A polynomial identity for the bilinear operation in Lie–Yamaguti algebras
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Abstract
AbstractWe use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie–Yamaguti algebra. This identity is a consequence of the defining identities for Lie–Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit form of this identity as an alternating sum over all permutations of the variables in a nonassociative polynomial with 8 terms. Our computations show that no such identities exist in degrees less than 8.Keywords: Lie–Yamaguti algebrasanticommutative algebraspolynomial identitiescomputer algebrarepresentation theory of the symmetric groupAMS Subject Classification: Primary 17A30Secondary 17-0417A3217A4017A5017B01 AcknowledgementsThis research was partially supported by a Discovery Grant from NSERC, the Natural Sciences and Engineering Research Council of Canada. The author thanks Pilar Benito and the Department of Mathematics and Computer Science at the University of La Rioja in Logroño for their hospitality in April and May 2013, and the referee for a number of helpful suggestions.
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