The Real K-Theory of Compact Lie Groups

Abstract

Abstract. Let G be a compact, connected, and simply-connected Lie group, equipped with a Lie group involution σ G and viewed as a G-space with the conjugation action. In this paper, we present a description of the ring structure of the (equivariant) KR-theory of (G, σ G ) by drawing on previous results on the module structure of the KR-theory and the ring structure of the equivariant K-theory.

4-manifolds and topological modular forms

“…Therefore, part of the information about the equivariant TMF can be captured by the equivariant KO-theory. The coefficient ring of the latter has representation theoretic description [67][68][69]. Similarly to the non-equivariant case [62], The d-th group G-equivariant KO-group of a point KO d G (pt) = KO d,0 G (pt) can be obtained as the quotient of the Groethendieck groups of G-equivariant modules of the Clifford algebras of dimensions d and d + 1.…”

Real representations of C2-graded groups: The antilinear theory

“…The quotient algebra F[̟] (1+̟ 2 ) is just the tensor product F ⊗ R C. It follows that, similarly to (11), we have an isomorphism of extensions (12)…”

“…Dyson used explicit matrix calculations to describe the structure of antilinear representations [9]. A number of studies over the years made progress on the split case (where G = G ⋊ C 2 , a semidirect product) [3,13,5,21,11]. As pointed out by the referee, a mathematician only interested in the split case still needs the general case because C 2 -graded subgroups of a split C 2 -graded group are not split, in general (see [10] where this fact plays an important role in KR-theory).…”

Equivariant de Rham cohomology: theory and applications

“…[30,Theorem 1.3.4], states that the de Rham cohomology of G is an exterior algebra on generators of odd degree. The fact that the number of generators equals the rank of G can be proven by various means; see [30,Theorem 3.33] for an argument using rational homotopy theory, or [31] for a more elementary argument using the degree of the squaring map G → G; g → g 2 . It follows that the T -action on G by conjugation is equivariantly formal.…”