International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 10 Issue: 08 | Aug 2023
p-ISSN: 2395-0072
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CONSTRUCTION OF NONASSOCIATIVE GRASSMANN ALGEBRA Dr. Saad Salman Ahmed University of Technology and Applied Sciences-Muscat Branch Department of Computing and Mathematics Section of mathematics AL-KHUWAIR 133, Box 74, Muscat, Oman --------------------------------------------------------------------------------***--------------------------------------------------------------------Abstract: One of the main sources of algebraic structures that illustrates basic concepts of modern algebra, is theoretical physics where there are many well-known algebraic structures used as the Lie groups, rotation groups, and “Algebra of color” proposed by Domokos as a candidate for the algebra obeyed by quantized field describing quark and leptons. We wish to investigate the nonassociative Grassmann and symmetric algebras. Z. OZIEWIOZ and C. SITARCZYK in [8], (non-published) started with the pair of the mutually dual Grassmann and symmetric algebras with the full set of the interior and exterior Grassmann and symmetric (bosonic) multiplications. They investigate the non-associative Grassmann algebra by combining the exterior and interior products for the Grassmann and symmetric algebra in one product on the Z-graded vector space (with no positive gradation).In this paper, we derived a new multiplication rule for this algebra and gave the complete tables of multiplication in dimensions 3, 7, and 15.
Keywords: Grassmann Algebra, nonassociative Algebra, Z-graded vector spaces, The p th exterior power V of p
finite–dimensional vector space.
Literature Review: Herman Grassmann was a schoolteacher in StettinGermany who did remarkable work in mathematics and the theory of languages. He created an algebra of Geometry which we are concerned with here. Grassmann wrote two books on the subject, one in 1844 and the second one in 1862. They were “The Theory of Extension”(meaning extension to higher dimensions in space. These two books received little notice before his death in 1877 when he received an Honorary Ph.D. a year before his death. Nevertheless, since then many good mathematicians have taken notice of Grassmann’s work. William Clifford knew it and wrote a variant now called Clifford Algebra [3]. Alfred North Whitehead studied it and wrote “A Treatise on Universal Algebra which was an exposition of Grassmann’s work in English. Elie Caratan use Grassmann’s exterior product to develop differential forms and exterior derivatives and applied them to differential geometry. Willard Gibbs knew and admired Grassmann’s algebra but helped devise a different vector Analysis that became the mainstay of technical education for over a century. John Browne was doing an engineering Ph.D. when he discovered Grassmann Algebra and fell in love with it. He switched to the mathematics department and his thesis to Grassmann algebra. It was voted the best thesis of the year. John spent the rest of his life deepening, extending, and clarifying Grassmann's algebra where he wrote three books, [14,15,16]. John died in June 2021.
Introduction: Let V be a vector space over a field F. The Grassmann algebra of V, denoted by V , is the linear span (over F) of products v1 v 2 ..... v r where vi V . Each term in a member of V has a degree, which is the number of vectors in the product: deg( v1 v 2 ... v r ) r. We agree that, by definition, the elements of the field F will have degree equal to 0. The product is associative and bilinear, also it is very nilpotent: v v 0
for all v V . Furthermore, the product is anti-
commutative that is v w w v for all v, w V . We wish to investigate the nonassociative Grassmann and symmetric algebra. We start with the pair of the mutually dual Grassmann and symmetric algebras with the full set of the interior and exterior Grassmann and symmetric (bosonic)multiplication. The symmetric algebras are related by the permanent, in the analogy to the Grassmann case where the mutually dual Grassmann algebras are related by the determinant. Our study is motivated by the symplectic Clifford algebras which has been introduced by Albert Crumeyrolle in 1975, in full analogy to the usual presentation of Clifford algebras for the pseudo-Riemannian structures.
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