Free Nonassociative Algebra Research Articles

Primitive elements of free non-associative algebras over finite fields

The representation of elements of free non-associative algebras as a set of multidimensional tables of coefficients is defined. An operation for finding partial derivatives for elements of free non-associative algebras in the same form is considered. Using this representation, a criterion of primitivity for elements of lengths 2 and 3 in terms of matrix ranks, as well as a primitivity test for elements of arbitrary length, is derived. This test makes it possible to estimate the number of primitive elements in free non-associative algebras with two generators over a finite field. The proposed representation allows us to optimize algorithms for symbolic computations with primitive elements. Using these algorithms, we find the number of primitive elements of length 4 in a free non-associative algebra of rank 2 over a finite field.

Primitive Elements of Free Non-associative Algebras over Finite Fields

Primitive Elements of Free Non-associative Algebras over Finite Fields

The Lie algebra of classical mechanics

Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket. This Lie algebra, which is usually infinite dimensional, is useful in analyzing the system, as well as in geometric numerical integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy, regardless of the dimension of the system. Therefore, we study the universal object in this setting, the 'Lie algebra of classical mechanics' modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. We show that it is the direct sum of an abelian algebra \begin{document}$ \mathfrak{X} $\end{document} , spanned by 'modified' potential energies and isomorphic to the free commutative nonassociative algebra with one generator, and an algebra freely generated by the kinetic energy and its Poisson bracket with \begin{document}$ \mathfrak{X} $\end{document} . We calculate the dimensions \begin{document}$ c_n $\end{document} of its homogeneous subspaces and determine the value of its entropy \begin{document}$ \lim_{n\to\infty} c_n^{1/n} $\end{document} . It is \begin{document}$ 1.8249\dots $\end{document} , a fundamental constant associated to classical mechanics. We conjecture that the class of systems with Euclidean kinetic energy metrics is already free, i.e., that the only linear identities satisfied by the Lie brackets of all such systems are those satisfied by the Lie algebra of classical mechanics.

Free braided nonassociative Hopf algebras and Sabinin τ-algebras

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.

Almost Primitive Elements of Free lie Algebras of Small Ranks

Let K be a field, X = {x1, . . . , xn}, and let L(X) be the free Lie algebra over K with the set X of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free, A. I. Shirshov proved that subalgebras of free Lie algebras are free. A subset M of nonzero elements of the free Lie algebra L(X) is said to be primitive if there is a set Y of free generators of L(X), L(X) = L(Y ), such that M ⊆ Y (in this case we have |Y | = |X| = n). Matrix criteria for a subset of elements of free Lie algebras to be primitive and algorithms to construct complements of primitive subsets of elements with respect to sets of free generators have been constructed. A nonzero element u of the free Lie algebra L(X) is said to be almost primitive if u is not a primitive element of the algebra L(X), but u is a primitive element of any proper subalgebra of L(X) that contains it. A series of almost primitive elements of free Lie algebras has been constructed. In this paper, for free Lie algebras of rank 2 criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed.

Homogeneous almost primitive elements of free non-associative (anti-) commutative algebras

Criteria for homogeneous elements to be almost primitive are obtained in the paper for free non-associative commutative and anti-commutative algebras of any rank.

Polynomial Identities for Tangent Algebras of Monoassociative Loops

We introduce degree n Sabinin algebras, which are defined by the polynomial identities up to degree n in a Sabinin algebra. Degree 4 Sabinin algebras can be characterized by the polynomial identities satisfied by the commutator, associator, and two quaternators in the free nonassociative algebra. We consider these operations in a free power associative algebra and show that one of the quaternators is redundant. The resulting algebras provide the natural structure on the tangent space at the identity element of an analytic loop for which all local loops satisfy monoassociativity, a 2 a ≡ aa 2. These algebras are the next step beyond Lie, Malcev, and Bol algebras. We also present an identity of degree 5 which is satisfied by these three operations but which is not implied by the identities of lower degree.

An envelope for left alternative algebras

Let A be the free non-associative algebra and let T be the T -ideal generated by the identity (x, y, z)+( y, x, z). Given an ideal J ⊇ T , then CJ = A/J is a left alternative algebra. We construct an ideal ΓJ and we define a universal enveloping algebra of CJ as A/ΓJ . We introduce a linear map ω : CJ → A/ΓJ , such that ω((a, b, c)) = (a, b, c)−(b, a, c). As a conjecture we state that ω is injective. The injection of CJ into A/ΓJ is similar to the injection of a Lie algebra into an associative algebra by [a, b ]= ab − ba; moreover we show how to construct a spanning set of A/ΓJ from a basis of CJ and we define a universal property analogously like in the case of Lie algebras.

Almost primitive elements of free nonassociative (anty)commutative algebras of small rank

Criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed for free nonassociative commutative and anticommutative algebras of rank 1 and 2.

Almost primitive elements of free nonassociative algebras of small ranks

Let K be a field, X = {x 1 ,…, x n }, and let F(X) be the free nonassociative algebra over the field K with the set X of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free. A subset M of nonzero elements of the algebra F(X) is said to be primitive if there is a set Y of free generators of F(X), F(X) = F(Y ), such that M ⊆ Y (in this case we have |Y| = |X| = n). A nonzero element u of the free algebra F(X) is said to be almost primitive if u is not a primitive element of the algebra F(X), but u is a primitive element of any proper subalgebra of F(X) that contains it. In this article, for free nonassociative algebras of rank 1 and 2 criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed. New examples of almost primitive elements of free nonassociative algebras of rank 3 are constructed.

Finite-dimensional non-associative algebras and codimension growth

Let A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded. Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One of the main tools of independent interest is the construction in the free non-associative algebra of multialternating polynomials satisfying special properties.

On Some Ideals in the Free Nonassociative Algebra

The free Lie, right, and left Leibniz algebras are obtained as quotients of the free nonassociative algebra by suitable ideals. In this article, we prove some remarkable properties of these ideals.

Primitive elements of free nonassociative algebras

Improved algorithms to construct complements of primitive systems of elements of free nonassociative algebras with respect to free generating sets and algorithms to realize the rank of a system of elements are constructed and implemented.

On some properties of Hall elements in the free nonassociative algebra

We study certains aspects of a particular Hall set constructed with respect to the alpabetical order. In our main result we show how this Hall set leads to the construction of a family of generators of the kernel of “from right to left Lie bracketing” mapping. This construction is based on certain remarkable properties of these generators.

Free nonassociative supercommutative algebras

In the present work, we prove that homogeneous subalgebras of free nonassociative supercommutative algebras are free. As a consequence, we show that the group of automorphisms of a free nonassociative supercommutative algebra of finite rank is generated by elementary automorphisms.

Dimension Formulas for the Free Nonassociative Algebra

ABSTRACT The free nonassociative algebra has two subspaces which are 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. Using a theorem of Shestakov, we give a recursive formula for the dimension of the Akivis elements. Using a theorem of Shestakov and Umirbaev, we prove a closed formula for the dimension of the primitive elements. These results generalize the Witt dimension formula for the Lie elements in the free associative algebra.

Totally multiplicatively prime algebras

We introduce the totally multiplicatively prime algebras as those normed algebras for which there exists a positive number K such that K‖F‖‖a‖ ≤ ‖WF,a‖ for all F in M(A) (the multiplication algebra of A) and a in A, where WF,a denotes the operator from M(A) into A defined by WF,a(T) = FT(a) for all T in M(A). These algebras are totally prime and their multiplication algebra is ultraprime. We get the stability of the class of totally multiplicatively prime algebras by taking central closure. We prove that prime H*-algebras are totally multiplicatively prime and that the ℓ1-norm is the only classical norm on the free non-associative algebras for which these are totally multiplicatively prime.

Totally multiplicatively prime algebras

We introduce the totally multiplicatively prime algebras as those normed algebras for which there exists a positive number K such that K‖F‖‖a‖ ≤ ‖WF,a‖ for all F in M(A) (the multiplication algebra of A) and a in A, where WF,a denotes the operator from M(A) into A defined by WF,a(T) = FT(a) for all T in M(A). These algebras are totally prime and their multiplication algebra is ultraprime. We get the stability of the class of totally multiplicatively prime algebras by taking central closure. We prove that prime H*-algebras are totally multiplicatively prime and that the ℓ1-norm is the only classical norm on the free non-associative algebras for which these are totally multiplicatively prime.

Free Akivis Algebras, Primitive Elements, and Hyperalgebras

Free Akivis algebras and primitive elements in their universal enveloping algebras are investigated. It is proved that subalgebras of free Akivis algebras are free and that finitely generated subalgebras are finitely residual. Decidability of the word problem for the variety of Akivis algebras is also proved.The conjecture of K. H. Hofmann and K. Strambach (Problem 6.15 in [Topological and analytic loops, in “Quasigroups and Loops Theory and Applications,” Series in Pure Mathematics (O. Chein, H. O. Pflugfelder, and J. D. H. Smith, Eds.), Vol. 8, pp. 205–262, Heldermann Verlag, Berlin, 1990]) on the structure of primitive elements is proved to be not valid, and a full system of primitive elements in free nonassociative algebra is constructed.Finally, it is proved that every algebra B can be considered as a hyperalgebra, that is, a system with a series of multilinear operations that plays a role of a tangent algebra for a local analytic loop, where the hyperalgebra operations on B are interpreted by certain primitive elements.

Automorphic Orbits in Free Non-associative Algebras

We consider finitely generated free non-associative algebras, free commutative non-associative algebras, and free anti-commutative non-associative algebras. We study orbits of elements of these algebras under the action of automorphism groups. Using free differential calculus we obtain matrix criteria for a system of elements to have given rank (or to be primitive). It gives us a possibility to construct fast algorithm to recognize primitive systems of elements. We show that if an endomorphism of a free algebra preserves the automorphic orbit of a nonzero element, then it is an automorphism of this algebra. In particular, endomorphisms preserving primitivity of elements are automorphisms.