nLab equivariant K-theory

Contents

Context

Representation theory

representation theory

geometric representation theory

Contents

Idea

Equivariant K-theory is the equivariant cohomology version of the generalized cohomology theory K-theory.

To the extent that K-theory is given by equivalence classes of virtual vector bundles (topological K-theory, operator K-theory), equivariant K-theory is given by equivalence classes of virtual equivariant bundles or generalizations to noncommutative topology thereof, as in equivariant operator K-theory, equivariant KK-theory.

Properties

Bott periodicity

The Bott periodicity of plain K-theory generalizes to equivariant K-theory:

Complex equivariant K-theory is invariant under smashing with representation spheres of complex representations (Atiyah 68, Theorem 4.3), while real equivariant K-theory is invariant under smashing with representation spheres of real 8d reps with spin structure (Atiyah 68, Theorem 6.1).

Review in Karoubi 05, Section 5.

Complex orientation

Equivariant complex K-theory is an equivariant complex oriented cohomology theory (Greenlees 01, Sec. 10).

Proposition

(equivariant K-theory of projective G-space)

For $G$ an abelian compact Lie group, let

$\underset{i}{\oplus} \, \mathbf{1}_{V_i} \;\; \in \;\; G Representations_{\mathbb{C}}^{fin}$

The $G$-equivariant K-theory ring $K_G(-)$ of the corresponding projective G-space $P(-)$ is the following quotient ring of the polynomial ring in one variable $L$ over the complex representation ring $R(G)$ of $G$:

(1)$K_G \Big( P \big( \underset{i}{\oplus} \, \mathbf{1}_{V_i} \big) \Big) \;\; \simeq \;\; R(G) \big[ L \big] \big/ \underset{i}{\prod} \big( 1 - 1_{{}_{V_i}} L \big) \,,$

where

Corollary

(equivariant complex orientation of equivariant K-theory)

For $G$ an abelian compact Lie group and $\mathbf{1}_V \,\in\, G Representations_{\mathbb{C}}^{fin}$ a complex 1-dimensional linear representation, the corresponding representation sphere is the projective G-space $S^{\mathbf{1}_V} \,\simeq\, P\big( \mathbf{1}_V \oplus \mathbf{1} \big)$ (this Prop.) and so, by Prop. ,

\begin{aligned} \widetilde K_G \big( S^{\mathbf{1}_V} \big) & \simeq\, K_G \big( P( \mathbf{1}_V \oplus \mathbf{1} ) ; \, \underset{ \simeq \, pt }{ \underbrace{ P( \mathbf{1} ) } } \big) \\ & \simeq ker \Big( R(G)\big[L\big] \big/ (1 - 1_{{}_{V}} L) (1 - L) \longrightarrow \underset{ \simeq \, R(G) }{ \underbrace{ R(G)\big[L\big] \big/ (1 - L) } } \Big) \\ & \simeq (1 - L) \cdot R(G)\big[L\big] \big/ (1 - 1_{{}_{V}} L) (1 - L) \end{aligned}

is generated by the Bott element $(1 - L)$ over $P\big( \mathbf{1}_V \oplus \mathbf{1} \big)$. By the nature of the tautological equivariant line bundle, this Bott element is the restriction of that on infinite complex projective G-space $P\big(\mathcal{U}_G\big)$. The latter is thereby exhibited as an equivariant complex orientation in equivariant complex K-theory.

Relation to operator K-theory of crossed product algebras

The Green-Julg theorem identifies, under some conditions, equivariant K-theory with operator K-theory of corresponding crossed product algebras.

Relation to representation theory

Equivariant $KU$ and the complex representation ring

The representation ring of $G$ over the complex numbers is the $G$-equivariant K-theory of the point, or equivalently by the Green-Julg theorem, if $G$ is a compact Lie group, the operator K-theory of the group algebra (the groupoid convolution algebra of the delooping groupoid of $G$):

(2)$R_{\mathbb{C}}(G) \simeq KU^0_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G)) \,.$

The first isomorphism here follows immediately from the elementary definition of equivariant topological K-theory, since a $G$-equivariant vector bundle over the point is manifestly just a linear representation of $G$ on a complex vector space.

Chern classes of linear representations

Under the identification (2) and the Atiyah-Segal completion map

$R_{\mathbb{C}}(G) \simeq KU_G^0(\ast) \overset{ \widehat{(-)} }{\longrightarrow} KU(BG)$

one may ask for the Chern character of the K-theory class $\widehat{V} \in KU(B G)$ expressed in terms of the actual character of the representation $V$. For more see at Chern class of a linear representation.

There is a closed formula at least for the first Chern class (Atiyah 61, appendix):

For 1-dimensional representations $V$ their first Chern class $c_1(\widehat{V}) \in H^2(B G, \mathbb{Z})$ is their image under the canonical isomorphism from 1-dimensional characters in $Hom_{Grp}(G,U(1))$ to the group cohomology $H^2_{grp}(G, \mathbb{Z})$ and further to the ordinary cohomology $H^2(B G, \mathbb{Z})$ of the classifying space $B G$:

$c_1\left(\widehat{(-)}\right) \;\colon\; Hom_{Grp}(G, U(1)) \overset{\simeq}{\longrightarrow} H^2_{grp}(G,\mathbb{Z}) \overset{\simeq}{\longrightarrow} H^2(B G, \mathbb{Z}) \,.$

More generally, for $n$-dimensional linear representations $V$ their first Chern class $c_1(\widehat V)$ is the previously defined first Chern-class of the line bundle $\widehat{\wedge^n V}$ corresponding to the $n$-th exterior power $\wedge^n V$ of $V$. The latter is a 1-dimensional representation, corresponding to the determinant line bundle $det(\widehat{V}) = \widehat{\wedge^n V}$:

$c_1(\widehat{V}) \;=\; c_1(det(\widehat{V})) \;=\; c_1( \widehat{\wedge^n V} ) \,.$

More explicitly, via the formula for the determinant as a polynomial in traces of powers (see there) this means that the first Chern class of the $n$-dimensional representation $V$ is expressed in terms of its character $\chi_V$ as

(3)$c_1(V) = \chi_{\left(\wedge^n V\right)} \;\colon\; g \;\mapsto\; \underset{ { k_1,\cdots, k_n \in \mathbb{N} } \atop { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n } }{\sum} \underoverset{ l = 1 }{ n }{\prod} \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l ! } \left(\chi_V(g^l)\right)^{k_l}$

For example, for a representation of dimension $n = 2$ this reduces to

$c_1(V) = \chi_{V \wedge V} \;\colon\; g \;\mapsto\; \frac{1}{2} \left( \left( \chi_V(g)\right)^2 - \chi_V(g^2) \right)$

$\,$

Equivariant $KO$ and the real representation ring

An isomorphism analogous to (2) identifies the $G$-representation ring over the real numbers with the equivariant orthogonal $K$-theory of the point in degree 0:

$R_{\mathbb{R}}(G) \;\simeq\; KO_G^0(\ast) \,.$

But beware that equivariant KO, even of the point, is much richer in higher degree (Wilson 16, remark 3.34).

In fact, equivariant KO-theory of the point subsumes the representation rings over the real numbers, the complex numbers and the quaternions:

$KO_G^n(\ast) \;\simeq\; \left\{ \array{ 0 &\vert& n = 7 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{R}}(G) &\vert& n = 6 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 5 \\ R_{\mathbb{H}}(G) \phantom{/ R_{\mathbb{R}}(G) } &\vert& n = 4 \\ 0 &\vert& n = 3 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{H}}(G) &\vert& n = 2 \\ R_{\mathbb{R}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 1 \\ R_{\mathbb{R}}(G) \phantom{/ R_{\mathbb{R}}(G)} &\vert& n =0 } \right.$

Accordingly the construction of an index (push-forward to the point) in equivariant K-theory is a way of producing $G$-representations from equivariant vector bundles. This method is also called Dirac induction.

Specifically, applied to equivariant complex line bundles on coadjoint orbits of $G$, this is a K-theoretic formulation of the orbit method.

Relation to K-theory of homotopy quotient spaces (Borel constructions)

For $X$ a topological space equipped with a $G$-action for $G$ a topological group, write $X//G$ for the homotopy type of the corresponding homotopy quotient. A standard model for this is the Borel construction

$X//G \simeq (X \times EG)/G \,.$

The ordinary topological K-theory of $X//G$ is also called the Borel-equivariant K-theory of $X$, denoted

$K_G^{Bor}(X) \coloneqq K(X//G) \,.$

There is a canonical map

$K_G(X) \to K_G^{Bor}(X)$

from the genuine equivariant K-theory to the Borel equivariant K-theory. In terms of the Borel construction this is given by the composite

$K_G(X) \to K_G(X \times E G) \simeq K((X \times E G) / G ) \simeq K_G^{Bor}(X) \,,$

where the first map is pullback along the projection $X \times E G \to X$ and the first equivalence holds because the $G$-action on $X \times E G$ is free.

This map from genuine to Borel equivariant K-theory is not in general an isomorphism.

Specifically for $X$ the point, then $K_G(\ast) \simeq R(G)$ is the representation ring and $K_G^{Bor}(\ast) \simeq K(B G)$ is the topological K-theory of the classifying space $B G$ of $G$-principal bundles. In this case the above canonical map is of the form

$R(G) \to K(B G) \,.$

This is never an isomorphism, unless $G$ is the trivial group. But the Atiyah-Segal completion theorem says that the map identifies $K(B G)$ as the completion of $R(G)$ at the ideal of virtual representations of rank 0.

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point $\ast$
cohomology
of classifying space $B G$
(equivariant)
ordinary cohomology
HZBorel equivariance
$H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant)
complex K-theory
KUrepresentation ring
$KU_G(\ast) \simeq R_{\mathbb{C}}(G)$
Atiyah-Segal completion theorem
$R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant)
complex cobordism cohomology
MU$MU_G(\ast)$completion theorem for complex cobordism cohomology
$MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant)
algebraic K-theory
$K \mathbb{F}_p$representation ring
$(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$
Rector completion theorem
$R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant)
stable cohomotopy
$K \mathbb{F}_1 \overset{\text{Segal 74}}{\simeq}$ SBurnside ring
$\mathbb{S}_G(\ast) \simeq A(G)$
Segal-Carlsson completion theorem
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

Rationalization

Incarnations of rational equivariant K-theory:

cohomology theorydefinition/equivalence due to
$\simeq K_G^0\big(X; \mathbb{C} \big)$rational equivariant K-theory
$\simeq H^{ev}\Big( \big(\underset{g \in G}{\coprod} X^g\big)/G; \mathbb{C} \Big)$delocalized equivariant cohomologyBaum-Connes 89, Thm. 1.19
$\simeq H^{ev}_{CR}\Big( \prec \big( X \!\sslash\! G\big);\, \mathbb{C} \Big)$Chen-Ruan cohomology
of global quotient orbifold
Chen-Ruan 00, Sec. 3.1
$\simeq H^{ev}_G\Big( X; \, \big(G/H \mapsto \mathbb{C} \otimes Rep(H)\big) \Big)$Bredon cohomology
with coefficients in representation ring
Ho88 6.5+Ho90 5.5+Mo02 p. 18,
Mislin-Valette 03, Thm. 6.1,
Szabo-Valentino 07, Sec. 4.2
$\simeq K_G^0\big(X; \mathbb{C} \big)$rational equivariant K-theoryLück-Oliver 01, Thm. 5.5,
Mislin-Valette 03, Thm. 6.1

Equivariant Chern-character

There is an equivariant Chern character map from equivariant K-theory to rational equivariant ordinary cohomology above

References

General

The idea of equivariant topological K-theory and the Atiyah-Segal completion theorem goes back to

and for algebraic K-theory to

• Robert Thomason, Algebraic K-theory of group scheme actions, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563

with construction via permutative categories in

Introductions and surveys:

• John Greenlees, Equivariant version of real and complex connective K-theory, Homology Homotopy Appl. Volume 7, Number 3 (2005), 63-82. (Euclid:1139839291)

• N. C. Phillips, Equivariant K-theory for proper actions, Pitman Research Notes in Mathematics Series 178, Longman, Harlow, UK, 1989.

• Bruce Blackadar, section 11 of K-Theory for Operator Algebras

• Alexander Merkujev, Equivariant K-theory (pdf)

• Zachary Maddock, An informal discourse on equivariant K-theory (pdf)

• Dylan Wilson, Equivariant K-theory, 2016 (pdf, pdf)

Textbook accounts:

On Bott periodicity in equivariant K-theory:

Basic computations:

• Yimin Yang, On the Coefficient Groups of Equivariant K-Theory, Transactions of the American Mathematical Society

Vol. 347, No. 1 (Jan., 1995), pp. 77-98 (jstor:2154789)

• Max Karoubi, Equivariant K-theory of real vector spaces and real vector bundles, Topology and its Applications, 122, (2002) 531-456 (arXiv:math/0509497)

The equivariant Chern character is discussed in

• German Stefanich, Chern Character in Twisted and Equivariant K-Theory (pdf)

Discussion relating to K-theory of homotopy quotients/Borel constructions is in

Discussion of the twisted ad-equivariant K-theory of compact Lie groups:

Discussion of K-theory of orbifolds is for instance in section 3 of

Discussion of differential K-theory of orbifolds:

Discussion of combined twisted and equivariant and real K-theory

Discussion in the context of equivariant complex oriented cohomology theory:

For formulation and proof of the McKay correspondence:

Representing equivariant spectrum

That $G$-equivariant topological K-theory is represented by a topological G-space is due to:

This is enhanced to a representing naive G-spectrum in:

In its incarnation (under Elmendorf's theorem) as a Spectra-valued presheaf on the $G$-orbit category this is discussed in

• James Davis, Wolfgang Lück, Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory, K-Theory 15:201–252, 1998 (pdf)

Review:

Rational equivariant K-theory

Discussion of rational equivariant K-theory (see also the references at equivariant Chern character):

and with emphasis of commutative ring-structure:

For D-brane charge on orbifolds

The proposal that D-brane charge on orbifolds is given by equivariant K-theory (see at D-brane charge quantization in K-theory) goes back to

but it was pointed out that only a subgroup or quotient group of equivariant K-theory can be physically relevant, in

For further references see at fractional D-brane.

• Atiyah 61, Appendix

• L. Evens, On the Chern classes of representations of finite groups, Trans. Am. Math. Soc. 115, 180-193 (1965) (doi:10.2307/1994264)

• F. Kamber, Ph. Tondeur, Flat Bundles and Characteristic Classes of Group-Representations, American Journal of Mathematics Vol. 89, No. 4 (Oct., 1967), pp. 857-886 (doi:10.2307/2373408)

• Peter Symonds, A splitting principle for group representations, Comment. Math. Helv. (1991) 66: 169 (doi:10.1007/BF02566643)

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