Cross products, invariants, and centralizers

JOURNAL OF ALGEBRA

卷:500

Article; Proceedings Paper

Benkart, Georgia¹  Elduque, Alberto^2,3 

[1] Univ Wisconsin Madison, Dept Math, Madison, WI 53706 USA

[2] Univ Zaragoza, Dept Matemat, E-50009 Zaragoza, Spain

[3] Univ Zaragoza, Inst Univ Matemat & Aplicac, E-50009 Zaragoza, Spain

关键词: Cross product;    Invariant map;    3-tangle;    G(2);    Kaplansky superalgebra;    Centralizer algebra;   

DOI  :  10.1016/j.jalgebra.2016.11.013

来源: Elsevier

【 摘 要 】

An 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. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free