By Francesco Costantino

We identify a calculus for branched spines of 3-manifolds by way of branched Matveev-Piergallini strikes and branched bubble-moves. We in short point out a few of its attainable purposes within the examine and definition of State-Sum Quantum Invariants.

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All simple CG-modules are one-dimensional if and only if G is abelian. Proof. We have seen that the (finite-dimensional) simple modules for a commutative C-algebra are one-dimensional in the previous section, so suppose that every simple CG-module is one-dimensional. The regular module CG is isomorphic to a direct sum of simple modules by Maschke’s theorem: let f : CG → S1 ⊕ · · · ⊕ Sn be a module isomorphism where each Si is simple. The Si are one-dimensional, and for any ai ∈ Si we have g · ai = ri (g)ai for some group homomorphism ri : G → C× .

This is an isomorphism of vector spaces, and it is a module homomorphism: ev(g · m)(α) = α(g · m) and (g · ev(m))(α) = ev(m)(g −1 · α) = (g −1 · α)(m) = α((g −1 )−1 · m) = α(g · m). 21. If M is irreducible then so is M ∗ . Proof. Suppose M ∗ has a proper non-zero submodule N . Then M ∗ = N ⊕ C for some proper non-zero submodule C of M ∗ by Maschke, and so M∼ = M ∗∗ ∼ = (N ⊕ C)∗ ∼ = N ∗ ⊕ C∗ contradicting irreducibility of M , because dim N ∗ = dim N . 6 Tensor products Let V and W be two finite-dimensional complex vector spaces, and fix bases v1 , .

An ) = ai mi . i This is easily checked to be a surjective module homomorphism. Since the regular module A is semisimple, so is A⊕n , and we can write A⊕n ∼ = S1 ⊕ · · · ⊕ SN for some N . The previous lemma applies to f and shows M = im f is isomorphic to a direct sum of some of the Si and is therefore semisimple. 2 Semisimple algebras are direct sums of matrix algebras Here we prove that every semisimple C-algebra is isomorphic to a direct sum of matrix algebras. The strategy is to relate the structure of A to homA (A, A), use the fact that the regular module is isomorphic to a direct sum of simples, then use the fact that Schur’s lemma lets us understand homomorphisms between simple modules.

### A calculus for branched spines of 3-manifolds by Francesco Costantino

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