Seminars in Jordan and Axial Algebras

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Fernando Montaner: Pairs of quotients of Jordan pairs

4th November, 2024

In this talk, we expose ongoing joint work with I Paniello on systems of quotients (in a sense partially extending the localization theory of Jordan algebras, which in turn is inspired by the localization theory of associative algebras). Localization theory in associative algebras originated in the purpose of extending the construction of fields of quotients of integral domains, and therefore in the purpose of defining ring extensions in which a selected set of elements become invertible. As it is well known in associative theory that led to Goldie's theorems, and these in turn to more general localization theories for which the denominators of the fraction-like elements of the extensions are (one-sided) ideals taken in a class of filters (Gabriel filters). These ideas have been partially extended to Jordan algebras by several authors (starting with Zelmanov's version of Goldie theory in the Jordan setting, and its extension by Fernandez López-García Rus and Montaner) and Paniello and Montaner (among others) definition of algebras of quotients of Jordan algebras. Following the development of Jordan theory, a natural direction for extending these results is considering the context of Jordan pairs. This is the objective of the research presented here. Since obviously, a Jordan pair cannot have invertible elements unless it is an algebra, and in this case, we are back in the already developed theory, the kind of quotients that would make a significative (proper) extension of the case of algebras should be based in a different notion of the quotient. An approach that seems to be promising is considering the Jordan extension of Fountain and Gould notion of local order, as has been adapted to Jordan algebras by the work of Fernández López, and more recently by Montaner and Paniello with the notion of local order, in which the bridge between algebras and pairs is established by local algebras following the ideas of D'Amour and McCrimmon. In the talk, this idea is exposed, together with the state of the research, and the open problems that it raises.

Louis Rowen: Weakly primitive axial algebras

6th May, 2024

In earlier work, we studied the structure of primitive axial algebras of Jordan type (PAJs), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying a more general fusion rule, bringing in interesting new examples such as the band semigroup algebras and various non-commutative examples. Also, we broaden our investigation to the case of 2-generated algebras for which only one axis satisfies the fusion rules. As an example we describe precisely the 2-dimensional axial algebras and the 3-dimensional and 4-dimensional weakly primitive axial algebras of Jordan type (weak PAJs), and we see, in contrast to the case for PAJs, that there are higher-dimensional weak PAJs generated by two axes. We also prove a theorem that enables us to reduce weak PAJs to uniform components.

Michael Turner: Skew Axial Algebras of Monster Type

4th March, 2024

Given a 2-generated primitive axial algebra of Monster Type, it has been shown that it has an axet which is regular or skew. With all the known examples being regular, it was proposed if any axial algebra were skew and if so, can they be classified. We will begin by defining axial algebras and axets, before producing examples of axial algebras with skew axets. We will finish by stating the complete classification of these skew axial algebras and mention how it was proven.

Sergey Shpectorov: Solid subalgebras in algebras of Jordan type half

18th December, 2023

Algebras of Jordan type η generalize in the axial context the class of Jordan algebras generated by primitive idempotents. In addition to these examples, arising for η = 1/2, the class of algebras of Jordan type includes the Matsuo algebras, constructed in terms of 3-transposition groups for all values of η. The classification of algebras of Jordan type for η ≠ 1/2 was completed by Hall, Rerhen and Shpectorov in 2015, with a correction by Hall, Segev and Shpectorov in 2018. The case of η = 1/2 remains open. Among the known results about algebras of Jordan type half are the classification, in the above mentioned paper from 2015, of 2-generated algebras, the classification of 3-generated algebras by Gorshkov and Staroletov in 2020, and the recent (from 2023) result by De Medts, Rowen and Segev bounding the dimension of 4-generated algebras by 81. In the talk we will discuss another recent (in preparation, 2023) result on the subject, by Gorshkov, Staroletov and Shpectorov. A 2-generated subalgebra B of an algebra A of Jordan type half is called solid if every primitive idempotent from B is an axis in the entire A. Surprisingly, it turns out that, at least in characteristic zero, almost all 2-generated subalgebras are solid. More, precisely, a non-solid 2-generated subalgebra is necessarily of type 3C(1/2). Consequently, if a finite-dimensional algebra of Jordan type half has a finite automorphism group then it is either a Matsuo algebra or a factor of Matsuo algebra. The above result hints of a possibility of a geometric theory of algebras of Jordan type half.

Tom De Medts: Primitive axial algebras of Jordan type and 3-transposition groups

18th December, 2023

The classification of 3-transposition groups has a long history. In particular, it is a highly non-trivial fact that finitely generated 3-transposition groups are finite. We provide an alternative viewpoint on this question using the corresponding 'Matsuo algebras', a class of non-associative algebras. These are instances of primitive axial algebras of Jordan type. We prove that primitive 4-generated axial algebras of Jordan type are at most 81-dimensional (and this bound is sharp).

Olivier Mathieu: On free Jordan Algebras

18th December, 2023

The free Jordan algebra J(m) on m generators is an elusive object. It has been determined when m = 1 (folklore) and m = 2 (Shirshov's Theorem). Some partial informations are known in the case m = 3, namely the space of Jordan polynomial with three variables which are linear on the last one. We will present two conjectures. Conjecture 1, which determines combinatorially the structure of the homogenous components of J(m) is elementary but mysterious. Then we present Conjecture 2 about Lie algebra cohomology of a class of free Lie algebras in a certain category. Conjecture 2 is natural, but not elementary. Our main result is that Conjecture 2 implies Conjecture 1. The proof, which is quite long, is based on the cyclicity of the Jordan operad. Conjecture 1 has been checked up to degree 15 for m = 2, up to degree 7 for m = 3 and up to degree 6 for m > 3. In the case m = 1, the conjecture is equivalent to Jacobi triple identity. For conjecture 2, the vanishing of the cohomology has been proved up to degree 3 using polynomial functors. In recent work with J. Germoni, we found two new special identities in degree 8 and 4 variables. These identities have been checked by computer, but the interesting point is that they were predicted by our conjecture.

Askar Dzhumadil’daev: Rota-Baxter algebras with non-zero weights

18th December, 2023

For an associative commutative algebra A with Rota-Baxter operator R : A → A with weight λ denote by AR an algebra with linear space A and multiplication a ◦ b = aR(b). Let AR− and AR+ be the algebra AR under Lie and Jordan commutators. If λ = 0, then the algebra AR = (A, ◦) is Zinbiel, AR+ is associative, and AR− is Tortkara. We find polynomial identities of algebras AR, AR− and AR+ in the case λ ≠ 0. We prove that AR− is Tortkara. AR+ satisfies an identity of degree 5. In the case λ ≠ 0, the algebra AR is not associative-admissible.

Amir Fernández Ouaridi: On the simple transposed Poisson algebras and Jordan superalgebras

4th December, 2023

We prove that a transposed Poisson algebra is simple if and only if its associated Lie bracket is simple. Consequently, any simple finite-dimensional transposed Poisson algebra over an algebraically closed field of characteristic zero is trivial. Similar results are obtained for transposed Poisson superalgebras. An example of a non-trivial simple finite-dimensional transposed Poisson algebra is constructed by studying the transposed Poisson structures on the modular Witt algebra. Furthermore, we show that the Kantor double of a transposed Poisson algebra is a Jordan superalgebra, that is, we prove that transposed Poisson algebras are Jordan brackets. Additionally, a simplicity criterion for the Kantor double of a transposed Poisson algebra is obtained.

Hader Elgendy: On Jordan quadruple systems

25th September, 2023

We present the recent results on Jordan quadruple systems. We show the Peirce decomposition for a Jordan quadruple system with respect to a quadripotent. We extend the notions of the orthogonality, primitivity, and minimality of tripotents in a Jordan triple system to that of quadripotents in a Jordan quadruple system. We show the relation between minimal and primitive quadripotents in a Jordan quadruple system. We also discuss the results on complemented subsystems of Jordan quadruple systems.

Justin McInroy: Classifying quotients of the Highwater algebra

13th February, 2023

Axial algebras are a class of non-associative algebras with a strong natural link to groups and have recently received much attention. They are generated by axes which are semisimple idempotents whose eigenvectors multiply according to a so-called fusion law. Of primary interest are the axial algebras with the Monster type (α,β) fusion law, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. By previous work of Yabe, and Franchi and Mainardis, any symmetric 2-generated axial algebra of Monster type (α,β) is either in one of several explicitly known families, or is a quotient of the infinite-dimensional Highwater algebra H, or its characteristic 5 cover Ĥ. We complete this classification by explicitly describing the infinitely many ideals and thus quotients of the Highwater algebra (and its cover). As a consequence, we find that there exist 2-generated algebras of Monster type (α,β) with any number of axes (rather than just 1,2,3,4,5,6,∞ as we knew before) and of arbitrarily large finite dimension. In this talk, we will begin with a reminder of axial algebras which were introduced last week.