Tag - Jordan algebras

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.

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.

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.