Let $F$ be a field $\{\otimes,\oplus,R\_i\}$ with the one-element $R\_1$ (the null-
element $R\_0$ isn't very relevant). Let $T\_{ijk}$ be a "triangle condition" 
(symmetric in the indices and integer-valued) which is used to define 
${R\_i}\otimes{R\_j}=\oplus\_k T\_{ijk}\*R\_k$. Clearly $T\_{0jk}=0, T\_{1jk}=\delta^j\_k$
(although I wouldn't even guarantee for the latter when multiplicities
greater than 1 occur...) and the associativity of $\otimes$ forces a lot of further conditions on the $T\_{ijk}$. Call this setup a confusion category. :-)


You now have two possibilities: a) finite number of $R\_i$ (giving a fusion category -
my definition may be nonstandard or incomplete!); b) infinitely many $R\_i$ which form
the irreps of some Lie group. c): The Vogel plane shows that there might be a 
third possibility - infinitely many $R\_i$ which are NOT the irreps of some Lie group 
(but maybe generalize them somehow). Question 1: Do all the irreps of the Vogel plane
indeed form, provenly, a confusion category? (He only gave the lowest CG expansions,
and the computations were hard enough, so it's not obvious to me the process can
be continued ad infinitum.)


Question 2: It is probably easier to generalize the $BCD\_n$ series in an
analogous way, has this been done? 


At the moment, I'm after the symmetrized generalization of the $E\_7$ series.
The first Clebsch-Gordan expansions are $V\otimes{V}=1+A\_1+A\_2+A\_3, V\otimes{A\_1}=
V+B+C\_2+C\_3+D\_1$ (and cyclic). (I already have computed all CG coefficients,
Casimirs and dimensions with maximally one B, C or D occurring.) 
Since it was enormous work to get even that far and already $A\_1\otimes{A\_1}$
will have about 15 terms, including a double $A\_1$ (aaargh!) - can this form a
confusion category too? (If yes, one could do the Reshitikhine-Turaev 6j
construction to get a symmetrized 3-variable knot polynomial, and after
30 years I could finally rest in peace :-)



Infinite fusion categories (sort of)
For BCD what you want is called the BMW category.


For the Vogel plane, E7, exceptional, etc. everything is totally open.


