Burnside ring for $G$-vector spaces? It's known that the Burnside ring is the Grothendieck group of a category of $G$-sets, that must be an abelian monoid, but does exist a similar version of the Burnside ring for $G$-vector spaces? **To clarify:** By $G$-vector space I mean a $K$-vector spaces where $G$ acts.

Yes, there's no obstruction to making a similar definition for $G$-vector spaces. The ring you get (if you restrict to finite-dimensional vector spaces) is called the representation ring of $G$ (over $K$). To be clear, the isomorphism classes of finite-dimensional $G$-vector spaces form a semiring, with addition given by direct sum and multiplication given by tensor product. The representation ring is then obtained by formally adjoining additive inverses.