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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.

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.

Victor Petrogradsky: Growth in Lie algebras

26th August, 2022

Different versions of the Burnside Problem ask what one can say about finitely generated periodic groups under additional assumptions. For associative algebras, Kurosh type problems ask similar questions about properties of finitely generated nil (more generally, algebraic) algebras. Similarly, one considers finitely generated restricted Lie algebras with a nil p-mapping. Now we study an oscillating intermediate growth in nil restricted Lie algebras.

Namely, for any field of positive characteristic, we construct a family of 3-generated restricted Lie algebras of intermediate oscillating growth. We call them Phoenix algebras, because of the following. a) For infinitely many periods of time the algebra is 'almost dying' by having a quasi-linear growth, namely the lower Gelfand-Kirillov dimension is 1, more precisely, the growth is of type n (ln ⋯ ln n)κ (ln q times), where q ∈ ℕ, κ > 0 are constants. b) On the other hand, for infinitely many n the growth function has a rather fast intermediate behaviour of type exp(n/(ln n)λ), λ being a constant determined by characteristic, for such periods the algebra is 'resuscitating'. c) Moreover, the growth function is bounded and oscillating between these two types of behaviour. d) These restricted Lie algebras have a nil p-mapping.

We also construct nil Lie superalgebras and nil Jordan superalgebras of similar oscillating intermediary growth over arbitrary field.

Pavel Etingof: Weak Jordan algebras in characteristic 5 and tensor categories

16th June, 2022

We propose a new algebraic structure, called a weak Jordan algebra, which we define outside of characteristics 2,3. Any Jordan algebra is a weak Jordan algebra, and the converse holds in characteristics different from 5. However, in characteristic 5 there are many examples of simple weak Jordan algebras which are not Jordan, and not even power associative - they are only power associative up to degree 5 (note that by a theorem of Albert, an algebra in characteristic 0 or ≥ 7 which is power associative up to degree 5 and even 4 is power associative in all degrees). These algebras correspond (via a version of the Kantor-Koehler-Tits construction) to Lie algebras in the Fibonacci tensor category Fib in characteristic 5, which can be obtained from Lie algebras in characteristic 5 with a derivation d such that d5 = 0 by the procedure of semisimplification. This allows one to view the notion of a weak Jordan algebra as an example from a new subject that may be called 'Lie theory in tensor categories'.

Antonio Peralta: How can we apply Jordan structures to reinterpret Wigner-Uhlhorn theorem?

1st April, 2022

So far to date, much has been written about E. Wigner's and U. Uhlhron's theorems and their importance for physics and mathematics. For the sake of conciseness, let us go straight to some of the starring results. There are six mathematical models employed in quantum mechanics, among them we have: 1. The C-algebra B(H) of bounded operators; 2. The Jordan algebra B(H)sa of bounded self-adjoint operators; 3. The orthomodular lattice L of closed subspaces of H, equivalently, the lattice of all projections in B(H), where H is a complex Hilbert space.

The natural automorphisms of these mathematical models (i.e., the bijections on these sets preserving the corresponding relevant structure: associative product and involution, Jordan product, and orthogonality and order between subspaces or projections) represent the symmetry groups of quantum mechanics and are endowed with natural topologies induced by the probabilistic structure of quantum mechanics. It is known that these symmetry groups are all isomorphic when dim(H) ≥ 3. The last restriction exclude rank two, where there are no more than two orthogonal projections. This equivalence can be seen as the celebrated Wigner unitary-antiunitary theorem.

By replacing the set of projections P(H) by the wider set PI(H) = U(B(H)), of all partial isometries on H, L. Molnár proved the following result: Let be a complex Hilbert space with dim(H) ≥ 3. Suppose that Φ: U(B(H)) → U(B(H)) is a bijective transformation which preserves the natural partial ordering and the orthogonality between partial isometries in both directions. If Φ is continuous (in the operator norm) at a single element of U(B(H)) different from 0, then Φ extends to a real linear triple isomorphism.

During this talk we shall present new results, obtained in collaboration with Y. Friedman, showing that an extension of the previous results is possible in the case of a bijection between the lattices of tripotents of two Cartan factors and atomic JBW-triples non-containing rank-one Cartan factors. These new result provide new models to understand the quantum models. We shall also see how the results provide new alternatives to complement recent studies by J. Hamhalter proving that the set of partial isometries with its partial order and orthogonality relation is a complete Jordan invariant for von Neumann algebras.

Michel Racine: Lie Algebras afforded by Jordan algebras

3rd March, 2022

Given a (quadratic) Jordan algebra J over a ring k, one obtains three Lie algebras, the derivation algebra, the structure algebra, and the Tits algebra. We are particularly interested in the case where J is an Albert algebra.

Holger Petersson: Octonions and Albert algebras over commutative rings

24th February, 2022

In the first part of the lecture, I will focus on two properties of octonion algebras that are known to hold over fields but fail over arbitrary commutative rings: their enumeration by means of the Cayley-Dickson construction, and the norm equivalence theorem. In the second part, I will describe a new approach to the first Tits construction of Albert algebras that, even over fields, is more general than the classical one and sheds some new light on the classification problem for reduced Albert algebras over commutative rings.

Plamen Koshlukov: Gradings on upper triangular matrices

22nd February, 2022

The upper triangular matrix algebras are important in Linear Algebra, and represent a powerful tool in Ring Theory. They also appear in the theory of PI algebras.

In addition to the usual associative product, one can consider the Lie bracket and also the symmetric (Jordan) product on the upper triangular matrices.

We discuss the group gradings on the upper triangular matrices viewed as an associative, Lie and Jordan algebra, respectively. Valenti and Zaicev proved that the associative gradings are, in a sense, given by gradings on the matrix units. Di Vincenzo, Valenti and Koshlukov classified such gradings. Later on, Yukihide and Koshlukov, described the Lie and the Jordan gradings. In this talk we recall some of these results as well as a new development in a rather general setting, obtained by Yukihide and Koshlukov.