  Given a Lie superalgebra \g, we introduce several variants of the
representation ring, built as subrings and quotients of the ring R_{\Z_2}(\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 \Z_{2}-graded ring in the complex case and a
\Z_{8}-graded ring in the real case.
  Our primary result is a six-term periodic exact sequence relating the rings
R^{*}_{\Z_2}(\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.
