Non-associative algebraic structures: classification and structure

Abstract

These are detailed notes for a lecture on "Non-associative Algebraic Structures: Classification and Structure" which I presented as a part of my Agrega\c{c}\~ao em Matem\'atica e Applica\c{c}\~oes (University of Beira Interior, Covilh\~a, Portugal, 13-14/03/2023).

Transposed Poisson structures on the Lie algebra of upper triangular matrices

“…The algebraic and geometric classification of 3-dimensional transposed Poisson algebras was given in [2]. Also, see [8,Section 7.3], and the references therein for similar studies. For the list of actual open questions on transposed Poisson algebras, see [3].…”

The algebraic and geometric classification of nilpotent binary and mono Leibniz algebras

“…The algebraic classification (up to isomorphism) of algebras of dimension n from a certain variety defined by a certain family of polynomial identities is a classic problem in the theory of non-associative algebras. There are many results related to the algebraic classification of small-dimensional algebras in the varieties of Jordan, Lie, Leibniz, Zinbiel, and many other algebras [2,5,6,29,38] and references in [30,36]. Geometric properties of a variety of algebras defined by a family of polynomial identities have been an object of study since 1970's (see, [5,6,13,16,21,22,28,31,33,49] and references in [30]).…”

“…There are many results related to the algebraic classification of small-dimensional algebras in the varieties of Jordan, Lie, Leibniz, Zinbiel, and many other algebras [2,5,6,29,38] and references in [30,36]. Geometric properties of a variety of algebras defined by a family of polynomial identities have been an object of study since 1970's (see, [5,6,13,16,21,22,28,31,33,49] and references in [30]). Gabriel described the irreducible components of the variety of 4-dimensional unital associative algebras [16].…”