nLab 6j symbol




6j6j symbols are pieces of combinatorial data which can be associated to fusion categories, encoding their associator with a finite amount of algebraic data. The 6j6j symbols along with the fusion ring is enough data to uniquely reconstruct the category.

6j6j symbols are only defined for multiplicity free fusion categories. That is, fusion categories in which all of the fusion coefficients are 00 or 11. For fusion categories with larger coefficients, one has to replace 6j6j symbols (which are complex numbers) with higher-dimensional matrices.

The name “6j6j symbol” comes from the fact that the symbol has 66 indicies associated to it.

The 6j6j symbols depend on a local choice of basis of hom-spaces of your fusion category, and hence they are not bona-fide invariants. Choosing a different choice of local basis corresponds to a gauge transformation.


Let 𝒞\mathcal{C} be a multiplicity free fusion category. Let

α:(−⊗−)⊗−→−⊗(−⊗−)\alpha: (- \otimes -)\otimes - \to - \otimes (-\otimes -)

be its associator. We wish to encode the associator with a finite amount of algebraic data. That is, with a finite quantity of complex numbers. We do this by studying the action of α\alpha with respect to the (finite) set ℒ\mathcal{L} of isomorphism classes of simple objects of 𝒞\mathcal{C}.

For every triple A,B,CA,B,C of simple objects of 𝒞\mathcal{C}, choose a distinguished generator

This will be a non-zero morphism whenever dimHom(A⊗B,C)≥1\dim \text{Hom}(A\otimes B,C)\geq 1 and 00 otherwise. Here, we are using the graphical language of string diagrams. There generators are chosen arbitrarily, except for the conditions that

where ρ A\rho_A denotes the right unitor and λ A\lambda_A denotes the left unitor.

Note the key use of the fact that 𝒞\mathcal{C} is multiplicity free. If fusion coefficients could be ≥2\geq 2 then we would not have one-element bases of the spaces Hom(A⊗B,C)\text{Hom}(A\otimes B,C). Instead, we would have to choose larger bases of these spaces are get matrices for our 6j6j symbols instead of complex numbers.

For all quadruples A,B,C,DA,B,C,D of simple objects, there is a basis of Hom((A⊗B)⊗C,D)\text{Hom}((A\otimes B)\otimes C,D) given by the symbols

and there is a basis of Hom(A⊗(B⊗C),D)\text{Hom}(A\otimes (B\otimes C),D).

where in both cases NN ranges over representatives of isomorphism classes of simple objects.


The associator α\alpha induces a map

α A,B,C:Hom((A⊗B)⊗C,D)→Hom(A⊗(B⊗C),D)\alpha_{A,B,C}:\text{Hom}((A\otimes B)\otimes C,D)\to \text{Hom}(A\otimes (B\otimes C),D)

for all simple objects A,B,C,D∈𝒞A,B,C,D\in\mathcal{C}. Expressing the source and target of this map in the distinguished bases above, we can canonically identify α A,B,C\alpha_{A,B,C} with a matrix.

The 6j symbol F d;n,m a,b,cF^{a,b,c}_{d;n,m} is defined to be the (N,M)(N,M)th coefficient of this matrix, where a,b,c,d,n,ma,b,c,d,n,m are isomorphism classes of simple objects. That is, the 6j6j symbols are the unique complex numbers making the identity


Use of Yoneda perspective

It is a key observation that one does not use the 6j6j symbols to directly encode α\alpha. Instead, the 6j6j symbols are used to describe the action of the hom-space hom(−,D)\text{hom}(-,D).

They key point is that objects are not associated vector spaces in a linear category: hom-spaces are. Hence, when encoding maps as matrices one has to take a dual perspective from objects to morphisms.

The Yoneda lemma exactly guarantees that the data encoded is enough to recover α\alpha. Namely, knowing the action of α\alpha on the hom-space hom((A⊗B)⊗C,D)→hom(A⊗(B⊗C),D)\text{hom}((A\otimes B)\otimes C,D)\to \text{hom}(A\otimes (B\otimes C),D) for all simple objects DD one gets the full data of a natural transformation hom((A⊗B)⊗C,−)→hom(A⊗(B⊗C),−)\text{hom}((A\otimes B)\otimes C,-)\to \text{hom}(A\otimes (B\otimes C),-) by the semisimplicity of 𝒞\mathcal{C}. Because the Yoneda embedding is fully faithful, we can uniquely determine what map α\alpha must have induces this natural transformation.

Last revised on July 19, 2023 at 16:41:15. See the history of this page for a list of all contributions to it.