## Non-associative algebra and Lie theory, Mexico City, February 27 - March 3, 2017

Titles and abstracts sent to the conference organisers or submitted through the registration form appear in this page. We urge the speakers who have not submitted their titles and abstracts to do that as soon as possible.

Murray Bremner (University of Saskatchewan)

Title: Quadratic nonsymmetric quaternary operads

Abstract: We use computational linear algebra and commutative algebra to study spaces of relations satisfied by quadrilinear operations. The relations are analogues of associativity in the sense that they are quadratic (every term involves two operations) and nonsymmetric (every term involves the identity permutation of the arguments). We focus on determining those quadratic relations whose cubic consequences have minimal or maximal rank. We approach these problems from the point of view of the theory of algebraic operads. This is a joint work with Juana Sánchez-Ortega, to appear in Linear and Multilinear Algebra, arXiv:1512.02880.

Bruno Cisneros de la Cruz (National Autonomous University of Mexico)

Title: Algebraic and topological perspectives on virtual braids

Abstract: In this talk I will present different approaches to virtual braids (combinatorial, algebraic and topological) and how these can help us to solve some problems.

Askar Dzhumadildaev (Instutute of Mathematics, Academy of Sciences of Kazakhstan)

Title: Pfaffians, Baxter operators and related algebraic structures

Abstract: We give construction of pfaffians and hafnians by Dyck paths. We study symmetries of pfaffians and hafnians. We construct Baxter operator on pfaffians and show that pfaffians have natural Zinbiel structure.

Vyacheslav Futorny (University of Sao Paulo)

Title: Gelfand-Tsetlin representations and Gelfand-Graev continuaton

Abstract: Gelfand-Tsetlin representations form an important class of representations of gl(n). This is the largest known subcategory of weight modules (with respect to a fixed Cartan subalgebra) where the classification problem of irreducible modules can be addressed successfully. Explicit construction of such modules was always a challenge. After the discovery of a Gelfand-Tsetlin basis for finite dimensional modules Gelfand and Graev suggested a certain method which allows to construct explicitly such modules. Recently this method was improved and a large new family of such modules was obtained in a joint work with L.E. Ramirez and J.Zhang. We will discuss the state of the art in this theory.

Alexandre Grichkov (University of Sao Paulo)

Title: Antisymmetric cohomology and its applications

Abstract: We define some kind of non-standard cohomology over the field of two elements and apply it to description of two-step nilpotent Bol loops of exponent two.

Isabel Hernández (CONACYT-CIMAT Mérida)

Title: Closure Orbits of Complex (2,2)-Dimensional Lie Superalgebras.

Abstract: In this talk we discuss some results about the affine variety of complex Lie superalgebras. In particular we establish the list of GL

_{2}×GL

_{2}-orbits under the action by "change of basis", and we provide the Zariski closure of these orbits. Joint with Alejandra Alvarez.

Natalia Iyudu (University of Edinburgh)

Title: Sklyanin algebras via Gröbner bases and finiteness conditions for potential algebras

Abstract: I will discuss how some questions on Sklyanin algebras can be solved using combinatorial techniques, namely, the theory of Groebner bases, and elements of homological algebra. We calculate the PoincarĂ© series, prove Koszulity, PBW, Calabi-Yau, etc., depending on the parameters of the Sklyanin algebras. There was a gap in the Artin-Schelter classification of algebras of global dimension 3, where Koszulity and the PoincarĂ© series for Sklyanin algebras were proved only generically. It was filled in the Grothendieck Festschrift paper of Artin, Tate and Van den Bergh, using the geometry of elliptic curves. Our point is that we recover these results by purely algebraic, combinatorial means. We use similar methods for generalized Sklyanin algebras, and for other potential algebras.

Dmitri Kaledin (Steklov Mathematics Institute and CINVESTAV)

Title: Roots of unity, positive characteristic, and deformation quantization

Abstract: There is a well-known and trivial observation that "p-th root of unity is p-adically close to unity", and it leads to some non-trivial consequences both in representation theory and in sympletic algebraic geometry. A representation theory example is Lusztig's quantum group at p-th root of unity: it is defined over the integers, and when you reduce it mod p, it stops being quantum. This has been exploited by Lusztig and others to relate representation theory of the quantum group, a char 0 theory, and representation theory of the usual universal enveloping algebra in char p. I am going to discuss algebraic geometry underlying this representation theory pattern; this includes both my old paper with Bezrukavnikov on deformation quantization in char p and newer results.

Iryna Kashuba (University of Sao Paulo)

Title: Indecomposable representations of Jordan superalgebras JP

_{2}and Kan(n)

Abstract: We will construct indecomposable representations of Jordan superalgebras JP

_{2}and Kan(n). Our main tool is the famous Tits-Kantor-Köcher construction. The representations of JP

_{2}and Kan(n) are given in terms of the Ext quiver algebras of the category of representations with the short grading for Lie superalgebras P(3) and H(n+3) respectively.

This is joint result with Vera Serganova.

Ernesto Lupercio (CINVESTAV)

Title: Quantum toric geometry, complex systems, and mirror symmetry.

Abstract: In this talk I will survey our investigations regarding quantum toric varieties (Katzarkov, Lupercio, Meersseman, Verjovsky), its relation to sandpiles, tropical geometry and complex systems (Guzman, Kalinin, Lupercio, Prieto, Shkolnikov) and Mirror Symmetry (Katzarkov, Kerr, Lupercio, Meerssemann).

Martin Markl (Institute of Mathematics, Czech Academy of Sciences)

Title: Open-closed modular operads, Cardy condition and string field theory

Abstract: We prove that the modular operad of diffeomorphism classes of Riemann surfaces with both `open' and `closed' boundary components, in the sense of string field theory, is the modular completion of its genus 0 part quotiented by the Cardy condition. We also provide a finitary presentation of a version of this modular two-colored operad and characterize its algebras via morphisms of Frobenius algebras. This is joint work with Martin Doubek.

Olivier Mathieu (University Lyon 1)

Title: Classification of modules of the intermediate series

Abstract: This is based on joint work with C. Martinez and E. Zelmanov.

Zbigniew Oziewicz (National Autonomous University of Mexico)

Title: Non-associative Frobenius algebra

Abstract: Frobenius algebra is formulated within the abelian monoidal category of operad of graphs. The new concept of solvable Frobenius algebra is proposed as an analogy for solvable Lie algebra. I am looking for necessary and sufficient conditions on an algebra to be solvable Frobenius algebra. It is pointed that the semi-simple Lie algebra as well as Jordan algebras are examples of non-associative Frobenius algebra. Frobenius algebra give rise to the Frobenius camel-coalgebra.

José-Antonio de la Peña (National Autonomous University of Mexico)

Title: A Partition Formula for Fibonacci Numbers

Abstract: We are going to present a partition formula for the even Fibonacci numbers. The formula is motivated by the appearance of these numbers in the representation theory of the socalled 3-Kronecker quiver, i.e., the oriented graph with two vertices and three arrows in the same direction. We greatly generalize this example using Coxeter elements of Weyl groups of graphs.

Michael Polyak (Technion and CINVESTAV)

Title: Milnor's μ-invariants via configuration spaces and dialgebras

Abstract: Milnor's μ-invariants of link-homotopy are higher versions of linking numbers (related to Massey products in the link complement).

We describe a new construction of these invariants using configuration spaces and a tree part of a certain graph complex. This enables us to extract some simple combinatorial formulas for these invariants in terms of link diagrams and weighted tree counting.

Invariance is then reformulated in terms of Loday's dialgebras, with two types of crossings in a link diagram corresponding to the dialgebra left and right multiplications. Milnor's invariants can then be regarded as certain morphisms from an operad of tree tangles to the dialgebra operad.

Raul Quiroga (CIMAT, Guanajuato)

Title: Some applications of Lie theory to operator theory

Abstract: Lie theory has among its relevant features a very well developed and interesting representation theory. For noncompact semisimple Lie groups of Hermitian type some of these representations act on Hilbert spaces of holomorphic functions. This has led to interesting applications of Lie theory to the operator theory on function spaces. We will describe some of these applications with particular emphasis on Toeplitz operators acting on Bergman spaces.

Maria Ronco (University of Talca)

Title: Algebraic structures on graph associahedra

Abstract: We want to motivate the study of algebraic structures on the space spanned by graph associahedra. In many examples, combinatorics provide an important tool in the study of free objects for new algebraic theories, as well as in the understanding of the relations between them. In the last years, an important amount of work has been done on generalisations of associahedra, on one hand, and on combinatorial descriptions of operads related to it, on the other side. Graph associahedra was introduced by M. Carr and S. Devadoss. They defined a convex polytope KΓ of dimension n associated to any finite simple graph Γ with n + 1 nodes. When Γ is the complete graph K

_{n+1}, the polytope K

_{n+1}they get is the usual permutohedron of dimension n, while the line graph L

_{n+1}gives rise to the associahedron. Our aim is to introduce algebraic theories on the spaces spanned by the faces of graph associahedra. We show that it is posible to define a coalgebra structure, as well as an associative product, on the vector space spanned by all faces of graph associahedra, which restricts to well-know combinatorial Hopf algebras when we consider certain special families of graphs. There exists a notion of suspension of graphs which induces associative products on the vector spaces spanned by the faces of previously studied polytopes, as the stellohedra and the pterohedra. On the other hand, we prove that substitution induces natural operations on graph associahedra, which gives a generalization of non-symmetric operads.

Larissa Sbitneva (Autonomous University of the State of Morelos)

Title: Methods of S. Lie in the study of some classes of smooth loops: differential equations and integrability conditions.

Abstract: The original approach of S. Lie to Lie groups based on differential equations allowed to obtain characterizations of smooth loops with the right Bol identity and the integrability conditions which lead to the binary-ternary algebra as a proper infinitesimal object, which turns out to be a Bol algebra. There exists an analogous theory for Moufang loops.

We will consider the differential equations of smooth M-loops, generalising smooth Bol loops, and the further generalisations introduced by Lev Sabinin. The corresponding geometry is related to generalised symmetric spaces.

Further examination of the integrability conditions for the differential equations allows to introduce proper infinitesimal objects for the class of loops under consideration. This development leads to a finite-dimensional Lie algebra g and a subalgebra h⊂g with the decomposition g=h⊕m, which is hyporeductive in this case. There is a theory which allows to introduce a proper infinitesimal object as for the hyporeductive and pseudoreductive loops (Lev V. Sabiinin, Smooth Quasigroups and Loops, Kluwer Academic Publisher, 1999)

As well we will present the consideration of Lev Sabinin, who initiated the study of smooth loops actions suitable for applications to Mathematical Physics. The investigation of corresponding differential equations leads to some analogus of Lie theorems.

Ivan Shestakov (University of Sao Paulo)

Title: Non-matrix varieties for some classes of non-associative algebras

Abstract: see this PDF file.

Alexander Stolin (University of Gothenburg)

Title: Classification of quantum groups and Galois cohomology

Abstract:

Gregor Weingart (National Autonomous University of Mexico)

Title: Kählerian normal coordinates

Abstract: Exponential coordinates for Kähler manifolds are not holomorphic coordinates, for the reason the do not fit well into the framework of complex analysis on Kähler manifolds. Nevertheless around every point in a Kähler manifold there exists a distinguished holomorphic coordinate system as observed by Bochner. In the talk I will study the infinite order jet of these Kählerian normal coordinates using a mix of differential geometric intuition and multilinear algebra. In particular I will describe the Spencer connection on the Kähler normal potential completely, moreover I will show that Kählerian normal coordinates generalize to a much larger class of complex affine manifolds.

Gregory P. Wene (University of Texas at San Antonio)

Title: Semifields antiisomorphic to themselves

Abstract: see this PDF file.

James Wilson (Colorado State University)

Title: Group isomorphism by nonassociative methods

Abstract: Using filters we summarize how Lie, Jordan, star, and pair algebras have recently emerged to solve hard instances of the group isomorphism problem. This is a consequence of a general Galois triple correspondence between multilinear algebra, algebraic geometry, and non-associative operator domains. Among the outcomes, we discover the standard Whitney tensor product is not universal, but rather tensor products over Lie algebras are universal. These new tensor products characterize many nonassociative simple algebras and modules, and supports generalizations of Morita theory and Rosenberg-Zelinsky sequences. But all progress stops when we encounter nonsingular tensors.

This talk reports on individual and joint research with Uriya First and Josh Maglione