arXiv subjects
Publications and source records attributed to Javier Rández-Ibáñez.
Composition algebras in symmetric tensor categories are defined. The classical results on standard unital composition algebras, as well as the classification of the unital composition superalgebras, are reviewed in light of this definition, and the unital composition algebras in the category $Ver_4^+$ (the counterpart to the category of vector superspaces in characteristic 2) are classified.
The double extension method allows the construction of quadratic Lie algebras from smaller ones. Using this approach, in this work we describe all solvable indecomposable quadratic Lie algebras of dimension at most $7$ over arbitrary fields of characteristic not $2$. In particular, we get complete classifications up to isometric isomorphism over algebraically and real closed fields.
A new class of nonassociative algebras, Vidinli algebras, is defined based on recent work of Coşkun and Eden. These algebras are conic (or quadratic) algebras with the extra restriction that the commutator of any two elements is a scalar multiple of the unity. Over fields of characteristic not 2, Vidinli algebras may be considered as generalizations of the Jordan algebras of Clifford type. However, in characteristic 2, the class of Vidinli algebras is much larger and include the unitizations of anticommutative algebras.
Every non-solvable and non-semisimple quadratic Lie algebra can be obtained as a double extension of a solvable quadratic Lie algebra. Thanks to a partial classification of nilpotent Lie algebras and this result, we can design different techniques to obtain any quadratic Lie algebra whose (solvable) radical ideal is nilpotent. To achieve this, we propose two alternative methods, both involving the use of quotients. In addition to their mathematical description, both approaches introduced in this paper have been computationally implemented and are publicly available to use for generating these algebras.
The procedure of double extension of vector spaces endowed with non-degenerate bilinear forms allows us to introduce the class of generalized $\mbK$-oscillator algebras over any arbitrary field $\mbK$. Starting from basic structural properties of such algebras and the canonical forms of skew-adjoint endomorphisms, we will proceed to classify the subclass of quadratic nilpotent algebras and characterize those algebras in the class with quadratic dimension 2. This will enable us to recover the well-known classification of real oscillator algebras, also known as Lorentzian algebras, given by Alberto Medina in 1985.