Solvability of universal bol 2-loops

ABSTRACT

Let (A, ^?) be a structurable algebra. Then the opposite algebra (A ^op , ^?) is structurable, and we show that the triple system B ^op _A(x, y, z):=V^opx,y(z)=x(y¯z)+z(y¯x)?y(x¯z), x, y, z ∈ A, is a Kantor triple system (or generalized Jordan triple system of the second order) satisfying the condition (A). Furthermore, if A=𝔸₁?𝔸₂ denotes tensor products of composition algebras, (^?) is the standard conjugation, and (^∧) denotes a certain pseudoconjugation on A, we show that the triple systems B ^op _𝔸1?𝔸2 ( x , y¯^∧, z) are models of compact Kantor triple systems. Moreover these triple systems are simple if (dim𝔸₁, dim𝔸₂) ≠ (2, 2). In addition, we obtain an explicit formula for the canonical trace form for compact Kantor triple systems defined on tensor products of composition algebras.     

Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients

This paper is devoted to strictly hyperbolic systems and equations with non‐smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate Gårding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of the Cauchy problem (or of a transmission problem in the piecewise regular case) exists, the generalised solution is shown to be associated with the classical solution (or the piecewise classical solution satisfying the appropriate transmission conditions).     

Split-By-Nilpotent Extensions Algebras and Stratifying Systems

Let Γ and Λ be artin algebras such that Γ is a split-by-nilpotent extension of Λ by a two sided ideal I of Γ. Consider the change of rings functors G: =_ΓΓ_Λ ?_Λ ? and F: =_ΛΛ_Γ ?_Γ ?. In this article, by assuming that I _Λ is projective, we find the necessary and sufficient conditions under which a stratifying system (Θ, ≤) in modΛ can be lifted to a stratifying system (GΘ, ≤) in mod(Γ). Furthermore, by using the functors F and G, we study the relationship between their filtered categories of modules; and some connections with their corresponding standardly stratified algebras are stated (see Theorem 5.12, Theorem 5.15 and Theorem 5.18). Finally, a sufficient condition is given for stratifying systems in mod(Γ) in such a way that they can be restricted, through the functor F, to stratifying systems in mod(Λ).     

Compact Exceptional Simple Kantor Triple Systems Defined on Tensor Products of Composition Algebras∗

In this article we give the classification of compact exceptional simple Kantor triple systems defined on tensor products of composition algebras A =  ₁?  ₂ such that their Kantor algebras ?(φ, A) are real forms of exceptional simple Lie algebras.     

Direct Unions of Lie Tori (Realization of Locally Extended Affine Lie Algebras)

In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by “direct unions.” Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A₁, B, C, and BC.     

Clifford analysis in two and several vector variables

In this paper we present some basic function theoretic results and ideas concerning nullsolutions of several Dirac type operators. In particular we consider plane wave solutions, axial symmetry, zonal functions and irreducible representations of the group SO (m).     

Nichols Algebras of Unidentified Diagonal Type

Abstract

The groups having exactly one normalizer are well-known. They are the Dedekind groups. All finite groups having exactly two normalizers were classified by M. D. Pérez-Ramos and, in a recent paper, S. Camp-Mora generalized that result to locally finite groups. In this paper, we will characterize arbitrary groups with a finite number of normalizers and we will investigate the properties of arbitrary groups with two, three and four normalizers.     

Étude des ω-PI Algèbres Commutatives de Degré 4: I. Algèbres Non Barycentriques Non Invariantes par Gamétisation

We define and study the properties of baric algebras defined by ω-polynomial identities, called ω-PI algebras. We show that every finite dimensional baric algebra is ω-PI. Next we introduce the study of ω-PI algebras of degree 4 with one indeterminate. By gametization we reduce their study to four types. We study the first type corresponding to algebras that are neither barycentric nor invariant by gametization. We show that the variety of these algebras is partitioned into only two subvarieties admiting an unique ponderation, an idempotent, and verifying ω-monomial identities of degrees > 4.     

Three-dimensional zeropotent algebras over an algebraically closed field of characteristic two

Abstract

We previously classified three-dimensional zeropotent algebras over an algebraically closed field of any characteristic except for two. The exceptional case of characteristic two is special because some of the previous transformation matrices to verify isomorphism are unavailable. In this paper, we give new transformation matrices peculiar to characteristic two and then achieve classification in the exceptional case. We thus accomplish a classification of three-dimensional zeropotent algebras over an algebraically closed field of any characteristic.     

Free Structures in Division Rings

Makar–Limanov's conjecture states that, if a division ring D is finitely generated and infinite dimensional over its center k, then D contains a free k-subalgebra of rank 2. In this work, we will investigate the existence of such structures in D, the division ring of fractions of the skew polynomial ring L[t; σ], where t is a variable and σ is a k-automorphism of L. For instance, we prove Makar-Limanov's conjecture when either L is the function field of an abelian variety or the function field of the n-dimensional projective space.     

Hom-Lie quadratic and Pinczon Algebras

Presenting the structure equation of a hom-Lie algebra 𝔤, as the vanishing of the self commutator of a coderivation of some associative comultiplication, we define up to homotopy hom-Lie algebras, which yields the general hom-Lie algebra cohomology with value in a module. If the hom-Lie algebra is quadratic, using the Pinczon bracket on skew symmetric multilinear forms on 𝔤, we express this theory in the space of forms. If the hom-Lie algebra is symmetric, it is possible to associate to each module a quadratic hom-Lie algebra and describe the cohomology with value in the module.     

Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for infinite matrices

In this paper we extend the work of Kawamura, see [K. Kawamura, The Perron-Frobenius operators, invariant measures and representations of the Cuntz-Krieger algebras, J. Math. Phys. 46 (2005)], for Cuntz-Krieger algebras OA for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of OA. We use these representations to describe the Perron-Frobenius operator, associated to a nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples.     

Domestic canonical algebras and simple Lie algebras

For each simply-laced Dynkin graph Δ we realize the simple complex Lie algebra of type Δ as a quotient algebra of the complex degenerate composition Lie algebra of a domestic canonical algebra A of type Δ by some ideal I of that is defined via the Hall algebra of A, and give an explicit form of I. Moreover, we show that each root space of has a basis given by the coset of an indecomposable A-module M with root easily computed by the dimension vector of M. Dedicated to Professor Claus Michael Ringel on the occasion of his 60th birthday.     

On Fomin–Kirillov algebras for complex reflection groups

In this note, we apply classification results for finite-dimensional Nichols algebras to generalizations of Fomin–Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger–Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of semisimple group type can be used to find that these algebras are infinite-dimensional for many non-exceptional complex reflection groups in the Shephard–Todd classification.     

On a Remarkable Class of Left-symmetric Algebras and its Relationship with the Class of Novikov Algebras

We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G, G] is acting by translations. In other words, we consider left-symmetric algebras satisfying the identity [x, y]·z = 0. We derive some basic characterizations of such left-symmetric algebras, and we highlight their relationships with the so-called Novikov algebras and derivation algebras.     

Row and Column Removal Theorems for Homomorphisms of Specht Modules and Weyl Modules

We prove a q-analogue of the row and column removal theorems for homomorphisms between Specht modules proved by Fayers and the first author [16]. These results can be considered as complements to James and Donkin’s row and column removal theorems for decomposition numbers of the symmetric and general linear groups. In this paper we consider homomorphisms between the Specht modules of the Hecke algebras of type A and between the Weyl modules of the q-Schur algebra.This research was supported by ARC grant DP0343023. The first author was also supported by a Sesqui Research Fellowship at the University of Sydney.     

A Construction Method for Albert Algebras over Algebraic Varieties

We construct cubic Jordan algebras over an integral proper scheme X such that 2, 3 ∈ H ⁰(X, 𝒪 _X ), generalizing a construction by B. N. Allison and J. R. Faulkner. In the process, we obtain admissible cubic algebras and pseudocomposition algebras over X. Results on the structure of these algebras are obtained, as well as examples over elliptic curves.