Mini-Workshop “Non-associative algebras and hom-algebra structures”

Mälardalen University, Västerås, October 23, 2025

Organiser: Professor Sergei Silvestrov External link., Division of Mathematics and Physics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

Workshop program

Location: Mälardalen University, Västerås, Room: U2-158 (Second floor, U-building)

13:00-13:15 Opening of the workshop (preparing equipment for talks)

Time: 13:15-14:00

Title: A New Perspective on the Cayley–Dickson Construction: Flipped Polynomial Rings

Speaker: Per Bäck, Mälardalen University, Västerås, Sweden

Abstract: In this talk, I will shed new light on the mysterious Cayley–Dickson construction. We introduce a new class of polynomial rings equipped with a “flipped” multiplication, from which all Cayley–Dickson algebras arise naturally as quotients. This framework extends the classical realization of the complex numbers as a quotient of a polynomial ring, and the quaternions as a quotient of a skew polynomial ring, to the octonions and beyond.

This is joint work with Masood Aryapoor.

Time: 14:00-14:45

Title: Gradings associated to certain nilpotent elements

Speaker: Esther García González, Universidad Rey Juan Carlos, Madrid

Abstract: An element in a ring is nilpotent last-regular if it is nilpotent of certain index and its last nonzero power is regular von Neumann. This type of element naturally arises when studying certain inner derivations in the Lie algebra Skew(R,*) of a ring R with involution * whose indices of nilpotence differ when considering them acting as derivations on Skew(R,*) and on the whole R. When moving to the symmetric Martindale ring of quotients we still obtain inner derivations with the same indices of nilpotence on the symmetric Martindale ring of quotients and on its skew-symmetric part, but with the extra condition of being generated by a nilpotent last-regular element. This condition strongly determines the structure of the symmetric Martindale ring of quotients. I will review the Jordan canonical form of nilpotent last-regular elements, their associated matrix units, and show how to get gradings in associative algebras (with and without involution) when they have such elements.

Time: 14:50-15:35

Title: Gradings on Heisenberg-type Algebras

Speaker: Cristina Draper Fontanals, Universidad de Málaga, Spain

Abstract: Over the twentieth century and beyond, there has been growing interest in group gradings on Lie-theoretic structures. Gradings reveal structural properties and arise naturally in a variety of contexts, including superalgebras, loop algebras, Lie color algebras, finite-order automorphisms, symmetric and generalized symmetric spaces, and in contractions and deformations of Lie algebras. In the complex case, fine gradings on simple Lie algebras are a natural generalization of the root-space decomposition relative to a Cartan subalgebra, and they have had significant impact across many areas of mathematics. In physics, fine gradings yield maximal families of quantum observables with additive quantum numbers.

Although the classification of gradings on simple finite-dimensional Lie algebras over algebraically closed fields is not yet fully complete, it is largely understood (see the monograph [1]). By contrast, much less is known for nilpotent and solvable Lie algebras. Some references in this direction are [2-5]. Our work [6] advances this line by describing the fine (group) gradings on the Heisenberg (nilpotent) algebras Hn, on the Heisenberg superalgebras Hn,m and on the twisted Heisenberg (solvable) algebras Hₙλ , as well as their Weyl groups (symmetry groups of the gradings). The focus on these structures is motivated by the central role of the Heisenberg group in several branches of mathematics.

In this talk, I will survey gradings on Lie algebras, with emphasis on the results of [6], and on possible directions for future work. Complementing the analysis of specific cases (e.g., Heisenberg algebras), our recent contribution [7] uses gradings to construct new nilpotent Lie algebras. This broadens the family landscape – an especially fruitful approach since a complete classification of nilpotent Lie algebras is out of reach.

References

• Elduque and M. Kochetov, Gradings on Simple Lie Algebras, Mathematical Surveys and Monographs, Vol. 189, American Mathematical Society, 2013.
• A. Bahturin, Group gradings on free algebras of nilpotent varieties of algebras, Serdica Mathematical Journal, 38(1–3), 2011, pp. 1–6.
• A. Bahturin, M. Goze, and E. Remm, Group gradings on filiform Lie algebras, Communications in Algebra, 44(1), 2016, pp. 40–62.
• Hakavuori, V. Kivioja, T. Moisala, F. Tripaldi, Gradings for nilpotent Lie algebras, Journal of Lie Theory, 32(2), 2022, pp. 383–412.
• Shimoji, Morgan's mixed Hodge structures on p-filiform Lie algebras and low-dimensional nilpotent Lie algebras, arXiv:2504.08571, 2025.
• J. Calderón, C. Draper, C. Martín, and T. Sánchez, Gradings and Symmetries on Heisenberg-type Algebras, Linear Algebra and Its Applications, 458, 2014, pp. 463–502.
• Cuenca Carrégalo and C. Draper, New Lie algebras over the group ℤ₂³, arXiv:2501.02492, 2025.

Time: 15:40-16:25

Title: From algebras to superalgebras via tensor categories

Speaker: Alberto Elduque, Universidad de Zaragoza, Spain

Abstract: The symmetric tensor category of representations of the cyclic group of order p over a field of characteristic p is not semisimple. Its semisimplification is the Verlinde category Ver p, which contains a full subcategory equivalent to the category of vector superspaces. This allows us to obtain a superalgebra out of any algebra over a field of characteristic p endowed with an automorphism of order p (or a nilpotent derivation of degree at most p). This process will be explained with some detail and examples of interesting Lie and Jordan superalgebras obtained in this way will be shown.

Time: 16:30-17:15

Title: Strong hom-associativity

Speaker: Lars Hellström, Mälardalen University, Västerås, Sweden

Abstract: In the universal algebra approach to studying some variety of algebras, the first step is to determine what the free algebra looks like. In the case of hom-associative algebras, this is not yet known, and indeed it appears to be very complicated. (Surprisingly, a free hom-associative algebra always has zero divisors.) A rather large set of non-obvious identities are direct consequences of the hom-associativity axiom, but directly generating these does not seem feasible. On the other hand, their forms suggest considering the slightly stronger axiom system of the "canyon identities", which are taken as the definition of strong hom-associativity.

It turns out the previously known constructions of hom-associative algebras do in fact produce strongly hom-associative algebras. Moreover, the system of canyon identifies can be proved confluent (in the rewriting sense), so they determine a normal form for the free strongly hom-associative algebra. Hence the strongly hom-associative algebras may be a more natural variety to study.

Time: 17:15-18:00

Title: Hom-Lie Structures of generalized sl(2)-type

Speaker: Stephen Mboya, University of Nairobi, Kenya. Mälardalen University, Sweden

Abstract: In this talk, I will discuss the properties and structures of Hom-Lie algebras of generalized sl(2)-type. We construct classes of linear twisting maps that turn a skew-symmetric algebra of generalized sl(2)-type into Hom-Lie algebras, and we further identify subclasses that yield multiplicative Hom-Lie algebras. Our study explores ideals, Hom-ideals, sub-algebras, and Hom-sub-algebras, with particular attention to derived series, central descending series, as well as nilpotency and solvability properties. Furthermore, we highlight the differences between non-multiplicative and multiplicative cases.