Identities of the Model Algebra of Multiplicity 2

We construct an additive basis of the free algebra of the variety generated by the model algebra of multiplicity 2 over an infinite field of characteristic not 2 and 3. Using the basis we remove a restriction on the characteristic in the theorem on identities of the model algebra (previously the same was proved in the case of characteristic 0). In particular, we prove that the kernel of the relatively free Lie-nilpotent algebra of index 5 coincides with the ideal of identities of the model algebra of multiplicity 2.     

Identities of PI-Algebras Graded by a Finite Abelian Group

We consider associative PI-algebras over an algebraically closed field of zero characteristic graded by a finite abelian group G. It is proved that in this case the ideal of graded identities of a G-graded finitely generated PI-algebra coincides with the ideal of graded identities of some finite dimensional G-graded algebra. This implies that the ideal of G-graded identities of any (not necessary finitely generated) G-graded PI-algebra coincides with the ideal of G-graded identities of the Grassmann envelope of a finite dimensional (G × ?₂)-graded algebra, and is finitely generated as GT-ideal. Similar results take place for ideals of identities with automorphisms.     

Degenerate alternative algebras

We prove that each degenerate alternative algebra of characteristic ≠ 2 contains a nonzero ideal with an additive basis consisting of the absolute zero divisors of an arbitrary large order. As a corollary we establish the existence of infinitely many nonisomorphic commutative prime alternative algebras and existence of infinite series of strict and nonstrict exceptional alternative algebras with different sets of proper identities.     

Spectral Theory of Super-Differential Operators of Quaternion and Octonion Variables

The article is devoted to spectral theory of super-differential operators over the quaternion skew field and the octonion algebra. An existence of their resolvent functions is proved, their spectra are investigated. It is shown, that spectra are contained in general in the quaternion skew field or the octonion algebra and can not be reduced to the field of complex numbers.     

From polynomial to formal identities

Polynomial identities, when multiplied in the group algebra of the symmetric group by a two sided ideal of non identities, imply various formal (combinatorial) identities.     

The alternative Clifford algebra of a ternary quadratic form

We prove that the alternative Clifford algebra of a nondegenerate ternary quadratic form is an octonion ring whose center is the ring of polynomials in one variable over the field of definition.     

Capelli elements of the group algebra

Inspired by the Capelli-type identities for group determinants researched by Tôru Umeda, we give Capelli identities for irreducible representations of any finite group, and Capelli elements of the group algebra associated with these identities. These elements construct a basis of the centre of the group algebra.     

具体算子代数的内理想刻画及同构定理

本文绘出了具有单位元的具体算子代数的内理想的一系列本质刻画和具体算子代数之间的一个同构定理．     

Simple finite-dimensional algebras without finite basis of identities

In 1993, Shestakov posed a problem of existence of a central simple finite-dimensional algebra over a field of characteristic 0 whose identities cannot be defined by a finite set (Dniester Notebook, Problem 3.103). In 2012, Isaev and the author constructed an example that gave a positive answer to this problem. In 2015, the author constructed an example of a central simple seven-dimensional commutative algebra without finite basis of identities. In this article we continue the study of Shestakov’s problem in the case of anticommutative algebras. We construct an example of a simple seven-dimensional anticommutative algebra over a field of characteristic 0 without finite basis of identities.     

The structure group of an alternative algebra

The structure group of an alternative algebra and various canonical subgroups are defined and investigated. Using the principle of triality, natural sets of generators for these groups in the case of octonion algebras are exhibited. Horst Tietz zur Vollendung des achtzigsten Lebensjahres gewidmet     

Differentiable functions of Cayley-Dickson numbers and line integration

We investigate the superdifferentiability of functions defined on domains of the real octonion (Cayley) algebra and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analogue of the Cauchy integral and criteria for functions of an octonion variable to be analytic. In particular, the octonion-valued exponential and logarithmic functions are considered. Moreover, superdifferentiable functions of variables belonging to Cayley-Dickson algebras (containing the octonion algebra as a proper subalgebra) of finite and infinite dimensions are investigated. Among the main results, the Cayley-Dickson-algebra analogues of the Cauchy, Hurewicz, Mittag-Löffler, Rouche, and Weierstrass theorems and the argument principle are proved. Examples of special functions such as the beta and gamma functions of Cayley-Dickson numbers are studied. Applications to the study of zeros of polynomials of Cayley-Dickson numbers are given.     

Partial Maps with Domain and Range: Extending Schein's Representation

Let A be a *-algebra. An additive mapping E : A → A is called a Jordan *-derivation if E(X²) = E(x)x^*+xE(x) holds, for all x 6 A. These mappings have been extensively studied in the last 6 years by Bresar, Semrl, Vukman and Zalar because they are closely connected with the problem of representability of quadratic functionals by sesquilinear forms. This study was, however, always in the setting of associative rings. In the present paper we study Jordan *-derivations on the Cayley-Dickson algebra of octonions, which is not associative. Our first main result is that every Jordan *-derivation on the octonion algebra is of the form E(x)=ax*-xa. In the terminology of earlier papers this means that every Jordan *-derivation on the octonion algebra is inner. This generalizes the known fact that Jordan *-derivations on complex and quaternion algebras are inner. Our second main result is a representation theorem for quadratic functionals on octonion modules. Its proof uses the result mentioned above on Jordan *-derivations.     

Symmetric composition algebras over algebraic varieties

Let ${(X,\mathcal{O}_X)}$ be a locally ringed space. We investigate the structure of symmetric composition algebras over X obtained from cubic alternative algebras ${\mathcal{A}}$ over X generalizing a method first presented by J. R. Faulkner. We find examples of Okubo algebras over elliptic curves which do not have any isotopes which are octonion algebras and of an octonion algebra which is a Cayley-Dickson doubling of a quaternion algebra but does not contain any quadratic étale algebras.     

Octonion algebras obtained from associative algebras with involution

A natural octonion algebra structure on the symmetric elements of trace 0 of central simple associative algebras of degree 3 with involution of the second kind is obtained.

     

Feynman integration over octonions with application to quantum mechanics

The article is devoted to Gaussian quasi‐measures and Feynman integrals on infinite‐dimensional spaces with values in the octonion algebra. Their characteristic functionals are studied. Products and convolutions of characteristic functionals and quasi‐measures are investigated. Theorems about properties of octonion‐valued Gaussian quasi‐measures and Feynman integrals are proved. Applications of the Feynman integration over octonions to quantum mechanics and partial differential equations are outlined. Copyright © 2009 John Wiley & Sons, Ltd.     

Identities of representations of nilpotent lie algebras

Lret N be a nilpotent of class 2 Lie algebra with one-dimensional centre C = Kc over an infinite field K and let p : N → End_k:(V) be a representation of N in a vector space V such that p(c) is invertible in End_k(V). We find a basis for the identities of the representation p. As consequences we obtain a basis for all the weak polynomial identities of the pair (M₂:(K), s1₂(K)) over an infinite field K of characteristic 2 and describe the identities of the regular representation of Lie algebras related with the Weyl algebra and its tensor powers.     

On the Ternary Approach to Clifford Structures and Ising Lattices

We continue to modify and simplify the Ising-Onsager-Zhang procedure for analyzing simple orthorhombic Ising lattices by considering some fractal structures in connection with Jordan and Clifford algebras and by following Jordan-von Neumann-Wigner (JNW) approach. We concentrate on duality of complete and perfect JNW-systems, in particular ternary systems, analyze algebras of complete JNW-systems, and prove that in the case of a composition algebra we have a self-dual perfect JNW-system related to quaternion or octonion algebras. In this context, we are interested in the product table of the sedenion algebra.     

Eigenvalue problem for symmetric 3×3 octonionic matrix

The eigenvalue problem of symmetric 3×3 octonionic matrix has been analyzed. We have especially proved explicitly first that octonionic eigenfunctions have six independent solutions in general with four degeneracy each, and second that for different eigenvalues they satisfy a cubic orthogonality relation under some conditions, which has been previously discovered by Dray and Manogue by computer use. For these, the close relationship between the octonion algebra and a Clifford algebra plays a significant role.