Non-associative rings – Page 11

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'.

David Jordan: Skew derivations of quantum spaces

3rd June, 2022

Let n be a positive integer and let Q = (q_ij) be a multipicatively antisymmetric n × n matrix over a field 𝕂, that is q_ii = 1 for 1 ≤ i ≤ n and, for 1 ≤ i,j ≤ n, q_ij ≠ 0 and q_ji=q_ij^-1. The quantized (co-ordinate ring of) quantum n-space R=𝒪_Q(𝕂ⁿ) is the 𝕂-algebra generated by x₁,x₂, . . .,x_n subject to the relations x_ix_j = q_ijx_jx_i for 1 ≤ i < j ≤ n.

Although the space of derivations Der(R) is well understood through work of Alev and Chamarie in 1982, less is known about the space Der_σ(R) of σ-derivations of R. The only case in the literature where the σ-derivations of R are determined appears to be when n = 2 and, for some λ ∈ 𝕂*, σ(x₁)=λx₁ and σ(x_2)=λ^-1 x₂. This case appears in a 2018 paper by Almulhem and Brzeziński that was motivated by differential geometry. This talk will discuss the classification of the σ-derivations of R for all n when σ is toric, that is each x_i is an eigenvector for σ, with a view to applications to iterated Ore extensions of 𝕂. Any such classification must include the inner σ-derivations of R, that is those for which there exists a ∈ R such that δ_a(r)=ar-σ(r)a for all r ∈ R.

The methods are based on two of the classical methods of non-commutative algebra, namely localization and grading, in this case by ℤⁿ. Localization at the set {x₁^d₁x₂^d₂ . . . x_n^d_n} yields the quantum n-torus T=𝒪_Q((𝕂*)ⁿ) to which σ and all σ-derivations extend. A σ-derivation δ of T is homogeneous, of weight (d₁,d₂, . . ., d_n), if δ(x_i) ∈ 𝕂x₁^d₁x₂^d₂ . . .,x_i^d_i+1. . . x_n^d_n for 1 ≤ i ≤ n and every σ-derivation of T is a unique linear combination of homogeneous σ-derivations. It turns out that if δ is a homogeneous σ-derivation of T then either the automorphism σ is inner or the σ-derivation δ is inner and the Ore extension T[x ; σ,δ] can, by a change of variables, be expressed as an Ore extension of either automorphism type or derivation type. This dichotomy influences the space Der_σ(R) which can be identified with { δ ∈ Der_σ(T) : δ(R) ⊆ R }. The most obvious σ-derivations included here are the homogeneous σ-derivations of weight (d₁,d₂, . . ., d_n) where each d_i ≥ 0, but more interesting are those for which one d_i = -1. There are two types of these, depending on whether σ or δ is inner on T. In the latter case we are in a common situation where a σ-derivation of a ring R is not inner on R but becomes inner on the localization of R at the powers of a normal element of R, giving rise to a distinguished normal or central element of the Ore extension R[x ; σ,δ].

Efim Zelmanov: Automorphism groups and Lie algebras of vector fields on affine varieties

26th May, 2022

Let V be an affine algebraic variety over a commutative ring K and let A be the K-algebra of regular (polynomial) functions on V.

The group of automorphisms of V, namely Aut(A), is, generally speaking, not linear. We will discuss the following two questions: which properties of linear groups extend to Aut(A), and which properties of finite-dimensional Lie algebras extend to the Lie algebra Der(A) of vector fields on V?

In particular, we will focus on analogues of classical theorems of Selberg, Burnside, and Schur for Aut(A) and an analogue of the Engel theorem for Der(A). In order to achive natural degree of generality and to include some interesting non-commutative cases we prove the theorems for PI-algebras.

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.

Jef Laga: Arithmetic statistics and graded Lie algebras

17th March, 2022

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier work of Thorne. This gives a uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves.

Alexander Stolin: 40 years of Lie bialgebras: From definition to classification

17th March, 2022

The history of Lie bialgebras began with the paper where the Lie bialgebras were defined: V. G. Drinfeld, "Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations".

The aim of my talk is to celebrate 40 years of Lie bialgebras in mathematics and to explain how these important algebraic structures can be classified. This classification goes "hand in hand" with the classification of the so-called Manin triples and Drinfeld doubles also introduced in Drinfeld's paper cited above.

The ingenious idea how to classify Drinfeld doubles associated with Lie algebras possessing a root system is due to F. Montaner and E. Zelmanov. In particular, using their approach the speaker classified Lie bialgeras, Manin triples and Drinfeld doubles associated with a simple finite-dimensional Lie algebra 𝔤 (the paper was based on a private communication by E. Zelmanov and it was published in Comm. Alg. in 1999).

Further, in 2010, F. Montaner, E. Zelmanov and the speaker published a paper in Selecta Math., where they classified Drinfeld doubles on the Lie algebra of the formal Taylor power series 𝔤[[u]] and all Lie bialgebra structures on the polynomial Lie algebra 𝔤[u].

Finally, in March 2022 S. Maximov, E. Zelmanov and the speaker published an arXiv preprint, where they made a crucial progress towards a complete classification of Manin triples and Lie bialgebra structures on 𝔤[[u]].

Of course, it is impossible to compress a 40 years history of the subject in one talk but the speaker will try his best to do this.

David Galban: Cohomology and Representation Theory for Lie Superalgebras

14th March, 2022

This talk will consist of two parts. In the first, I will describe the cohomology groups for the subalgebra 𝔫⁺ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu, essentially with these calculations essentially providing the first steps towards an analogue of Kostant’s theorem for Lie superalgebras. In the second part, based on joint work with Nakano, I will analyze the sheaf cohomology groups R^I ind_B^G L_𝔣(λ), where L_𝔣(λ) is an irreducible representation for the detecting subalgebra 𝔣, providing analogues for the BBW theorem and Kempf’s vanishing theorem for sufficiently large λ.

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.

Robert Spencer: (Some) Gram Determinants for A_n nets

1st March, 2022

The nets giving a diagrammatic description of the category of (tensor products of) fundamental representations of 𝔰𝔩_n form a cellular category. We can then ask about the natural inner form on certain cell modules. In this talk, we will calculate the determinant of some of these forms in terms of certain traces of clasps or magic weave elements (for which there is a conjectured formula due to Elias). The method appears moderately general and gives a result which is hopefully illuminating and applicable to other monoidal, cellular categories.

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.

Tag - Non-associative rings