Tag - Axial algebras

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).

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.

Clara Franchi: Axial algebras of Monster type

6th February, 2023

Extending earlier work by Ivanov on Majorana algebras, axial algebras of Monster type were introduced in 2015 by Hall, Rehren and Shpectorov in order to axiomatize some key features of certain classes of algebras related to large families of finite simple groups, such as the weight-2 components of OZ-type vertex operator algebras, Jordan algebras, and Matsuo algebras. In this talk, I'll review the definition of axial algebras and the major examples. Then I'll discuss the general classification problem of the 2-generated objects and, time permitting, show its applications in some special cases related to the Monster.

Louis Rowen: Finitely generated axial algebras

14th October, 2021

This lecture is a continuation of the general talk given at the Drensky conference last month, on axial algebras, which are (not necessarily commutative, not necessarily associative) algebras generated by semisimple idempotents. After a review of the definitions, we investigate the key question, being, "Under what conditions must an axial algebra be finite-dimensional?" Krupnik showed that 3 idempotents can generate arbitrarily large dimensional associative algebras (and thus infinite-dimensional algebras via an ultraproduct argument), so some restriction is needed. We consider 'primitive' axes, in which the left and right eigenspaces having eigenvalue 1 are one-dimensional.

Hall, Rehren, Shpectorov solves obtained a positive answer for commutative axial algebras of 'Jordan type' λ ≠ 1/2, although the proof relies on the classification of simple groups and the given bound of the dimension is rather high. Gorshkov and Staroletov provided a sharp bound for 3-generated commutative axial algebras of 'Jordan type'. Our objective in this project is give a non-commutative version and indicate how to investigate 4-generated commutative axial algebras of 'Jordan type', in terms of the regular representation.

Our method is to build an associative algebra from the adjoint algebra of A, which has a strictly larger dimension which nevertheless also is finite-dimensional.

Sergey Shpectorov: 2-generated algebras of Monster type

29th April, 2021

The class of non-associative axial algebras was introduced in 2015 as a broad generalization of Majorana algebras of Ivanov that were modelled after the properties of the Griess algebra, the algebra whose automorphism group is the Monster sporadic simple group. Sakuma's theorem classifies 2-generated Majorana algebras, which in axial terms correspond to algebras of Monster type (1/4,1/32). The quest to classify all 2-generated algebras of arbitrary Monster type (α,β) was started by Rehren who proved an upper bound on the dimension and generalised the Norton-Sakuma algebras to arbitrary (α,β). Recently, new results emerged from the work of Franchi, Mainardis and the speaker, and independently, of Yabe, who classified symmetric 2-generated algebras of Monster type. Several new classes of algebras have been found.