Mathematics – Representation Theory
Scientific paper
2004-03-12
K-Theory 36 (2005), no. 1-2, 115-168
Mathematics
Representation Theory
36 pages, 1 figure, uses Payl Taylor's diagrams package. Updated with minor corrections
Scientific paper
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.
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