## LieJor Online Seminar

(Thursday's, 14:00, Room: to be announced 15 minutes before the beginning of the talk)Please, when joining the Seminar, turn off camera and mute the microfone. Turn on camera and microfone only if you want to ask a question.

The talks will be recorded.

#### Future seminars

- 13 Aug, 2020.
*Viacheslav Artamonov (Moscow State University, Russia)*

**Polynomially complete quasigroups and their application.**Polynomial completeness of a universal algebra A means that any operation on A is a composition of basic operations with a specialization of some variables. In the talk we consider algebraic properties of finite polynomially complete quasigroups, the problem of recognition of its completeness un terms of its Latin square. Basing on this approach we can construct polynomially complete quasigroups of any order greater than 32 which is a power of 2. As an application we consider cryptosystems based on quasigroups.

- 20 Aug, 2020.
*Andrea Solotar (Universidad de Buenos Aires, Argentina)*

**A cup-cap duality in Koszul calculus.**In this talk I will introduce a cup-cap duality in the Koszul calculus of \(N\)-homogeneous algebras following https://arxiv.org/abs/2007.00627. As an application of this duality, it follows that the graded symmetry of the Koszul cap product is a consequence of the graded commutativity of the Koszul cup product. I will also comment on a conceptual approach to this problem that may lead to a proof of the graded commutativity, based on derived categories in the framework of DG algebras and DG bimodules. This is joint work with Roland Berger.

- 27 Aug, 2020.
*Albert Schwarz (UC Davis, USA)*

**Some questions on Jordan algebras inspired by quantum theory.**One can formulate quantum theory taking as a starting point a convex set (the set of states) or a convex cone (the set of non-normalized states.) Jordan algebras are closely related to homogeneous cones, therefore they appear naturally in this formulation. There exists a conjecture that superstring can be formulated in terms of exceptional Jordan algebras. In my purely mathematical talk I'll formulate some results and conjectures on Jordan algebras coming from these ideas.

- 03 Sep, 2020.
*Jacob Mostovoy (Cinvestav, Mexico)*

**TBA.**

#### Past seminars

- 06 Aug, 2020.
*Drazen Adamovic (University of Zagreb, Croatia)*(watch the talk here)

**On logarithmic and Whittaker modules for affine vertex algebras.**Simple affine vertex algebras at admissible levels are semi-simple in the category O, but beyond the category O they contain interesting categories of representations with many new research challenges. We will first present our explicit lattice realizations of simple affine VOA \(L_k(sl(2))\) at arbitrary admissible level \(k\), and their modules in certain categories. Then we discuss the existence and explicit realization of logarithmic modules which appear as extensions of weight modules. Next natural task is to include Whittaker modules in the representation category. Although Whittaker modules are constructed using standard Lie-theoretic constructions, we will show that in order to understand the structure of affine Whittaker modules, one needs to apply vertex-algebraic techniques. We present explicit realization of Whittaker modules for some vertex algebras. We will discuss our recent efforts to generalize this realization in higher rang cases.

Show abstract - 30 Jul, 2020.
*Mikhail Chebotar (Kent State University, USA)*(watch the talk here)

**On polynomials over nil rings.**We will discuss some recent results related to polynomials over nil rings. In particular, we will present solutions of problems posed by Beidar, Puczylowski and Smoktunowicz, Greenfeld, Smoktunowicz and Ziembowski.

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(Joint work with Wen-Fong Ke, Pjek-Hwee Lee and Edmund Puczylowski.) - 23 Jul, 2020.
*Matej Brešar (University of Ljubljana and University of Maribor, Slovenia)*(watch the talk here)

**Images of noncommutative polynomials.**Let \(f=f(X_1,\dots,X_m)\) be a noncommutative polynomial with coefficients in a field \(F\). We will discuss various questions concerning the image of \(f\) in an \(F\)-algebra \(A\), which is defined to be the set \(f(A)=\{f(a_1,\dots,a_m)\,|\,a_1,\dots,a_m\in A\}\). A special emphasis will be on the Waring type problem, asking about the existence of a positive integer \(N\) (independent of \(f\), provided that \(f\) is neither an identity nor a central polynomial of \(A\)) such that every element, or at least every commutator, in \(A\) is a linear combination of \(N\) elements from \(f(A)\). We are primarily interested in the case where \(A=M_n(F)\), but some other algebras will also be considered.

Show abstract - 16 Jul, 2020.
*Patrick Le Meur (Université Paris Diderot, France)*(watch the talk here)

**Equivariant models for graded algebras and application to Calabi-Yau duality.**Skew Calabi-Yau algebras are generalisations of Calabi-Yau algebras due to Reyes, Rogalski, and Zhang. Within the graded (associative and unital) algebras over a field \(k\), they form the non-commutative analogues of the regular algebras. As a special feature, such an algebra \(A\) is equipped with its so-called Nakayama automorphism \(\phi\). The talk will present ongoing investigations on the presentations of these algebras by generators and relations taking into account their homological specificities. Such presentations are well-known for Calabi-Yau algebras (after Ginzburg, Bocklandt and van den Bergh) and also for Koszul skew Calabi-Yau algebras (after Bocklandt, Wemyss and Schedler). The general situation involves the interaction of the \(A\)-infinity Yoneda algebra \(E(A):=Ext_A(k,k)\) with the Nakayama automorphism \(\phi\), and also the \(A\)-infinity Yoneda algebra \(E(A[x,\phi])\) of the Ore extension \(A[x,\phi]\) of A by \(\phi\). More precisely, one is particularly intereseted in minimal models of these \(A\)-infinity algebras.

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After having presented all these concepts, I will discuss the relationship between these minimal models as well as consequences in terms of presentations of \(A\). - 09 Jul, 2020.
*( 12:00) Pavel Kolesnikov (Sobolev Institute of Mathematics, Russia)*(watch the talk here)

**Derived algebras and their identities.**In this talk we will consider a "differential counterpart" of the dendriform splitting procedure for operads. This problem has a very natural interpretation in the language of non-associative algebras. It is well-known that a (non-associative, in general) algebra equipped with a Rota--Baxter operator (a formalization of integration) gives rise to a system in a class of splitting algebras. The latter include dendriform (pre-associative), pre-Lie (left-symmetric), pre-Poisson, Zinbiel (pre-commutative) algebras, etc. What happens if we replace a Rota-Baxter operator with a derivation? The answer is well-known for associative commutative algebras: the resulting class of systems obtained in this way coincides with the variety Nov of Novikov algebras. We will show in general that for an arbitrary binary operad Var the variety of derived Var-algebras coincides with the Manin white product of operads Var and Nov. If we allow the initial multiplication(s) to leave in the language of a derived algebra then the same sort of description can be obtained just by replacement of Nov with \(GD^!\), the Koszul dual to the operad of Gelfand-Dorfman algebras. We will also discuss similar statements for the "integral" case of Rota--Baxter operators.

Show abstract - 09 Jul, 2020.
*( 10:00) Tomoyuki Arakawa (Kyoto University, Japan)*(watch the talk here)

**Urod algebra and translation for W-algebras.**In 2016 Bershtein, Feigin and Litvinov introduced the Urod algebra, which gives a representation theoretic interpretation of the celebrated Nakajima-Yoshioka blowup equations in the case that the sheaves are of rank two. In this talk we will introduce higher rank Urod algebras. This is done by constructing translation functors for affine W-algebras.

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This is a joint work with Thomas Creutzig and Boris Feigin. - 02 Jul, 2020.
*Yuly Billig (Univ Carleton, Canada)*(watch the talk here)

**Towards Kac-van de Leur Conjecture: polynomiality vs locality for superconformal algebras.**Superconformal algebras are graded Lie superalgebras of growth 1, containing a Virasoro subalgebra. They play an important role in Conformal Field Theory. In 1988 Kac and van de Leur made a conjectural list of simple superconformal algebras, which since has been amended with an exceptional superalgebra CK(6). It has been proposed to use conformal superalgebras to attack this conjecture, and Fattori and Kac established a classification of finite simple conformal superalgebras. It still needs to be proved that one can associate a finite conformal superalgebra to each simple superconformal algebra. In this talk we will show how to use the results of Billig-Futorny to prove that every simple superconformal algebra is polynomial, which implies that one can attach to it an affine conformal superalgebra. We will discuss the difference between finite and affine conformal algebras. We also introduce quasi-Poisson algebras and show how to use them to construct known simple superconformal algebras. Quasi-Poisson algebras may be viewed as a refinement of the notion of Novikov algebras. Quasi-Poisson algebras may be used for computations of automorphisms and twisted forms of superconformal algebras.

Show abstract - 25 Jun, 2020.
*Agatha Atkarskaya (Bar-Ilan University, Israel)*(watch the talk here)

**Group-like small cancellation theory for rings.**(joint work with Prof. A.Kanel-Belov, Prof. E.Plotkin, Prof. E.Rips) It is well known that small cancellation groups and especially their generalizations provide a very powerful technique for constructing groups with unusual, and even exotic, properties, like for example, infinite Burnside groups and Tarski monster groups. In the present work we develop a small cancellation theory for associative algebras with a basis of invertible elements. We introduce a list of small cancellation conditions for a presentation of such associative algebras. We construct an explicit linear basis for these algebras. In parallel we show that our algebras possesses algorithmic properties similar to ones for groups with small cancellation groups. Namely, the equality problem in these algebras is solvable with the use of a certain analogue of Dehn's algorithm. We do hope that being of interest as rings of new type by itself, these algebras will inherit useful practical properties known for small cancellation groups and, thus, they can be used for obtaining complicated algebras with the very specific properties.

Show abstract - 18 Jun, 2020.
*Liudmila Sabinina (Universidad Autonoma del Estado de Morelos, Mexico)*(watch the talk here)

**Burnside Problems in the theory of loops.**We prove that for positive integers \(m \geq 1, n \geq 1\) and a prime number \(p \neq 2,3\) there are finitely many finite \(m\)-generated Moufang loops of exponent \(p^n\). For groups this result was proved by Efim Zelmanov. My talk is based on the joint research with Alexander Grishkov and Efim Zelmanov.

Show abstract - 11 Jun, 2020.
*Vladislav Khartchenko (Universidad Nacional Autónoma de México, Mexico)*(watch the talk here)

**Drinfeld-Jimbo quantizations as quadratic-linear algebras.**We prove that in q-Weyl generators the multi-parameter quantizations of type \(A_n^+\) and \(B_n^+\) are quadratic-linear Koszul algebras.

Show abstract - 04 Jun, 2020.
*Vera Serganova (University of California, Berkeley, USA)*(watch the talk here)

**The celebrated Jacobson-Morozov theorem for Lie superalgebras via semisimplification functor for tensor categories.**The famous Jacobson-Morozov theorem claims that every nilpotent element of a semisimple Lie algebra \(g\) can be embedded into \(sl(2)\)-triple inside \(g\). Let \(g\) be a Lie superalgebra with reductive even part and \(x\) be an odd element of \(g\) with non-zero nilpotent \([x,x]\). We give necessary and sufficient condition when \(x\) can be embedded in osp\((1|2)\) inside \(g\). The proof follows the approach of Etingof and Ostrik and involves semisimplification functor for tensor categories. Next, we will show that for every odd x in g we can construct a symmetric monoidal functor between categories of representations of certain superalgebras. We discuss some properties of these functors and applications of them to representation theory of superalgebras with reductive even part. (Joint work with Inna Entova-Aizenbud).

Show abstract - 04 Jun, 2020.
*(12:00) Antonio Giambruno (Università di Palermo, Italy)*(watch the talk here)

**Polynomial identities and growth functions.**A way to measure a T-ideal of the free associative algebra in characteristic zero is through the sequence of codimensions introduced by Regev in 1972. It is known that for any non-trivial T-ideal such sequence either grows exponentially or is polynomially bounded. We shall describe what kind of functions arise and how they can determine some invariants of the T-ideals that can be explicitly computed in some cases.

Show abstract - 28 May, 2020.
*Dessislava Kochloukova (UNICAMP, Brazil)*(watch the talk here)

**Weak commutativity in groups.**We consider some recent results on the weak commutativity construction suggested by Said Sidki in 1980. By definition given a group \(G\) and an isomorphic copy \(H\) of it we can consider a new group \(X(G)\) that is a quotient of the free product of \(G\) and \(H\) by the normal closure of the the commutators \([g,h]\) where h is the image of \(g\) in \(H\) and \(g\) runs through the elements of \(G\). We will consider some recent results about the structure of \(X(G)\) due to Bridson-Kochloukova, Kochloukova-Sidki, Lima-Oliveira.

Show abstract - 28 May, 2020.
*(12:00) Claudio Procesi (Università di Roma, Italy)*(watch the talk here)

**T-ideals of Cayley-Hamilton trace algebras.**A classical Theorem of Amitsur states that any proper prime T-ideal in a free algebra is the ideal of polynomial identities of the algebra of \(n\times n\) matrices for some \(n\). Moreover the associated relatively free algebra has a very interesting structure which has been extensively studied. We give a similar Theorem in the context of trace identities classifying prime T ideals of the trace algebra containing the \(n\) Cayley-Hamilton identity and discuss the corresponding relatively free algebras from algebraic and geometric points of view. One has a one to one correspondence between these T ideals and semisimple e \(n\) Cayley-Hamilton algebras and also the strata of the so called Luna stratification for the quotient variety of \(k\) tuples of \(n\times n\) matrices under the conjugation action of the projective group.

Show abstract - 21 May, 2020.
*Marc Rosso (CNRS, France)*(watch the talk here)

**On Feigin homomorphisms for quantum shuffle algebras.**Feigin homomorphisms map the "upper triangular subalgebras" of quantum groups to some quantum (or twisted) polynomial algebras. They are important in the study of their skew fields of quotients. Several years ago, I gave a construction of these "quantum upper triangular subalgebras" as subalgebras of quantum shuffle algebras. More recently, the construction of Feigin homomorphisms has been extended to the whole quantum shuffle algebras by D. Rupel, with a computational proof. I shall explain another, quite direct approach, stressing the universal property of the quantum shuffle algebra, and putting quantum polynomial algebras naturally in this framework. All necessary background will be recalled.

Show abstract - 14 May, 2020.
*(15:30) João Fernando Schwarz (IME-USP)*(watch the talk here)

**Poisson birational equivalence and Coloumb branches of 3d N=4 SUSY gauge theories.**In this talk we discuss a notion of birational equivalence suitable for Poisson affine varieties: namely, that their function fields are isomorphic as Poisson fields. Some very interesting questions on noncommutative birational geometry, such as the Gelfand-Kirillov Conjecture, make perfect sense in the quasi-classical limit, and naturally leads one to consider the Poisson birational class of the algebras they quantize. In this setting, we study the behavior of Poisson birational equivalence on the quasi-classical limit of rings of differential operators. With this idea we solve a Poisson analogue of Noether's Problem, introduced by Julie Baudry and François Dumas, in a constructive fashion, for essentially all finite symplectic reflection groups. As applications of our method, we show the Poisson rationality of the Generalized Calogero-Moser spaces, introduced by Etingof and Ginzburg in 2002, and surprisngly for this author, all Coloumb branches of \(3d \, N=4\) SUSY gauge theories --- an important object in mathematical physics recently given a rigorous formulation by Nakajima in 2015, and later Nakajima, Braverman, Finkelberg in 2016.

Show abstract - 14 May, 2020.
*Ualbai Umirbaev (Wayne State University, USA)*(watch the talk here)

**A Dixmier theorem for Poisson enveloping algebras.**(Joint work with Viktor Zhelyabin) We consider a skew-symmetric \(n\)-ary bracket on the polynomial algebra \(K[x_1,\ldots,x_n,x_{n+1}]\) (\(n\geq 2\)) over a field \(K\) of characteristic zero defined by \(\{a_1,\ldots,a_n\}=J(a_1,\ldots,a_n,C)\), where \(C\) is a fixed element of \(K[x_1,\ldots,x_n,x_{n+1}]\) and \(J\) is the Jacobian. If \(n=2\) then this bracket is a Poisson bracket and if \(n\geq 3\) then it is an \(n\)-Lie-Poisson bracket on \(K[x_1,\ldots,x_n,x_{n+1}]\). We describe the center of the corresponding \(n\)-Lie-Poisson algebra and show that the quotient algebra \(K[x_1,\ldots,x_n,x_{n+1}]/(C-\lambda)\), where \((C-\lambda)\) is the ideal generated by \(C-\lambda\), \(0\neq \lambda \in K\), is a simple central \(n\)-Lie-Poisson algebra if \(C\) is a homogeneous polynomial that is not a proper power of any nonzero polynomial. This construction includes the quotients \(P(\mathrm{sl}_2(K))/(C-\lambda)\) of the Poisson enveloping algebra \(P(\mathrm{sl}_2(K))\) of the simple Lie algebra \(\mathrm{sl}_2(K)\), where \(C\) is the standard Casimir element of \(\mathrm{sl}_2(K)\) in \(P(\mathrm{sl}_2(K))\). It is also proven that the quotients \(P(\mathbb{M})/(C-\lambda)\) of the Poisson enveloping algebra \(P(\mathbb{M})\) of the exceptional simple seven dimensional Malcev algebra \(\mathbb{M}\) are central simple.

Show abstract - 07 May, 2020.
*Nicolás Andruskiewitsch (Universidad Nacional de Córdoba, Argentina)*(watch the talk here)

**The role of Nichols algebras in the cohomology of finite-dimensional Hopf algebras.** - 30 Apr, 2020.
*Olivier Mathieu (Université de Lyon, France) (The video of this talk will be adjusted)*

**On the chromatic polynomials.**In the past weeks, the connections between the mathematicians has been impaired by the strict isolation measures. Fortunately, the internet seminar, like this one, are good to rebuild them. During this unpleasant time, we need, in addition, to add some colours to our poor academic life. By definition, a colouring of the mathematicians means covering each mathematician with a colour, such that two connected mathematicians have a different colour.

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In the distinguished community of Graph Theory, the notion of the colouring of a graph \(\Gamma\) is the same, except that the mathematicians are called the vertices of the graph and the connections are called the edges of the graph.

There is a polynomial \(P_\Gamma\), called the chromatic polynomial, such that \(P_\Gamma( q)\) is the number of way of colouring the graph with q colour. There is a classical and old work of Whytney concerning the combinatorics of \(P_\Gamma\). Then we will describe Whythney combinatorics in the setting of a cohomology algebra. - 23 Apr, 2020.
*Efim Zelmanov (University of California, San Diego, USA)*(watch the talk here)

**Growth Functions.** - 16 Apr, 2020.
*Ivan Shestakov (IME-USP).*(watch the talk here)

**Coordination Theorems for certain non-associative algebras.**Coordinatization Theorems are very useful for classification problems. The classical Wedderburn Coordinatization Theorem claims that if a unital associative algebra \(A\) contains a matrix subalgebra \(M_n(F)\) with the same unit then \(A=M_n(B)\) for a certain subalgebra \(B\). The Jacobson Coordinatization Theorems in the structure theories of alternative and Jordan algebras state similar results for octonions and Albert algebras. Various coordinatization theorems were proved for noncommutative Jordan algebras, for commutative power associative algebras, for alternative and Jordan superalgebras, etc. In our talk, we consider three coordinatization theorems:

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1) for 2x2 matrices in the class of alternative algebras (Jacobson's problem),

2) for Jordan algebra of symmetric 2x2 matrices in the class of Jordan algebras,

3) for octonions in the class of right alternative algebras.