# Workshop on Non-associative Algebras and Applications

Monday 9th–Friday 13th July 2018

The first examples of non-associative rings and algebras appeared in the mid-19th century. Since then the theory has evolved into an independent branch of algebra, exhibiting many points of contact with other fields of mathematics and also with physics, mechanics, biology and other sciences. The aim of this workshop is to bring together mathematicians, physicists and other natural scientists interested in this field, with a focus on increasing the quality of research, promoting interaction among researchers and discussing new directions for the future.

## Schedule

MondayTuesdayWednesdayThursdayFriday
Management School LT1 Management School LT1 Management School LT1 PSC Building LT PSC Building LT
9.00 Pumpluen Mostovoy Albuquerque 9.00 - Moede
9.30 - Wesche
10.15 Fialowski Shestakov Makhlouf 10.15 - Paez
10.45 - Ceballos
11.15 Coffee break Coffee break Coffee break Coffee break
11.45 Registration Martinez Elduque Shpectorov 11.45 - Montaner
12.15 - Brox
12.45 Lunch Lunch Lunch Lunch Lunch
14.00 Lyudu Premet Benayadi 14.00 - Rua
14.30 - Barreiro
15.00 Holubowski Dotsenko McInroy 15.00 - Cabrera
15.30 - Mattarei
16.00 Coffee break Coffee break Coffee break Coffee break and end
16.30 Bavula Benkart Kulshammer
17.30 Shkarin Hurle Richter
19.00 Conference dinner

## Committees

### Scientific Committee

• Yuri Bahturin (Memorial University of Newfoundland)
• Georgia Benkart (University of Wisconsin-Madison)
• Murray Bremner (University of Saskatchewan)
• Alberto Elduque (University of Zaragoza)
• Rolf Farnsteiner (Kiel University)
• Pavel Kolesnikov (Sobolev Institute of Mathematics)
• Ottmar Loos (FernUniversität in Hagen)
• Consuelo Martinez (University of Oviedo)
• Justin McInroy (University of Bristol)
• Jacob Mostovoy (Cinvestav IPN)
• Alexander Premet (University of Manchester)
• Susanne Pumpluen (University of Nottingham)
• Ivan Shestakov (University of Sao Paulo)
• Sergey Shpectorov (University of Birmingham)
• Efim Zelmanov (UC San Diego)

We have limited funds to support UK-based PhD students who do not have access to Research Council or other institutional funding. This funding will pay for travel costs and accommodation and will be allocated on a first-come, first-served basis. Should you wish to be considered for this, please contact one of the organisers, giving an estimate of costs and the name of your supervisor(s), who will be asked to support your request.

## Abstracts

Poisson admissible algebras and Symmetric Leibniz algebras

G2-invariants from cross-product algebras

Benkart - abstract (PDF)

A unified framework for PBW type theorems

I shall explain how to put all existing PBW type theorems for universal enveloping algebras of various algebraic structures on the same ground, using the language of category theory. It turns out that the (appropriately defined) PBW property for universal enveloping T-algebras of ALL algebras of a given type S is equivalent to the simple property of the corresponding pair of algebraic structures S and T, naturally expressible in the language of modules over operads [or, more generally, monads]. I shall then use this framework to prove a new PBW type result that seems to be out of reach for classical methods, a PBW theorem for universal enveloping dendriform algebras of pre-Lie algebras. This talk is partly based on joint work with P.Tamaroff.

Alberto Elduque (University of Zaragoza)

The classification, up to equivalence, of the fine gradings by abelian groups on simple Lie algebras over an algebraically closed field of characteristic 0 has been achieved recently through the work of many authors. In order to extend this classification to semisimple Lie algebras, it is necessary to study the relationship between fine gradings by abelian groups on a direct sum of simple algebras and fine gradings on the simple components.

In this talk, it will be shown that the classification of gradings by abelian groups on finite direct sums of simple finite-dimensional nonassociative algebras over an algebraically closed field can be reduced, by means of the use of loop algebras, to the corresponding problem for simple algebras. This requires a good definition of (free) products of group-gradings.

Benedikt Hurle (Université de Haute-Alsace)

Alpha-type Cohomologies and Deformations of Hom-algebras

Hom-Lie algebras were first introduced in the study of q-deformations of Lie algebras appearing in physics. Here the Jacobi identity is twisted by a morphism called structure map. Later other types of Hom-algebras such as Hom-associative algebras have been defined and studied. In this talk, I want to define a generalization of the Hochschild cohomology for associative algebras to the Hom-setting. This new cohomology is called alpha-type Hochschild cohomology. It can be used to study formal deformations in the sense of Gerstenhaber, where the multiplication and the structure map are deformed.

Natalia Iyudu (Edinburgh University)

Hilbert series and Koszulity for quadratic algebras via the Groebner bases

We consider certain potential algebras: Sklyanin, contraction algebras, homology of moduli spaces of pointed curves $M_{0,6}$ of genus zero. The facts we prove for them, using in particular, Gr\"obner bases theory, include Koszulity, Hilbert series calculation, Calabi-Yau, PBW. This is joint work with A. Smoktunowicz [IMRN, 2018], and S.Shkarin [J.Algebra, 2017].

Julian Külshammer (Universität Stuttgart)

Understanding quasi-hereditary algebras using A-infinity algebras

A-infinity algebras are a class of (in general) non-associative graded algebras, which are however associative up to a system of higher homotopies. A particular example is provided by the Ext-algebra of a module over an associative algebra. In this talk, I will explain how a version of Koszul duality for these A-infinty algebras provides the construction of a unique exact Borel subalgebra of each quasi-hereditary algebra. Quasi-hereditary algebras are a class of finite dimensional associative algebras that frequently appear in representation theory, e.g. of Lie algebras and finite groups. This is based on joint work with S. Koenig, S. Ovsienko, and V. Miemietz.

Abdenacer Makhlouf (University of Haute Alsace)

Quantum Deformations, Twistings and Hom-Type algebraic structures

Makhlouf - abstract (PDF)

Consuelo Martinez (University of Oviedo)

Superconformal algebras and their representations

We will consider the known superconformal algebras, giving their constructions and the open conjectures. We will also discuss what is known about their representations and some work in progress.

Sandro Matterai (University of Lincoln)

Exponentials of derivations in prime characteristic

Taking the exponential of a derivation is a classical tool for producing an automorphism of an algebra, under suitable conditions. Over fields of positive characteristic modified exponentials such as a truncated exponential have been used, notably in the fundamental but rather specialized technique technique of toral switching for modular Lie algebras. A crucial aspect of toral switching is that the application of such modified exponentials to a derivation produces a linear map which transforms a certain grading of the algebra into another one.

I will analyze the mechanism which makes this work and describe how generalized exponentials based on certain Laguerre polynomials achieve a type of grading switching for nonassociative algebras which goes beyond the reach of toral switching.

Justin McInroy (Bristol University)

Code algebras, axial algebras and VOAs

We introduce a new class of non-associative algebras coming from binary linear codes, called code algebras. This is inspired by the link between binary codes and vertex operator algebras (VOAs), where the code governs the representation theory of the code VOA. VOAs themselves are important objects: Borcherds used them to show monstrous moonshine - the unexpected link between the Monster simple sporadic group and modular forms - which won him a Fields Medal, but they are also mysterious and difficult to work with.

Code algebras have several nice properties: except for one example, they are always simple algebras. There are some obvious idempotents, but we also give a construction for finding other idempotents. This allows us to show when they are axial algebras. These are another type of non-associative algebra, generated by idempotents, with a Peirce-like decomposition of the algebra satisfying some fusion law. When this fusion law is graded it leads to automorphisms of the algebra. We classify when our code algebra axial algebras are $\mathbb{Z}_2$-graded, discovering in the process an infinite family which are $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded - the first axial algebras with more than a $\mathbb{Z}_2$-grading.

Jacob Mostovoy (Cinvestav IPN)

Leibniz homology and differential graded Lie algebras

I will describe a pair of adjoint functors between the category of Leibniz algebras and the category of differential graded Lie algebras; a non-linear version of these functors relates the category of racks to that of "cubical products". These constructions shed some light on the nature of Leibniz homology.

Mária Pilar Páez-Guillán (University of Santiago de Compostela)

Pilar Paez-Guillan - abstract (PDF)

Alexander Premet (University of Manchester)

Lie algebras without strong degeneration and the Richardson property

Premet - abstract (PDF)

Susanne Pumpluen (University of Nottingham)

Nonassociative algebra obtained from skew polynomial rings and their applications

Starting from a skew polynomial ring, we define a class of unital nonassociative algebras introduced by Petit in 1966 but still basically unknown. These algebras are canonical generalizations of (associative) central simple algebras. Classical results from Albert, Amitsur and Jacobson can, therefore, be generalized to a nonassociative setting. Time permitting, we will also mention some applications to space-time block coding.

Stanislav Shkarin (Queen’s University, Belfast)

Construction of hypercyclic multiplication operators via factoring nonassociative normed algebras

The maximal ideal spectrum of a (unital) commutative Banach algebra is known to be a Hausdorff compact topological space, which is metrizable provided the algebra is separable. Conversely, every metrizable compact topological space is the spectrum of a separable commutative Banach algebra. Curiously enough, in the sea of examples of Banach algebras in the vast literature on the subject not a single example exists of a separable Banach algebra for which the underlying Banach space is reflexive, while the spectrum is locally infinite dimensional (every non-empty open subset is infinite dimensional). One of the objectives of the talk is to construct a separable unital commutative semisimple Banach algebra such that its spectrum is homeomorphic to the Hilbert cube and the underlying Banach space is isomorphic to a Hilbert space. The construction is done in two steps. First, we introduce a nonassociative normed algebra with certain properties of multiplication and with the underlying normed space being a Hilbert space. The required Banach algebra is obtained by factoring this nonassociative algebra (the commutativity and the associativity hold in the factor).

The Banach algebra $A$ obtained can be used to procure another very peculiar example. We take the direct limit of a sequence of very carefully chosen quotients of $A$ to get an infinite dimensional non-unital commutative Banach algebra $B$ whose undelying Banach space is a Hilbert space and for which there is an element $b$ such that the sequence $b(1+b)^n$ is dense in $B$. The existence of such an element forces $B$ to be radical and weakly amenable (seems to be the first example of a reflexive radical weakly amenable Banach algebra). Furthermore, multiplication by $1+b$ provides an example of a hypercyclic operator on a Hilbert space whose direct sum with itself is non-cyclic. This is not the first such example (Bayart and Matheron provided one a few years back), but it is certainly the first one with small spectrum (answers a question of Bayart and Matheron about the spectra of such operators).

Sergey Shpectorov (University of Birmingham)

Jordan algebras, Matsuo algebras, double axes

Starting from the properties of Jordan algebras, we will motivate the introduction of the class of axial algebras called algebras of Jordan type $\eta$. In addition to Jordan algebras, the key examples of these are Matsuo algebras which correspond to 3-transposition groups. The available results on algebras of Jordan type will be reviewed, including: (1) the classification of 2-generated algebras; (2) the classification of algebras with $\eta\neq\frac{1}{2}$; (3) algebras with non-bijective $\tau$-map; (4) existence of an associating bilinear form. After that we will turn to double axes, which are sums of two orthogonal single axes, and we will demonstrate that double axes satisfy similar fusion rules to the $2A$-axes in the Griess algebra for the Monster simple group. We will conclude with examples of algebras generated by at most three single and double axes and present a construction giving infinite series of examples.

Morten Wesche (TU Braunschweig)

Enumeration of class 2 associative algebras over finite fields

Wesche - abstract (PDF)

## Participants

• Helena Albuquerque (Coimbra, Portugal)
• Mashaer Alsaeedi (Birmingham, England)
• Majed Almubark (Sheffield, UK)
• Floriana Amicone (Manchester, UK)
• Alexander Baranov (Leicester, UK)
• Elisabeta Barreiro (Coimbra, Portugal)
• José Brox (Coimbra, Portugal)
• Alessandro Carotenuto (Trieste, Italy)
• Yolanda Cabrera (Malaga, Spain)
• Manuel Ceballos (Seville, Spain)
• Isabel Cunha (Covilhã, Portugal)
• Alberto Elduque (Zaragoza, Spain)
• Christof Ender (Vienna, Austria)
• Alice Fialowski (Eotvos Lorand, Hungary)
• Jan Gabrowski (Lancaster, UK)
• Pilar Páez (Santiago de Compostella, Spain)
• Waldemar Holubowski (Gliwice, Poland)
• Benedikt Hurle (Mulhouse, France)
• Natalia Iyudu (Edinburgh, UK)
• Julian Külshammer (Stuttgart, Germany)
• Paul Levy (Lancaster, UK)
• Mark MacDonald (Lancaster, UK)
• Sandro Mattarei (Lincoln, UK)
• Justin McInroy (Bristol, UK)
• Abdenacer Makhlouf (Mulhouse, France)
• Consuelo Martinez (Oviedo, Spain)
• Tobias Moede (Braunschweig, Germany)
• Fernando Montaner (Zaragoza, Spain)
• Jacob Mostovoy (Cinvestav, Mexico)