# Origin of notion of “split Grothendieck group”?

In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with split Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A}$ and writes $\langle \mathcal{A} \rangle$ for its split Grothendieck group: the free abelian group on objects $\langle A \rangle$ corresponding as usual to isomorphism classes, modulo sums$\langle C \rangle = \langle A \rangle + \langle B \rangle$ corresponding only to the situation $C \cong A \oplus B$.

This is a less familiar situation than the usual Grothendieck group with sums corresponding to short exact sequences which may or may not split.

Where does the notion of split Grothendieck group originate, and why?

This is mostly asked out of curiosity, but I'm also looking for further interesting examples.

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I was going to point out that this concept for finite groups is known as the Green ring. Then I discovered that you were the reviewer for: MR0537496 (80e:20012) Alperin, J. L. Projective modules for SL(2,2n). J. Pure Appl. Algebra 15 (1979), no. 3, 219--234. – Bruce Westbury Jan 7 '13 at 13:36
@Bruce: Yes, the idea is basically the same for Green rings though I had the impression that split Grothendieck groups were defined explicitly in a more geometric framework. I still have to do some rechecking about origins and motivation. – Jim Humphreys Jan 7 '13 at 13:49

These groups are mentioned in [Swan '68 - Algebraic K-Theory, p.69]. He constructs $K_0(\mathcal{A}, S)$ for a class $S$ of exact sequences in $\mathcal{A}$. Take the free abelian group mod the relations from sequences in $S$. For example the class of all exact sequences for the Grothendieck-group $K_0(\mathcal{A})$ or all exact split seqences for the group you mentioned. The name 'split-Grothendieck-group' does not appear.
He generalizes further to $K_0(\mathcal{A}, F)$ for a bifunctor $F:\mathcal{A} \times \mathcal{A} \to \mathcal{A}$ and obtains a generalized Picard-group.