Enveloping algebras of Malcev algebras

We first discuss the construction by Perez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the 8-dimensional division algebra of real octonions. We conclude with some brief remarks on tangent algebras of analytic Bol loops and monoassociative loops. This is a revised and expanded version of the survey talk given by the first author at the Second Mile High Conference on Nonassociative Mathematics at the University of Denver (Denver, Colorado, USA, June 22 to 26, 2009).Comment: 15 page

Bernstein algebras given by symmetric bilinear forms

AbstractLet (A, ω) be a finite dimensional Bernstein algebra and N the kernel of ω. We study the algebras where dim N2 is 1. The algebras fall into two general classes. For the first of these classes we give the multiplication tables for the complete set of nonisomorphic algebras. For the second of these classes we give the multiplication tables for what we call “complete algebras.” We show that any algebra of the second class can be embedded in a complete algebra. The multiplication in the complete algebras is easy to describe. The Bernstein algebras of the second class are then characterized as subalgebras of the complete algebras. For Jordan Bernstein algebras satisfying dim N2 = 1 we give the complete classification

The Nucleus of the Free Alternative Algebra

We use a computer procedure to determine a basis of the elements of degree 5 in the nucleus of the free alternative algebra. In order to save computer memory, we do our calculations over the field Z103. All calculations are made with multilinear identities. Our procedure is also valid for other characteristics and for determining nuclear elements of higher degree

Counterexamples in Nonassociative Algebra

We present a method of constructing counterexamples in nonassociative algebra. The heart of the computation is constructing a matrix of identities and reducing this matrix (usually very sparse) to row canonical form. The example is constructed from the entries in one column of this row canonical form. While this procedure is not polynomial in the degree of the identity, several shortcuts are listed which shorten calculations. Several examples are given

Minimal Identities of Bernstein Alegebras

We construct the minimal identities for Bernstein algebras, exceptional Bernstein algebras and normal Bernstein algebras. We use the technique of processing identities via the representation of the symmetric groups. The computer algorithms for creating the standard tableaus and the integral representations are summarized

Dimension formulas for the free nonassociative algebra

The free nonassociative algebra contains two subspaces closed under both the commutator and the associator: the Akivis elements and the primitive elements. Every Akivis element is primitive, but there are primitive elements which are not Akivis

O Ressarcimento do dano em acordos de leniência da lei anticorrupção

O estudo analisa controvérsias existentes em relação ao ressarcimento do dano em acordos de leniência regidos pela Lei n.º 12.846, de 2013, como a natureza jurídica dessa obrigação, a possibilidade de quitação, a metodologia de cálculo e a cobrança do ressarcimento em patamar adicional ao que convencionado por quem não participou do acordo. Este trabalho concluiu, ao final, que a obrigação de ressarcimento do dano exige aprimoramentos em relação à metodologia de apuração do dano, que atualmente não está definida normativamente e não é amplamente conhecida, haja vista a ausência de transparência dos acordos de leniência nesse aspecto. A partir do aprimoramento da metodologia, as outras controvérsias antes mencionadas poderão ser mitigadas, tudo a contribuir para a segurança jurídica dos acordos de leniência e o combate à corrupção.This paper examines controversies regarding the reimbursement of damages in leniency agreements under Anti-Corruption Act, such as its legal nature, its discharge, the methodologies that quantify the damages and the additional reimbursement requested by whom did not take part in the agreement. This paper concluded that the reimbursement of damages requires improvements concerning methodologies that quantify the damages, which is not set forth in statutory law and is not broad known since the leniency agreements are sealed as to this information. Once the methodologies that quantify the damages are improved, other above-mentioned controversies may be mitigated, which might contribute to legal certainty as well to the fight against corruption