Workshop on Nonassociative Algebras and Applications
Monday 9th–Friday 13th July 2018
The first examples of nonassociative rings and algebras appeared in the mid19th 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.
Speakers
 Helena Albuquerque (University of Coimbra), Algebras, Superalgebras and Quasialgebras
 Elisabete Barreiro (University of Coimbra), The structure of Leibniz superalgebras admitting a multiplicative basis
 Vladimir Bavula (Sheffield University), Classification of simple weight modules over the Schrödinger algebra
 Saïd Benayadi (University of Lorraine), Poisson admissible algebras and Symmetric Leibniz algebras
 Georgia Benkart (University of WisconsinMadison), G2invariants from cross product algebras
 Jose Brox (University of Coimbra), Nonassociative products in nonassociative algebras with involution
 Yolanda Cabrera (Universidad de Málaga), Evolution algebras
 Manuel Ceballos (Universidad Loyola Andalucía), Filiform Lie algebras
 Vladimir Dotsenko (Trinity College Dublin), A unified framework for PBW type theorems
 Alberto Elduque (University of Zaragoza), Gradings on semisimple algebras
 Alice Fialowski (Eotvos Lorand University), Lie algebras with invariant inner product, and their cohomology
 Waldemar Hołubowski (Silesian University of Technology), Ideals and derivations in Lie algebras of infinite matrices
 Benedikt Hurle (Université de HauteAlsace), Alphatype Cohomologies and Deformations of Homalgebras
 Natalia Iyudu (Edinburgh University), Hilbert series and Koszulity for quadratic algebras via the Groebner bases
 Julian Külshammer (Universität Stuttgart), Understanding quasihereditary algebras using Ainfinity algebras
 Abdenacer Makhlouf (University of Haute Alsace), Quantum Deformations, Twistings and HomType algebraic structures
 Sandro Matterai (University of Lincoln), Exponentials of derivations in prime characteristic
 Consuelo Martinez (University of Oviedo), Superconformal algebras and their representations
 Justin McInroy (Bristol University), Code algebras, axial algebras and VOAs
 Tobias Moede (TU Braunschweig), Searching for new simple Lie algebras in small characteristic
 Fernando Montaner (University of Zaragoza), Some Hersteinlike theorems on Tracers in associative systems
 Jacob Mostovoy (Cinvestav IPN), Leibniz homology and differential graded Lie algebras
 Mária Pilar PáezGuillán (University of Santiago de Compostela), On Whitehead’s quadratic functor for supermodules
 Alexander Premet (University of Manchester), Lie algebras without strong degeneration and the Richardson property
 Susanne Pumpluen (University of Nottingham), Nonassociative algebra obtained from skew polynomial rings and their applications
 Elizabeth Remm (Universite de Haute Alsace, Mulhouse), Geometrical Structures on Lie algebras
 Johan Richter (Lunds Universitet), Nonassociative Ore extensions
 Ignacio Rúa (University of Oviedo), Nonassociative rings and nonlinear codes
 Ivan Shestakov (University of Sao Paulo), The speciality problem for Malcev algebras
 Stanislav Shkarin (Queen’s University, Belfast), Construction of hypercyclic multiplication operators via factoring nonassociative normed algebras
 Sergey Shpectorov (University of Birmingham), Jordan algebras, Matsuo algebras, double axes
 Morten Wesche (TU Braunschweig), Enumeration of class 2 associative algebras over finite fields
Schedule
Monday  Tuesday  Wednesday  Thursday  Friday  

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 WisconsinMadison)
 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 UKbased 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 firstcome, firstserved 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.
Please contact the local organisers with any queries.
Local Organising Committee
Abstracts
Saïd Benayadi (University of Lorraine)
Poisson admissible algebras and Symmetric Leibniz algebras
Georgia Benkart (University of WisconsinMadison)
G2invariants from crossproduct algebras
Vladimir Dotsenko (Trinity College Dublin)
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 Talgebras 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 preLie algebras. This talk is partly based on joint work with P.Tamaroff.
Alberto Elduque (University of Zaragoza)
Gradings on semisimple algebras
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 finitedimensional 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 groupgradings.
Benedikt Hurle (Université de HauteAlsace)
Alphatype Cohomologies and Deformations of Homalgebras
HomLie algebras were first introduced in the study of qdeformations of Lie algebras appearing in physics. Here the Jacobi identity is twisted by a morphism called structure map. Later other types of Homalgebras such as Homassociative algebras have been defined and studied. In this talk, I want to define a generalization of the Hochschild cohomology for associative algebras to the Homsetting. This new cohomology is called alphatype 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, CalabiYau, PBW. This is joint work with A. Smoktunowicz [IMRN, 2018], and S.Shkarin [J.Algebra, 2017].
Julian Külshammer (Universität Stuttgart)
Understanding quasihereditary algebras using Ainfinity algebras
Ainfinity algebras are a class of (in general) nonassociative graded algebras, which are however associative up to a system of higher homotopies. A particular example is provided by the Extalgebra of a module over an associative algebra. In this talk, I will explain how a version of Koszul duality for these Ainfinty algebras provides the construction of a unique exact Borel subalgebra of each quasihereditary algebra. Quasihereditary 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 HomType algebraic structures
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 nonassociative 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 nonassociative algebra, generated by idempotents, with a Peircelike 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 nonlinear 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 PáezGuillán (University of Santiago de Compostela)
On Whitehead’s quadratic functor for supermodules
Pilar PaezGuillan  abstract (PDF)
Alexander Premet (University of Manchester)
Lie algebras without strong degeneration and the Richardson property
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 spacetime 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 nonempty 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 nonunital 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 noncyclic. 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 3transposition groups. The available results on algebras of Jordan type will be reviewed, including: (1) the classification of 2generated algebras; (2) the classification of algebras with $\eta\neq\frac{1}{2}$; (3) algebras with nonbijective $\tau$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
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)
 Vladimir Bavula (Sheffield, UK)
 Saïd Benayadi (Lorraine, France)
 Georgia Benkart (WisconsinMadison, USA)
 José Brox (Coimbra, Portugal)
 Alessandro Carotenuto (Trieste, Italy)
 Yolanda Cabrera (Malaga, Spain)
 Manuel Ceballos (Seville, Spain)
 Isabel Cunha (Covilhã, Portugal)
 Vladimir Dotsenko (Dublin, Ireland)
 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)
 Radu Iordanescu (Bucharest, Romania)
 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)
 Rubén Muñoz (Madrid, Spain)
 Adam Owen (Nottingham, UK)
 Alexander Premet (Manchester, UK)
 Susanne Pumpluen (Nottingham, UK)
 Antonio Ricciardo (Bologna, Italy)
 Johan Richter (Malarden, Sweden)
 Ignacio Fernández Rúa (Oviedo, Spain)
 Ivan Shestakov (Sao Paulo, Brazil)
 Stanislav Shkarin (Queens University, Belfast, UK)
 Hasan Shlaka (Leicester, UK)
 Sergey Shpectorov (Birmingham, UK)
 Daniel Thompson (Nottingham, UK)
 David Towers (Lancaster, UK)
 Guillermo Vera de Salas (Madrid, Spain)
 Morten Wesche (Braunschweig, Germany)