Cross products, invariants, and centralizers

Abstract

An algebra $V$ with a cross product $\times$ has dimension 3 or 7. In this work, we use 3-tangles to describe, and provide a basis for, the space of homomorphisms from $V^{\otimes n}$ to $V^{\otimes m}$ that are invariant under the action of the automorphism group $Aut(V,\times )$ of $V$, which is a special orthogonal group when $dimV=3$, and a simple algebraic group of type $G_2$ when $dimV=7$. When $m=n$, this gives a graphical description of the centralizer algebra $En{d}_{Aut(V,\times )}(V^{\otimes n})$, and therefore, also a graphical realization of the $Aut(V,\times )$-invariants in $V^{\otimes 2n}$ 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.

27 pages

References in corpus (3)

• Enumeration of non-positive planar trivalent graphs
• Invariants of the orthosymplectic Lie superalgebra and super Pfaffians
• Invariants of $G_2$ and $Spin(7)$ in positive characteristic

Cited by in corpus (2)

• Invariants of the special orthogonal group and an enhanced Brauer category
• Cellular subalgebras of the partition algebra