Tensor and Fusion Categories — Examples

By [F1] and [F2], the finite-dimensional subcategory is a rigid $k$-linear monoidal category.

Every finite-dimensional vector space is a finite direct sum of copies of $k$, so $k$ is the only simple isomorphism class and the category is finite semisimple. Its unit has endomorphism algebra $k$.

These clauses are exactly those of [F3], so ${\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_k^{\mathrm{f}\mathrm{d}}$ is fusion.

Dimension is additive on short exact sequences: a basis of a subspace, together with lifts of a basis of the quotient, is a basis of the middle space. Thus $d([V])={\dim }_{k}V$ respects every relation in [F2] and induces a group homomorphism $d:K_0\to \mathbb{Z}$. Split direct-sum sequences and [F1] give $[V]=({\dim }_{k}V)[k]$, including $[0]=0$. Hence $e:\mathbb{Z}\to K_0$, $m\mapsto m[k]$, satisfies $de=\mathrm{id}$ and $ed=\mathrm{id}$, proving the asserted additive bijection on all classes, including negative virtual classes.

Transport integer multiplication along this bijection: set $a\cdot b=e(d(a)d(b))$. This is a well-defined unital ring structure, with unit $e(1)=[k]$. Tensor products of finite bases give ${\dim }_k(V{\otimes }_{k}W)={\dim }_{k}V{\dim }_{k}W$, so $[V]\cdot [W]=e({\dim }_k(V{\otimes }_{k}W))=[V{\otimes }_{k}W]$. Thus the intended tensor multiplication in [F2] descends to the quotient; its bilinear extension is unique because object classes generate the group. The map $d$ is therefore an isomorphism of rings, with zero and unit preserved.

Let objects be $n\times n$ matrices $(V_{ij})$ and set $(V\otimes W{)}_{ij}={⨁}_r{V}_{ir}{\otimes }_k{W}_{rj}$. The matrix $I$ with $k$ on the diagonal and $0$ off it is a unit.

Entrywise semisimplicity and finite direct sums from [F1] give the finite semisimple rigid structure required in [F2].

But $I=E_{11}\oplus \cdots \oplus E_{nn}$ is a nontrivial direct sum when $n>1$. Hence this is multifusion, not fusion.

The unit isomorphism $k{\otimes }_{k}k\cong k$ gives $[k][k]=[k]$.

Comparing this with the defining expansion in [F2] for the supplied one-element family yields $N_{11}^1=1$.