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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.

Mikhail Kotchetov: Fine gradings on classical simple Lie algebras

8th July, 2021

Gradings by abelian groups have played an important role in the theory of Lie algebras since its beginning: the best known example is the root space decomposition of a semisimple complex Lie algebra, which is a grading by a free abelian group (the root lattice). Involutive automorphisms or, equivalently, gradings by the cyclic group of order 2, appear in the classification of real forms of these Lie algebras. Gradings by all cyclic groups were classified by V. Kac in the late 1960s and applied to the study of symmetric spaces and affine Kac-Moody Lie algebras.

In the past two decades there has been considerable interest in classifying gradings by arbitrary groups on algebras of different varieties including associative, Lie and Jordan. Of particular importance are the so-called fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism, although not in a unique way. If the ground field is algebraically closed and of characteristic 0, then the classification of fine abelian group gradings on an algebra (up to equivalence) is the same as the classification of maximal quasitori in the algebraic group of automorphisms (up to conjugation). Such a classification is now known for all finite-dimensional simple complex Lie algebras.

In this talk I will review the above mentioned classification and present a recent joint work with A. Elduque and A. Rodrigo-Escudero in which we classify fine gradings on classical simple real Lie algebras.

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.

Albert Schwarz: Some questions on Jordan algebras inspired by quantum theory

27th August, 2020

One can formulate quantum theory taking as a starting point a convex set (the set of states) or a convex cone (the set of non-normalized states.) Jordan algebras are closely related to homogeneous cones, therefore they appear naturally in this formulation. There exists a conjecture that superstring can be formulated in terms of exceptional Jordan algebras. In my purely mathematical talk I'll formulate some results and conjectures on Jordan algebras coming from these ideas.

Ivan Shestakov: Coordination Theorems for certain non-associative algebras

16th April, 2020

Coordinatization Theorems are very useful for classification problems. The classical Wedderburn Coordinatization Theorem claims that if a unital associative algebra A contains a matrix subalgebra Mn(F) with the same unit then A=Mn(B) for a certain subalgebra B. The Jacobson Coordinatization Theorems in the structure theories of alternative and Jordan algebras state similar results for octonions and Albert algebras. Various coordinatization theorems were proved for noncommutative Jordan algebras, for commutative power associative algebras, for alternative and Jordan superalgebras, etc. In our talk, we consider three coordinatization theorems:

1) for 2x2 matrices in the class of alternative algebras (Jacobson's problem),

2) for Jordan algebra of symmetric 2x2 matrices in the class of Jordan algebras,

3) for octonions in the class of right alternative algebras.