Cross products, invariants, and centralizers

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 $dim V = 3$, 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 $End_{Aut(V,\times)}(V^{\otimes n})$, 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.