (an educational talk) (Renato).
Neste seminário definiremos o conceito de módulos do tipo Verma a partir de uma subálgebra parabólica de \(\mathbb{g}\) contendo uma fixada subálgebra de Borel \(\mathbb{b}\). Então obteremos a definição de módulos de Verma clássico e imaginário.
Show abstract Handwritten drafts - 11.10.2012. Weyl modules for hyperalgebras (Angelo).
In this talk we construct the Weyl modules for hyperalgebras and discuss some differences between positive and zero characteristics.
Show abstract - 05.10.2012. Hopf algebras: Sweedler notation and duals (Evan).
In this talk we define Sweedler notation: a powerful way to do computations in a Hopf algebra, and the notion of the dual of a Hopf algebra.
Show abstract - 20.09.2012. Hopf algebras of the tensor and symmetric algebras (Evan).
In this talk we give the Hopf algebra structure of the tensor and symmetric algebras and their relation to the universal enveloping algebras of free Lie algebras.
Show abstract - 13.09.2012. Hall algebras: Introduction to Hopf algebras (Evan).
Hopf algebras are important structures which arise in many fields of mathematics. In this talk, we define Hopf algebras, and give some examples, including the universal enveloping algebra of a Lie algebra, the group ring of a finite group, quantum groups, and Hall algebras associated with representations of quivers.
Show abstract - 23.08.2012. Hall algebras: Ringel's Theorem (Kostyantyn).
We will state Ringel's theorem which connects the category of representations of a given quiver with the positive part of a certain quantum group. We aim to explain the construction of an isomorphism and basic ideas in proof.
Show abstract Handwritten drafts - 22.08.2012. Hall algebras: Quantum Kac-Moody algebras (Kostyantyn).
(10.00 am) In order to get deeper into the Ringel Theorem on this talk we will discuss basic notions about quantum Kac-Moody algebras \(U_v(g)\)
Show abstract Handwritten drafts - 16.08.2012. Hall algebras: Category of finite-dimensional representations of quivers (Evan).
In this talk we introduce the categories of representations of quivers over finite fields and their associated Hall algebras. We describe their Euler form and state Gabriel's theorem which provides some motivation for studying representations of quivers. We will also provide compute a simple case of Ringel's theorem, which connects the study of Hall algebras of quivers with the positive part of certain quantum groups.
Show abstract - 09.08.2012. Hall algebras: the First Touch (Kostyantyn).
In this talk we provide some preliminary notions about the finitary categories and the Hall algebras associated to them. We will discuss the case of the category of finite-dimensional representations of a given quiver over some finite field.
Show abstract Handwritten drafts - 21.06.2012. Groebner-Shirshov Bases and Weyl Modules for Loop Algebras (Evan).
In this talk, we define Groebner-Shirshov pairs, explain their relevance to the problem of finding bases of modules of Kac-Moody algebras, and Kang and Lee's algorithm for Groebner-Shirshov pairs. Finally, we define Weyl modules for loop algebras as quotients of the universal enveloping algebras, and discuss the problem of finding Groebner-Shirshov bases for these modules.
Show abstract - 20.06.2012. An Introduction to Groebner-Shirshov bases (Evan).
Groebner-Shirshov bases are an important computational tool, which allow one to, for example, solve the reduction problem for ideals. Groebner bases were introduced by Buchberger in 1965 for polynomial rings, while Shirshov had previously developed an analogous concept for Lie algebras in 1962. In 2000, Kang and Lee introduced the notion of Groebner pairs, in order to extend the methods of Groebner-Shirshov bases to study representations of non-commutative algebras (i.e. universal enveloping algebras of Kac-Moody algebras and Hecke algebras).
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In this talk, we define Groebner bases for (commutative) polynomial ideals, and introduce Buchberger's algorithm for finding Groebner bases, with examples. We also develop the theory of standard cyclic modules for Kac-Moody algebras from the viewpoint of quotients of the universal enveloping algebra modulo certain ideals. - 14.06.2012. Realizations of affine Lie algebras (Renato).
- 12.06.2012. Realization of \(\widehat{sl}(n)\) modules using crystal bases (Evan).
In this talk we define the notions of quantum groups and crystal bases and the realization of the crystals of \(\widehat{sl}(n)\) modules using the extended Young diagrams of Misra and Miwa. We then apply this realization to find the decomposition of the tensor product of \(\widehat{sl}(n)\) modules.
Show abstract Outlines - 31.05.2012. Graded modules over graded Lie algebras (Angelo).
We discuss some homological properties of graded modules with finite-dimensional components over a graded Lie algebra also having finite-dimensional components associated to a complex finite-dimensional simple Lie algebra. As application we shall present a graded character formula and a sort of truncated module related to Kirillov-Reshetikhin modules.
Show abstract - 24.05.2012. Representations of Posets in Linear and Unitary spaces (Kostiantyn).
Representations of posets in the category of linear spaces were introduced in the late sixties of XX century. Firstly I will recall basic definitions and results. We investigate a similar theory in the category of unitary spaces. There exist not many posets for which it is possible to classify all unitary representations up to the unitary equivalence, and the classification in that cases is rather simple (we will discuss these results). Therefore we impose the additional condition on representations (so-called orthoscalarity conditions). It turned out that orthoscalar unitary representations are strictly connected with stable linear representations. We will discuss some results for orthoscalar unitary representations which follow studying the stable linear representations of the posets. I will finish a talk with a few open questions (if time will permit).
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