Nonassociative Algebras Research Articles

Classification of five-dimensional nilpotent Jordan algebras

The paper is devoted to classify nilpotent Jordan algebras of dimension up to five over an algebraically closed field F of characteristic not 2. We obtained a list of 35 isolated non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras and 6 families of non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras depending either on one or two parameters over an algebraically closed field of characteristic ≠2,3. In addition to these algebras we obtained two non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras over an algebraically closed field of characteristic 3.

Bott type periodicity for the higher octonions

We study the series of complex nonassociative algebras On and real nonassociative algebras $O_{p,q}$ introduced in [10]. These algebras generalize the classical algebras of octonions and Clifford algebras. The algebras $O_{n}$ and $O_{p,q}$ with $p + q = n$ have a natural $Z_n^2$-grading, and they are characterized by cubic forms over the field $Z_2$. We establish a periodicity for the algebras $O_{n}$ and $O_{p,q}$ similar to that of the Clifford algebras Cln and $Cl_{p,q}$.

Linear idempotents in Matsuo algebras

Matsuo algebras are an algebraic incarnation of 3-transposition groups with a parameter $\alpha$, where idempotents takes the role of the transpositions. We show that a large class of idempotents in Matsuo algebras satisfy the Seress property, making these nonassociative algebras well-behaved analogously to associative algebras, Jordan algebras and vertex (operator) algebras. We calculate eigenvalues in the Matsuo algebra of ${\rm Sym}(n)$ for any $\alpha$, generalising some vertex algebra results for which $\alpha=\frac{1}{4}$. Finally, in the Matsuo algebra of the root system ${\rm D}_n$, we show $n-3$ conjugacy classes of involutions coming from the Weyl group are in natural bijection with idempotents in the algebra via their fusion rules.

Tensor products of nonassociative cyclic algebras

We study the tensor product of an associative and a nonassociative cyclic algebra. The condition for the tensor product to be a division algebra equals the classical one for the tensor product of two associative cyclic algebras by Albert or Jacobson, if the base field contains a suitable root of unity. Stronger conditions are obtained in special cases. Applications to space–time block coding are discussed.

Functorial constructions for non-associative algebras with applications to quasi-bialgebras

The aim of this paper is to establish a contravariant adjunction between the category of quasi-bialgebras and a suitable full subcategory of dual quasi-bialgebras, adapting the notion of finite dual to this framework. Various functorial constructions involving non-associative algebras and non-coassociative coalgebras are then carried out. Several examples illustrating our methods are expounded as well.

Polynomial identities and algebraic combinatorics on words

This survey is devoted to the construction of several counterexamples that can be given in combinatorial PI-theory when dealing with growth of varieties of non-associative algebras. These are examples of non-associative algebras over a field of characteristic zero constructed from Sturmian or periodic words.

Testing Nonassociative Quantum Mechanics.

The familiar concepts of state vectors and operators in quantum mechanics rely on associative products of observables. However, these notions do not apply to some exotic systems such as magnetic monopoles, which have long been known to lead to nonassociative algebras. Their quantum physics has remained obscure. This Letter presents the first derivation of potentially testable physical results in nonassociative quantum mechanics, based on effective potentials. They imply new effects which cannot be mimicked in usual quantum mechanics with standard magnetic fields.

Combinatorial Algebra and Its Antisymmetrized Algebra I

The simple non-associative algebra N(eAS, q, n, t)k and its simple subalgebras are defined in [1, 3, 5–7, 13]. In this work, we define the combinatorial algebra N(e𝔄p, n, t)k and its antisymmetrized algebra [Formula: see text] and their subalgebras. We prove that these algebras are simple. Some authors [2, 5–7, 10, 13, 14, 16, 17] found all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra. We find all the derivations of the subalgebra N(e±x1x2 ⋯xn, 0, n)[1] of N(e𝔄p, n, t)k and the Lie subalgebra [Formula: see text] of [Formula: see text].

Susumu Okubo

Susumu Okubo

The nonassociative algebras used to build fast-decodable space-time block codes

Let $K/F$ and $K/L$ be two cyclic Galois field extensions and $D=(K/F,\sigma,c)$ a cyclic algebra. Given an invertible element $d\in D$, we present three families of unital nonassociative algebras over $L\cap F$ defined on the direct sum of $n$ copies of $D$. Two of these families appear either explicitly or implicitly in the designs of fast-decodable space-time block codes in papers by Srinath, Rajan, Markin, Oggier, and the authors. We present conditions for the algebras to be division and propose a construction for fully diverse fast decodable space-time block codes of rate-$m$ for $nm$ transmit and $m$ receive antennas. We present a DMT-optimal rate-3 code for 6 transmit and 3 receive antennas which is fast-decodable, with ML-decoding complexity at most $\mathcal{O}(M^{15})$.

BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras

A BiHom-associative algebra is a (nonassociative) algebra $A$ endowed with two commuting multiplicative linear maps $\alpha,\beta\colon A\rightarrow A$ such that $\alpha (a)(bc)=(ab)\beta (c)$, for all $a, b, c\in A$. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).

Linearly triangularizable quadratic endomorphisms

In the article, we prove that any polynomial endomorphism of , where is a field of characteristic zero, of the form ‘identity + homogeneous component of degree two’ is linearly triangularizable when the Jacobian matrix of the quadratic term is nilpotent of index at most three. In particular, this proves the well-known Meisters and Olech Conjecture. In our approach, besides methods of the classical matrix theory, some elements of the theory of commutative non-associative algebras appear in the context of the so-called polarization algebras associated with quadratic homogeneous endomorphisms. A powerful role of polarization algebras in the investigation of polynomial endomorphisms is revealed in a recent paper of Umirbaev. His results and comments were the main inspiration for us.

On the non-vanishing property for real analytic solutions of the $p$-Laplace equation

By using a nonassociative algebra argument, we prove that $u\equiv0$ is the only cubic homogeneous polynomial solution to the $p$-Laplace equation $\mathrm{div} |Du|^{p-2}Du(x)=0 $ in $\mathbb{R}^n$ for any $n\ge2$ and $p\not\in\{0,2\}$.

Computing topological zeta functions of groups, algebras, and modules, II

Building on our previous work, we develop the first practical algorithm for computing topological zeta functions of nilpotent groups, possibly non-associative algebras, and modules. While we previously depended upon non-degeneracy assumptions, the theory developed here allows us to overcome these restrictions in various interesting cases.

Methods for parametrizing varieties of Lie algebras

A real n-dimensional anticommutative nonassociative algebra is represented by an element of ∧2(Rn)⁎⊗Rn. For each μ∈∧2(Rn)⁎⊗Rn, there is a unique subset Λ⊆{(i,j,k):1≤i<j≤n,1≤k≤n} so that the structure constant μijk with i<j is nonzero if and only if (i,j,k)∈Λ. The set of all μ∈∧2(Rn)⁎⊗Rn with subset Λ is denoted SΛ(R); these sets stratify ∧2(Rn)⁎⊗Rn. We describe how to smoothly parametrize the Lie algebras in SΛ(R) up to isomorphism, for Λ satisfying certain frequently seen hypotheses.

Classification of the Algebras 𝕆 p,q

We study a series of real nonassociative algebras 𝕆 p, q introduced in [5]. These algebras have a natural -grading, where n = p + q, and they are characterized by a cubic form over the field ℤ2. We establish all the possible isomorphisms between the algebras 𝕆 p, q preserving the structure of -graded algebra. The classification table of 𝕆 p, q is quite similar to that of the real Clifford algebras Cl p, q , the main difference is that the algebras 𝕆 n, 0 and 𝕆0, n are exceptional.

A new non-associative cryptosystem based on NTOW public key cryptosystem and octonions algebra

In this work, we present a public key cryptosystem, called OTWO, based on octonions algebra and NTWO cryptosystem [1] which is a multivariate version of NTRU [2]. Inherent security of this system relies on the difficulty of the shortest vector problem (SVP) in a certain type of lattices with a hybrid norm. Since the octonions are non-associative (power-associative) and alternative algebra, they do not have a matrix isomorphic representation. So, normally lattice attacks [3] against this cryptosystem are impossible. The only way to cryptanalysis and to find the private key for decryption in this cryptosystem is to expand the equation of public key as a linear system of equations and form a non-circular lattice. However, this type of attack seems to has no chance to succeed. We change the underlying algebraic structure of NTWO and use a different lattice for key generation and decryption that it increases complexity of decryption. Furthermore, the nonassociativity of underlying algebraic structure and existence of different lattice for key generation and decryption improve the security of cryptosystem markedly. Method: The octonion algebra can be consider over a field or any arbitrary commutative ring R [4]. In our work, we use the bivariate convolution polynomial ring R′ = Z[X]/(X − 1) = Z[x, y]/(x−1, y−1) which n is a fixed prime number. Hence, we define A, Ap and Aq as the three octonion algebras over the rings R′, R′ p = Zp[X]/(X −1) and R′ q = Zq[X]/(X −1), respectively with bilinear multiplication (denoted by the symbol ◦), as follows A := { f0(x, y) + ∑7 i=1 fi(x, y) · ei ∣∣ f0(x, y), . . . , f7(x, y) ∈ R′}, Ap := { f0(x, y) + ∑7 i=1 fi(x, y) · ei ∣∣ f0(x, y), . . . , f7(x, y) ∈ R′ p}, Aq := { f0(x, y) + ∑7 i=1 fi(x, y) · ei ∣∣ f0(x, y), . . . , f7(x, y) ∈ R′ q}. where {1, e1, e2, e3, e4, e5, e6, e7} are the basis of the algebras and they have the following rules ei = −1, i = 1, . . . , 7 ei · ej = −ej · ei i 6= j, i, j = 1, . . . , 7 ei · ej = ek → ei+1 · ej+1 = ek+1 i 6= j, i, j = 1, . . . , 7 ei · ej = ek → e2i · e2j = e2k i 6= j, i, j = 1, . . . , 7 and the indices greater than 7 should be reduced mod 7. For simplification we use the notation fi , fi(x, y), for i = 0, 1, 2, 3. We denote the conjugate and inverse of the octonion F by F ? = f0 − ∑7 i=1 fi · ei and F−1 = ( ∑7 i=1 fi )−1 · F , respectively. In OTWO, the public parameters (n, p, q, d) play the same role as the alternative parameters do in NTWO, i.e., n is an integer number such that n|(q − 1), p and q are two different prime numbers such that gcd(p, q) = gcd(n, q) = 1 and q p. The subsets Lf , Lg, Lφ and Lm of R′ contain small polynomials which are polynomials with coefficients with small Euclidian norm and small Hamming norm (defined as the number of nonzero coefficient of a polynomial). Let Jq = Qq + ∑7 i=1Qq · ei, where Qq = 〈σ = ∑ (a,b)∈T λ(a,b)〉 is an ideal generated by σ and λ(a,b)’s are Lagrange interpolators as follows

Codimension Growth of Algebras with Adjoint Unit

In this paper, numerical characteristics of identities of finite-dimensional nonassociative algebras are studied. Main attention is paid to the question on the change of the PI-exponent after adjoining an external unit element. We construct an example of a four-dimensional simple algebra such that its PI-exponent increases by one after adjoining an external unit element.

Role of division algebra in seven-dimensional gauge theory

The algebra of octonions 𝕆 forms the largest normed division algebra over the real numbers ℝ, complex numbers ℂ and quaternions ℍ. The usual three-dimensional vector product is given by quaternions, while octonions produce seven-dimensional vector product. Thus, octonionic algebra is closely related to the seven-dimensional algebra, therefore one can extend generalization of rotations in three dimensions to seven dimensions using octonions. An explicit algebraic description of octonions has been given to describe rotational transformation in seven-dimensional space. We have also constructed a gauge theory based on non-associative algebra to discuss Yang–Mills theory and field equation in seven-dimensional space.

Distributivity on the Gyrovector Spaces

Abstract. As a vector space provides a fundamental tool for the study of Euclideangeometry, a gyrovector space provides an algebraic tool for the study of hyperbolic ge-ometry. In general, the gyrovector spaces do not satisfy the distributivity with scalarmultiplication. In this article, we see under what condition the distributivity with scalarmultiplication is satis ed. 1. IntroductionIn order to provide an algebraic tool to study Einstein’s relativistic velocitysum, A. A. Ungar [2] has introduced a notion of gyrogroup and has developed to-gether the study of analytic hyperbolic geometry. The gyrogroup is a most naturalextension of a group into the nonassociative algebra. The associativity (and thecommutativity) of group operations is salvaged in a suitably modified form, calleda gyroassociativity (and a gyrocommutativity). In Section 2 we introduce a notionof (gyrocommutative) gyrogroup with its examples.Later on it is known that gyrocommutative gyrogroups are equivalent to Bruckloops (see [1]). To elaborate a precise language, we prefix a