In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold L, there exists a global trivialization of the tangent bundle, ...which defines a map ρp:l⟶TpL for each point p∈L, where l is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of l. Furthermore, flows of these vector fields give rise to a product between elements of l and L, which in turn induces a local loop structure (i.e. a non-associative analog of a group). Furthermore, we also define a generalization of a Lie algebra structure on l. We will describe the properties and examples of these constructions.
An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and ...isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.
We study realizable values of the length function for unital possibly nonassociative algebras of a given dimension. To do this we apply the method of characteristic sequences and establish sufficient ...conditions of realizability for a given value of length. The proposed conditions are based on binary decompositions of the value and algebraic constructions that allow to modify length function of an algebra. Additionally we provide a description of unital algebras of maximal possible length in terms of their bases.
An axial algebra A is a commutative non-associative algebra generated by primitive idempotents, called axes, whose adjoint action on A is semisimple and multiplication of eigenvectors is controlled ...by a certain fusion law. Different fusion laws define different classes of axial algebras.
Axial algebras are inherently related to groups. Namely, when the fusion law is graded by an abelian group T, every axis a leads to a subgroup of automorphisms Ta of A. The group generated by all Ta is called the Miyamoto group of the algebra. We describe a new algorithm for constructing axial algebras with a given Miyamoto group. A key feature of the algorithm is the expansion step, which allows us to overcome the 2-closedness restriction of Seress's algorithm computing Majorana algebras.
At the end we provide a list of examples for the Monster fusion law, computed using a magma implementation of our algorithm.
We study polynomial identities of algebras with involution of nonassociative algebras over a field of characteristic zero. We prove that the growth of the sequence of ⁎-codimensions of a ...finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional ⁎-PI-exponent. We also construct a family of infinite-dimensional algebras Cα such that exp⁎(Cα) does not exist.
In this article, we prove the nilpotency of commutative nonassociative finitely generated algebras satisfying an identity of type
with α + β ≠ 0. Our result requires characteristic ≠ 2, 3, 5.
We investigate extensions of Malcev algebras and give an explicit example of extended algebras. We present a new algebraic identity, which can be regarded as a generalization of the Jacobi identity ...or the Malcev identity. As applications to gravity, we demonstrate that the extended algebra can be linked with general relativity.
Extensions of Malcev algebras are investigated and an explicit example of extended algebras is given. A new algebraic identity is presented, which can be regarded as a generalization of the Jacobi identity or the Malcev identity. As applications to gravity, the author demonstrates that the extended algebra can be linked with general relativity.
Let R and S be nonassociative unital algebras. Assuming that either one of them is finite dimensional or both are finitely generated, we show that every derivation of R⊗S is the sum of derivations of ...the following three types: (a) adu where u belongs to the nucleus of R⊗S, (b) Lz⊗f where f is a derivation of S and z lies in the center of R, and (c) g⊗Lw where g is a derivation of R and w lies in the center of S.
Let V be a linear space over a field k with a braiding τ:V⊗V→V⊗V. We prove that the braiding τ has a unique extension on the free nonassociative algebra k{V} freely generated by V so that k{V} is a ...braided algebra. Moreover, we prove that the free braided algebra k{V} has a natural structure of a braided nonassociative Hopf algebra such that every element of the space of generators V is primitive. In the case of involutive braidings, τ2=id, we describe braided analogues of Shestakov–Umirbaev operations and prove that these operations are primitive operations. We introduce a braided version of Sabinin algebras and prove that the set of all primitive elements of a nonassociative τ-algebra is a Sabinin τ-algebra.
Enumerating semifields Aman, Kelly
Electronic notes in discrete mathematics,
05/2013, Volume:
40
Journal Article
In 1960, Kleinfeld published representatives for all of the isomorphism classes of 16 element semifields, E. Kleinfeld, Techniques for Enumerating Veblen-Wedderburn Systems, J. ACM 7 (1960) 330–337. ...It is not entirely clear how Kleinfeld generated some of his results, but it is likely that it was similar to the approach that Walker used in 1962 to generate representative for the isotopism classes of 32 element semifields, R. J. Walker, Determination of Division Algebras With 32 Elements, Proc. Sympos. Appl. Math. XV (1963) 8385. This paper introduces an alternative notation for publication which is both simple and practical, and which leads to an alternative method which was used to verify Kleinfeldʼs results.
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