Tag - Other non-associative rings

Antonio Viruel: Permutation representations of finite groups via evolution algebras

18th December, 2023

In the wake of the influential work by Elduque-Labra, it is known that every finite-dimensional evolution K-algebra X such that X2 = X, namely X is idempotent, has a finite group of automorphisms. Building on this foundation, works of Costoya et al. show that given any finite group G, there exists an idempotent finite-dimensional evolution algebra X such that Aut(X) ≅ G. Moreover, when the base field is sufficiently large in comparison to the group G, such an X can be selected to be simple. As a result, Sriwongsa-Zou propose that idempotent finite-dimensional evolution algebras can be classified based on the isomorphism type of their group of automorphisms and dimension. Within this context, we establish that the natural representation of highly transitive groups cannot be realized as the complete group of automorphisms of an idempotent finite-dimensional evolution algebra. For instance, for any sufficiently large integer n, there exists no evolution algebra X such that X2 = X, dim X = n, and Aut(X) is isomorphic to the alternating group An. However, we demonstrate that for any (not necessarily faithful) permutation representation ρ : G → Sn and any field K, there exists a finite-dimensional evolution K-algebra X such that X2 = X, Aut(X) ≅ G and the induced representation given by the Aut(X)-action on the natural idempotents of X is ρ.

Susanne Pumpluen: A way to generalize classical results from central simple algebras to the nonassociative setting

11th December, 2023

Recently, the theory of semiassociative algebras and their Brauer monoid was introduced by Blachar, Haile, Matri, Rein, and Vishne as a canonical generalization of the theory of associative central simple algebras and their Brauer group: together with the tensor product semiassociative algebras over a field form a monoid that contains the classical Brauer group as its unique maximal subgroup. We present classes of semiassociative algebras that are canonical generalizations of classes of certain central simple algebras and explore their behavior in the Brauer monoid. Time permitting, we also discuss some - hopefully interesting - particularities of this newly defined Brauer monoid.

Artem Lopatin: Polynomial invariants for 2-dimensional algebras

20th November, 2023

For every 2-dimensional non-associative algebra A we describe generators for the algebra I(A) of polynomial invariants of several copies of A. We also discuss Artin's conjecture on invariants, which claims that I(A) is generated by the traces of operators of left and right multiplication over the algebra A.

Petr Vojtěchovský: Solvability and nilpotence just beyond groups

30th October, 2023

Solvability and nilpotence arise naturally from the commutator theory in congruence modular varieties. In the presence of associativity, the resulting concepts agree with the classical concepts of group theory. But the two kinds of solvability differ in loops (= not necessarily associative groups) and it is a difficult question to determine the boundary where the two theories coincide. I will review the general theory and report on recent results, particularly in Moufang loops. For instance, we will prove the Odd Order Theorem for Moufang loops for the stronger notion of solvability.

Adam Chapman: Roots and Critical Points of Cayley-Dickson Algebras

18th September, 2023

We study the roots and critical points (i.e., points at which the formal derivative vanishes) of standard polynomials over Cayley-Dickson algebras. In the anisotropic real case, we prove that the critical points live inside the convex hull of the roots of the polynomial.

Bauyrzhan Sartayev: Binary perm algebras and alternative algebras

11th September, 2023

We describe the defining identities of a variety of binary perm algebras which is a subvariety of the variety of alternative algebras. Moreover, we construct a basis of the free binary perm algebra. In addition, we describe the subalgebras of binary perm algebras under commutator which has a connection with Malcev algebras.

David Towers: Zinbiel algebras are nilpotent

30th January, 2023

Zinbiel algebras were introduced by Loday in 1995. They are the Koszul dual of Leibniz algebras and Lemaire proposed the name of Zinbiel,
which is obtained by writing Leibniz backwards. In this talk I will introduce some of their main properties, including the fact that, over any field, they are nilpotent.

Seidon Alsaody: Brown algebras, Freudenthal triple systems and exceptional groups over rings

16th January, 2023

Exceptional algebraic groups are intimately related to various classes of non-associative algebras: for example, octonion algebras are related to groups of type G2 and D4, and Albert algebras to groups of type F4 and E6. This can be used, on the one hand, to give concrete descriptions of homogeneous spaces under these groups and, on the other hand, to parametrize isotopes of these algebras using said homogeneous spaces. The key tools are provided by the machinery of torsors and faithfully flat descent, working over arbitrary commutative rings (sometimes assuming 2 and 3 to be invertible). I will talk about recent work where we do this from Brown algebras and their associated Freudenthal triple systems, whose automorphism groups are of type E6 and E7, respectively. I will hopefully be able to show how algebraic and geometric properties come together in this picture.

Sergey Malev and Alexei Kanel-Belov: Evaluations of nonassociative polynomials on finite-dimensional algebras

5th August, 2021

Let p be a polynomial in several non-commuting variables with coefficients in an algebraically closed field K of arbitrary characteristic. It has been conjectured that for any n, for p multilinear, the image of p evaluated on the set Mn(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(K). In this talk we will discuss the generalization of this result for non-associative algebras such as Cayley-Dickson algebra (i.e. algebra of octonions), pure (scalar free) octonion Malcev algebra and basic low rank Jordan algebras.

José María Pérez Izquierdo: Some aspects of the free non-associative algebra

15th April, 2021

The free nonassociative algebra provides a simple combinatorial context to extend some constructions from the associative setting. In this talk, based on joint work with J. Mostovoy and I. P. Shestakov, I will briefly discuss three of them related to nonassociative Lie theory: the embedding of the free loop as nonassociative formal power series, a nonassociative extension of the Baker-Campbell-Hausdorff formula and a nonassociative version of Solomon's descent algebra.