### ${\mathbb{Z}}_{3}$-orthograded quasimonocomposition algebras with one-dimensional null component.

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We describe two constructions of a certain ${\mathbb{Z}}_{4}^{3}$-grading on the so-called Brown algebra (a simple structurable algebra of dimension $56$ and skew-dimension $1$) over an algebraically closed field of characteristic different from $2$. The Weyl group of this grading is computed. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types ${E}_{6}$, ${E}_{7}$ and ${E}_{8}$.

We present a short and rather self-contained introduction to the theory of finite-dimensional division algebras, setting out from the basic definitions and leading up to recent results and current directions of research. In Sections 2-3 we develop the general theory over an arbitrary ground field k, with emphasis on the trichotomy of fields imposed by the dimensions in which a division algebra exists, the groupoid structure of the level subcategories 𝒟ₙ(k), and the role played by the irreducible...

Leibniz algebras are a non-commutative version of usual Lie algebras. We introduce a notion of (pre)crossed Leibniz algebra which is a simultaneous generalization of notions of representation and two-sided ideal of a Leibniz algebra. We construct the Leibniz algebra of biderivations on crossed Leibniz algebras and we define a non-abelian tensor product of Leibniz algebras. These two notions are adjoint to each other. A (co)homological characterization of these new algebraic objects enables us to...

The weak radical, W-Rad(A) of a non-associative algebra A, has been introduced by A. Rodríguez Palacios in [3] in order to generalize the Johnson's uniqueness of norm theorem to general complete normed non-associative algebras (see also [2] for another application of this notion). In [4], he showed that if A is a semiprime non-associative algebra with DCC on ideals, then W-Rad(A) = 0. In the first part of this paper we give an example of a non-semiprime associative algebra A with DCC on ideals and...

In this paper we give a review on δ-structurable algebras. A connection between Malcev algebras and a generalization of δ-structurable algebras is also given.

We explicitly construct a particular real form of the Lie algebra ${\U0001d522}_{7}$ in terms of symplectic matrices over the octonions, thus justifying the identifications ${\U0001d522}_{7}\cong \mathrm{\U0001d530\U0001d52d}(6,\mathbb{O})$ and, at the group level, ${E}_{7}\cong \text{Sp}(6,\mathbb{O})$. Along the way, we provide a geometric description of the minimal representation of ${\U0001d522}_{7}$ in terms of rank 3 objects called cubies.