Colour algebras and colour traces

First year pure mathematics students learn about the trace of a matrix or a linear transformation, and in dealing with Lie superalgebras and quantum superalgebras, one also deals with the supertrace of a linear transformation, which is a generalisation of the usual trace.

A number of people (e.g. David McAnally and Peter Jervis) have looked at colour algebras, which are generalisations of superalgebras. Recall that a superalgebra starts with a Z2 graded vector space on which one defines a supertrace on linear transformations of the Z2-graded vector space. With a colour algebra, the idea is to start with a Zn-graded vector space and define a colour trace on linear transformations of the Zn-graded vector space.

Colour algebras didn’t go very far as they weren’t seen as very useful – however, I’ve been wondering if they are useful in obtaining new knot invariants, and I’ve been starting to look at Zn-graded vector spaces and colour traces again.

Colour traces are interesting and non-trival – even properly defining the phase factor one obtains when applying the permutation operation P to tensor products is not straightforward. If one fixes P2 = id, one might obtain the usual supertrace (I’m not certain about this) and I’ve started investigating what happens if you don’t fix P2 = id.

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