Representation Rings of Lie Superalgebras

Given a Lie superalgebra g, we introduce several variants of the representation ring, built as subrings and quotients of the ring RZ2(g) of virtual g-supermodules, up to (even) isomorphisms. In particular, we consider the ideal R+(g) of virtual g-supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring SR(g) on which the parity reversal operator takes the class of a virtual g-supermodule to its negative. We also construct representation groups built from ungraded g-modules, as well as degree-shifted representation groups using Clifford modules. The full super representation ring SR(g), including all degree shifts, is then a Z2-graded ring in the complex case and a Z8-graded ring in the real case. Our primary result is a six-term periodic exact sequence relating the rings R Z2(g), R • +(g), and SR(g). We first establish a version of it working over an arbitrary (not necessarily algebraically closed) field of characteristic 0. In the complex case, this six-term periodic long exact sequence splits into two three-term sequences, which gives us additional insight into the structure of the complex super representation ring SR(g). In the real case, we obtain the expected 24-term version, as well as a surprising six-term version, of this periodic exact sequence.