nLab
pivotal symbols

The pivotal symbols or 3j3j symbols are a certain collection of signs attached to a fusion category arising from analyzing how invariant vectors in a 33-particle state transform under rotation by a full revolution.

In a fusion category, the nature of duals for objects in the category can be encoded in principal ×\mathbb{C}^\times-bundle over the simple objects. Firstly, suppose one goes ahead and chooses an arbitrary dual object V i *V_i^* for every simple object V iV_i. Write

(1)Dual(V iV i *) Dual(V_i \dashv V_i^*)

for the set of ways that V i *V_i^* can be expressed as a right dual of V iV_i. Notice that Dual(V iV i *)Dual(V_i \dashv V_i^*) is a ×\mathbb{C}^\times-torsor (you can multiply the unit by a nonzero complex number, following which you are forced to multiply the counit by the reciprocal of this number if you want to obey the snake equations).

In a fusion category, if V i *V_i^* is a right dual of V iV_i, then it is also a left dual of V iV_i, though not in a canonical way. This gives us the principal ×\mathbb{C}^\times-bundle LL over the simple objects — the fiber above V iV_i is

(2)L V i={Dual(V iV i *)Dual(V i *V i)}, L_{V_i} = \left\{Dual(V_i \dashv V_i^*) \stackrel{\sim}\rightarrow Dual(V_i^* \dashv V_i) \right\},

the set of isomorphisms between Dual(V iV i *)Dual(V_i \dashv V_i^*) and Dual(V i *V i)Dual(V_i^* \dashv V_i).

A pivotal structure or even-handed structure is a section of this bundle satisfying some naturality properties. In other words, it is a consistent way to turn right duals into left duals. The most stringent condition is that the invariant vectors in a tensor product of three particles must either be totally ‘bosonic’ or totally ‘fermionic’ when you rotate them through a full rotation.

That is, the endomorphism

(3)T ijk:Hom(1,V iV jV k)Hom(1,V iV jV k) T_{ijk} : Hom(1,V_i \otimes V_j \otimes V_k) \rightarrow Hom(1,V_i \otimes V_j \otimes V_k)

(which can be proved to be an involution) given by sending

Picture explaining the concept of "pivotal symbols" in a fusion category

must be equal to ±id\pm id. Assuming this is satisfied (if not, there is no consistent way to turn right duals into left duals), we can call this collection of signs the 3j3j symbols or the pivotal symbols ϵ ijk\epsilon_{ijk} of the fusion category:

(4)T ijk=ϵ ijkid. T_{ijk} = \epsilon_{ijk} id.

That is to say, the 3j3j symbols are the signs arising from choosing a trivialization of the ‘duality bundle’ and analyzing how invariant 33-particle tensors transform when they go through a full revolution.

It turns out that one can express the concept of a pivotal structure directly in terms of the pivotal symbols ϵ ijk\epsilon_{ijk}. Namely, a pivotal structure on the category amounts to an ϵ\epsilon-twisted monoidal natural tranformation? of the identity on the category. That is to say, a collection of nonzero complex numbers t it_{i} satisfying

(5)t jt k=ϵ ijkt i t_j t_k = \epsilon_{ijk} t_i

whenever V iV_i appears in V jV kV_j \otimes V_k. If all the 3j3j symbols were equal to 11, this would just be an ordinary monoidal natural transformation of the identity. Perhaps this phenomenon can be understood in a more enlightening way using the language of planar algebras.

References

Revised on July 1, 2009 08:31:30 by Toby Bartels (71.104.230.172)