000070698 001__ 70698 000070698 005__ 20191127155455.0 000070698 0247_ $$2doi$$a10.1016/j.jalgebra.2016.11.013 000070698 0248_ $$2sideral$$a105220 000070698 037__ $$aART-2018-105220 000070698 041__ $$aeng 000070698 100__ $$aBenkart, G. 000070698 245__ $$aCross products, invariants, and centralizers 000070698 260__ $$c2018 000070698 5060_ $$aAccess copy available to the general public$$fUnrestricted 000070698 5203_ $$aAn algebra V with a cross product x has dimension 3 or 7. In this work, we use 3-tangles to describe, and provide a basis for, the space of homomorphisms from V-circle times n to V-circle times m that are invariant under the action of the automorphism group Aut(V, x) of V, which is a special orthogonal group when dim V = 3, and a simple algebraic group of type G(2) when dim V = 7. When m = n, this gives a graphical description of the centralizer algebra End(Aut(v, x))(V-circle times n), and therefore, also a graphical realization of the Aut(V, x)-invariants in V-circle times 2n equivalent to the First Fundamental Theorem of Invariant Theory. We show how the 3-dimensional simple Kaplansky Jordan superalgebra can be interpreted as a cross product (super)algebra and use 3-tangles to obtain a graphical description of the centralizers and invariants of the Kaplansky superalgebra relative to the action of the special orthosymplectic group. 000070698 536__ $$9info:eu-repo/grantAgreement/ES/DGA/FSE$$9info:eu-repo/grantAgreement/ES/MINECO/MTM2013-45588-C3-2-P 000070698 540__ $$9info:eu-repo/semantics/openAccess$$aby-nc-nd$$uhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/ 000070698 590__ $$a0.666$$b2018 000070698 591__ $$aMATHEMATICS$$b189 / 313 = 0.604$$c2018$$dQ3$$eT2 000070698 592__ $$a1.137$$b2018 000070698 593__ $$aAlgebra and Number Theory$$c2018$$dQ1 000070698 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/acceptedVersion 000070698 700__ $$0(orcid)0000-0002-6497-2162$$aElduque, A.$$uUniversidad de Zaragoza 000070698 7102_ $$12006$$2005$$aUniversidad de Zaragoza$$bDpto. Matemáticas$$cÁrea Algebra 000070698 773__ $$g500 (2018), 69-102$$pJ. algebra$$tJOURNAL OF ALGEBRA$$x0021-8693 000070698 8564_ $$s312286$$uhttps://zaguan.unizar.es/record/70698/files/texto_completo.pdf$$yPostprint 000070698 8564_ $$s69516$$uhttps://zaguan.unizar.es/record/70698/files/texto_completo.jpg?subformat=icon$$xicon$$yPostprint 000070698 909CO $$ooai:zaguan.unizar.es:70698$$particulos$$pdriver 000070698 951__ $$a2019-11-27-15:47:05 000070698 980__ $$aARTICLE