Cross products, invariants, and centralizers

An algebra V with a cross product × 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⊗n to V⊗m that are invariant under the action of the automorphism group Aut(V,×) of V, which is a special orthogonal group when dimV=3, and a simple algebraic group of type G2 when dimV=7. When m=n, this gives a graphical description of the centralizer algebra EndAut(V,×)(V⊗n), and therefore, also a graphical realization of the Aut(V,×)-invariants in V⊗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.