The Experts below are selected from a list of 147 Experts worldwide ranked by ideXlab platform
David P. Jacobs - One of the best experts on this subject based on the ideXlab platform.
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a course in computational Nonassociative Algebra
Journal of Symbolic Computation, 1997Co-Authors: David P. JacobsAbstract:Abstract This paper describes a two-semester graduate-level course designed during 1995 at Universidade Federal do Rio Grande do Sul (UFRGS). A major goal of the course was to introduce several problems in the theory of Nonassociative Algebra and to show how computer Algebra methods can be used to gain insight. The course utilized AXIOM as well as the special purpose program Albert.
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A Basis for Free Assosymmetric Algebras
Journal of Algebra, 1996Co-Authors: Irvin Roy Hentzel, David P. Jacobs, Luiz Antonio PeresiAbstract:Abstract Assosymmetric Algebras are Nonassociative Algebras, where ( xy ) z − x ( yz ) remains invariant under each permutation of x , y , z . In general, the free Nonassociative Algebra in a variety is difficult to describe. We show this is not the case for free assosymmetric Algebras having characteristic ≠2, 3. We exhibit a natural basis, describe how basis elements are multiplied and show how arbitrary elements can be expressed relative to this basis.
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ISSAC - The Albert Nonassociative Algebra system: a progress report
Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94, 1994Co-Authors: David P. JacobsAbstract:After four years of experience with the Nonassociative Algebra program Albert, we highlight its successes and drawbacks. Among its successes are the discovery of several new results in Nonassociative Algebra. Each of these results has been independently veri ed { either with a traditional mathematical proof or with an independent computation.
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the albert Nonassociative Algebra system a progress report
International Symposium on Symbolic and Algebraic Computation, 1994Co-Authors: David P. JacobsAbstract:After four years of experience with the Nonassociative Algebra program Albert, we highlight its successes and drawbacks. Among its successes are the discovery of several new results in Nonassociative Algebra. Each of these results has been independently veri ed { either with a traditional mathematical proof or with an independent computation.
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A dynamic programming method for building free Algebras
Computers & Mathematics With Applications, 1991Co-Authors: Irvin Roy Hentzel, David P. JacobsAbstract:Abstract We are interested in deciding if a given Nonassociative polynomial h is an identity for a variety of Nonassociative Algebras. We present an algorithm for constructing a certain homomorphic image of a free Nonassociative Algebra which is sufficient to answer the question. The algorithm resembles dynamic programming in that the Algebra is built by constructing a sequence of subspaces; the basis of each subspace is determined by the basis of previous subspaces. The number of arithmetic operations required to construct the Algebra is bounded by a polynomial in the degree of h and the dimension of the resulting Algebra.
Irvin Roy Hentzel - One of the best experts on this subject based on the ideXlab platform.
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identities relating the jordan product and the associator in the free Nonassociative Algebra
Journal of Algebra and Its Applications, 2006Co-Authors: Murray R Bremner, Irvin Roy HentzelAbstract:We determine the identities of degree ≤ 6 satisfied by the symmetric (Jordan) product a○b = ab + ba and the associator [a,b,c] = (ab)c - a(bc) in every Nonassociative Algebra. In addition to the commutative identity a○b = b○a we obtain one new identity in degree 4 and another new identity in degree 5. We demonstrate the existence of further new identities in degree 6. These identities define a variety of binary-ternary Algebras which generalizes the variety of Jordan Algebras in the same way that Akivis Algebras generalize Lie Algebras.
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dimension formulas for the free Nonassociative Algebra
Communications in Algebra, 2005Co-Authors: Murray R Bremner, Irvin Roy Hentzel, Luiz Antonio PeresiAbstract: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.
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dimension formulas for the free Nonassociative Algebra
Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo, 2004Co-Authors: Murray R Bremner, Irvin Roy Hentzel, Luiz Antonio PeresiAbstract: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.
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invariant Nonassociative Algebra structures on irreducible representations of simple lie Algebras
Experimental Mathematics, 2004Co-Authors: Murray R Bremner, Irvin Roy HentzelAbstract:An irreducible representation of a simple Lie Algebra can be a direct summand of its own tensor square. In this case, the representation admits a Nonassociative Algebra structure which is invariant...
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A Basis for Free Assosymmetric Algebras
Journal of Algebra, 1996Co-Authors: Irvin Roy Hentzel, David P. Jacobs, Luiz Antonio PeresiAbstract:Abstract Assosymmetric Algebras are Nonassociative Algebras, where ( xy ) z − x ( yz ) remains invariant under each permutation of x , y , z . In general, the free Nonassociative Algebra in a variety is difficult to describe. We show this is not the case for free assosymmetric Algebras having characteristic ≠2, 3. We exhibit a natural basis, describe how basis elements are multiplied and show how arbitrary elements can be expressed relative to this basis.
Vladimir G. Tkachev - One of the best experts on this subject based on the ideXlab platform.
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Spectral properties of Nonassociative Algebras and breaking regularity for nonlinear elliptic type PDEs
arXiv: Analysis of PDEs, 2019Co-Authors: Vladimir G. TkachevAbstract:In this note, we address the following question: Why certain Nonassociative Algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? The aim of the paper is twofold. Firstly, to give a survey of diverse examples on nonregular solutions to elliptic PDEs with emphasis on recent results on nonclassical solutions to fully nonlinear equations. Secondly, to define an appropriate Algebraic formalism which makes the analytic part of the construction of nonclassical solutions more transparent.
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On an extremal property of Jordan Algebras of Clifford type
Communications in Algebra, 2018Co-Authors: Vladimir G. TkachevAbstract:AbstractIf V is a finite-dimensional unital commutative (maybe Nonassociative) Algebra carrying an associative positive definite bilinear form 〈, 〉 then there exist a nonzero idempotent c ≠ e (e be...
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The universality of one half in commutative Nonassociative Algebras with identities
arXiv: Rings and Algebras, 2018Co-Authors: Vladimir G. TkachevAbstract:In this paper we will explain an interesting phenomenon which occurs in general Nonassociative Algebras. More precisely, we establish that any finite-dimensional commutative Nonassociative Algebra over a field satisfying an identity always contains $\frac12$ in its Peirce spectrum. We also show that the corresponding $\frac12$-Peirce module satisfies the Jordan type fusion laws. The present approach is based on an explicit representation of the Peirce polynomial for an arbitrary Algebra identity. To work with fusion rules, we develop the concept of the Peirce symbol and show that it can be explicitly determined for a wide class of Algebras. We also illustrate our approach by further applications to genetic Algebras and Algebra of minimal cones (the so-called Hsiang Algebras).
Murray R Bremner - One of the best experts on this subject based on the ideXlab platform.
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identities relating the jordan product and the associator in the free Nonassociative Algebra
Journal of Algebra and Its Applications, 2006Co-Authors: Murray R Bremner, Irvin Roy HentzelAbstract:We determine the identities of degree ≤ 6 satisfied by the symmetric (Jordan) product a○b = ab + ba and the associator [a,b,c] = (ab)c - a(bc) in every Nonassociative Algebra. In addition to the commutative identity a○b = b○a we obtain one new identity in degree 4 and another new identity in degree 5. We demonstrate the existence of further new identities in degree 6. These identities define a variety of binary-ternary Algebras which generalizes the variety of Jordan Algebras in the same way that Akivis Algebras generalize Lie Algebras.
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dimension formulas for the free Nonassociative Algebra
Communications in Algebra, 2005Co-Authors: Murray R Bremner, Irvin Roy Hentzel, Luiz Antonio PeresiAbstract: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.
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dimension formulas for the free Nonassociative Algebra
Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo, 2004Co-Authors: Murray R Bremner, Irvin Roy Hentzel, Luiz Antonio PeresiAbstract: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.
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invariant Nonassociative Algebra structures on irreducible representations of simple lie Algebras
Experimental Mathematics, 2004Co-Authors: Murray R Bremner, Irvin Roy HentzelAbstract:An irreducible representation of a simple Lie Algebra can be a direct summand of its own tensor square. In this case, the representation admits a Nonassociative Algebra structure which is invariant...
Alexandros Patsourakos - One of the best experts on this subject based on the ideXlab platform.
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on some ideals in the free Nonassociative Algebra
Communications in Algebra, 2010Co-Authors: Alexandros PatsourakosAbstract: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.
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On some properties of Hall elements in the free Nonassociative Algebra
Acta Mathematica Hungarica, 2008Co-Authors: Alexandros PatsourakosAbstract: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.