KR-theory of compact Lie groups with group anti-involutions

2014/05/31 by Chi-Kwong Fok
Mathematics · Medicine · #Advanced Algebra and Geometry #Algebra over a field #Automorphism #Combinatorics #Equivariant map #Formality #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Lie group #Mathematics #Ophthalmology and Eye Disorders #Pure mathematics #Ring (chemistry) #Symmetric space #math.AT #math.KT #msc:19L47 #msc:57T10

paper · pdf · doi:10.1016/j.topol.2015.10.008

published as Topology and its Applications (2016), pp. 50-59 · 11 pages. Accepted by Topology and its Applications

arxiv created 2015/10/17 · openalex publication_date 2015/11/10 · arxiv updated 2015/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let G be a compact, connected, and simply-connected Lie group, equipped with an anti-involution a_G which is the composition of a Lie group involutive automorphism σ_G and the group inversion. We view (G, a_G) as a Real (G, σ_G)-space via the conjugation action. In this note, we exploit the notion of Real equivariant formality discussed in \citeFo to compute the ring structure of the equivariant KR-theory of G. In particular, we show that when G does not have Real representations of complex type, the equivariant KR-theory is the ring of Grothendieck differentials of the coefficient ring of equivariant KR-theory over the coefficient ring of ordinary KR-theory, thereby generalizing a result of Brylinski-Zhang's (\citeBZ) for the complex K-theory case.

Citations

• Equivariant K-theory and completion
• The periodicity theorem for the classical groups and some of its applications
• BOTT PERIODICITY AND THE INDEX OF ELLIPTIC OPERATORS
• Surjectivity for Hamiltonian G-spaces in K-theory
• Vector bundles and homogeneous spaces