# G* Modules

# From Geometry to Algebra

Let GG be any Lie group, let g\mfg be its Lie algebra, and g~\tilde{\mfg} be the corresponding Lie superalgebra.

Definition

A GG^{\ast} algebra is a commutative superalgebra AA, together with a representation ρ\rho of GG as automorphisms of AA and an action of g~\tilde{\mfg} as (super)derivations of AA which are consistent in the sense that

ddtρ(exptξ)t=0=Lξρ(a)Lξρ(a1)=LAdaξρ(a)ıξρ(a1)=ıAdaξρ(a)dρ(a1)=d \begin{aligned} \frac{d}{dt} \rho(\exp t\xi)|_{t=0} &= L_{\xi} \\ \rho(a) L_{\xi}\rho(a^{-1}) &= L_{\Ad_{a}}\xi \\ \rho(a) \imath_{\xi} \rho(a^{-1}) &= \imath_{\Ad_{a}}\xi \\ \rho(a) d \rho(a^{-1}) &= d \end{aligned}

for all aGa \in G, ξg\xi \in \mfg.

A GG^{\ast} module is a supervector space AA together with a linear representation of GG on AA and a homomorphism g~End(A)\tilde{\mfg} \ra \End(A) such that the conditions above hold. So a GG^{\ast} algebra is a commutative superalgebra which is a GG^{\ast} module with the additional condition that GG acts as algebra automorphisms and g~\tilde{\mfg} acts as superderivations.

In order to make the set of GG^{\ast} modules and the set of GG^{\ast} algebras into category, we must first define what is meant by a morphism. So let AA and BB be GG^{\ast} modules and let f:ABf : A \ra B be a (continuous) linear map.

Definition

We say that ff is a morphism of GG^{\ast} modules if for all xAx \in A, aGa \in G, ξg\xi \in \mfg, we have

[ρ(a),f]=0[Lξ,f]=0[ıξ,f]=0[d,f]=0. \begin{aligned} [\rho(a), f] &= 0 \\ [L_{\xi}, f] &= 0 \\ [\imath_{\xi}, f] &= 0 \\ [d,f] &= 0. \end{aligned}

If, for all ii,

f:AiBi+k f: A_{i} \ra B_{i+k}

we say that ff has degree kk, with similar notation for the (Z/2Z)(\ZZ/2\ZZ)-graded case. We say that a morphism of degree k\bfk is even if k=0\bfk = \bf0 and odd if k=1\bfk = \bf1.

If the morphism is even (especially if it is of degree zero which will often be the case), one can say that ff preserves the GG^{\ast} action, or that the operators ρ,Lξ,ıξ\rho, L_{\xi}, \imath_{\xi}, and dd are equivariant with respect to ff.

The composition of two GG^{\ast} morphisms is again a GG^{\ast} module morphism, and hence the set of GG^{\ast} module morphisms is a category. Further, we define a morphism between GG^{\ast} algebras to be a map f:ABf : A \ra B which is an algebra morphism, and satisfies the four equations above. This makes the set of GG^{\ast} algebras into a category too. Similar analogous definitions exist of Z\ZZ-graded GG^{\ast} modules, algebras, and morphisms.

If we have a GG-action on a manifold MM, then Ω(M)\W(M) is a GG^{\ast} algebra in a canonical way. If MM and NN are GG-manifolds and F:MNF:M \ra N is a GG-equivariant smooth map, then the pullback map F:Ω(M)Ω(N)F^{\ast} : \W(M) \ra \W(N) is a morphism of GG^{\ast} algebras. Thus the category of GG^{\ast} algebras can be considered as an algebraic generalisation of the category of GG-manifolds.

# Cohomology

By definition, the element dd acts as a derivation of degree +1+1 with d2=0d^{2} = 0 on AA. So AA is a cochain complex. We define H(A):=H(A,d)H(A) := H(A,d) to be the cohomology of AA relative to the differential dd. In the case when A=Ω(M)A = \W(M), de Rham's theorem says that this is equal to H(M)H^{\ast}(M).

Remark

(to be added - page 19)

# Acyclicity

If MM is contractible, the de Rham complex (Ω(M),d)(\W(M), d) is acyclic, i.e., A=Ω(M)A = \W(M) satisfies

Hk(A,d)={F,k=0,0,k0, H^{k}(A,d) = \begin{cases} \FF,\quad &k = 0, \\ 0,\quad &k \neq 0, \end{cases}

where F\FF is the ground field, which is C\CC in our case. We take this as the definition of acyclicity for a general AA.

# Chain Homotopies

Definition

Let AA and BB be two GG^{\ast} modules. A linear map

Q:AB Q : A \ra B

is called a chain homotopy if it is odd, GG-equivariant, and satisfies

ıξQ+Qıξ=0,for all ξg. \imath_{\xi} Q + Q \imath_{\xi} = 0,\qquad \text{for all } \xi \in \mfg.

If AA and BB are Z\ZZ-graded (as is typical) we require that QQ be of degree 1-1 in the Z\ZZ-gradation. The GG-equivariance implies that

LξQQLξ=0,for all ξg. L_{\xi} Q - Q L_{\xi} = 0,\qquad \text{for all } \xi \in \mfg.

Proposition

If Q:ABQ : A \ra B is a chain homotopy, then

τ:=dQ+Qd \tau := dQ + Qd

is a morphism of GG^{\ast} modules.

Definition

We say that two morphisms τ0,τ1:AB\tau_{0}, \tau_{1} : A \ra B are chain homotopic (notation: τ0τ1\tau_{0} \simeq \tau_{1}) if there is a chain homotopy Q:ABQ : A \ra B such that

τ1τ0=Qd+dQ. \tau_{1} - \tau_{0} = Qd + dQ.

Remark

This implies that the induced maps on cohomology are equal:

τ0τ1τ0=τ1. \tau_{0} \simeq \tau_{1} \implies \tau_{0\ast} = \tau_{1\ast}.

Definition

(finish chain homotopy relative to... - page 22).

# Free and Locally Free Actions

Definition

An action of GG on MM is said to be locally free if the corresponding infinitesimal action of g\mfg is free, i.e., if, for every ξ0g\xi \neq 0 \in \mfg, the vector field ξM\xi_{M}^{-} generating the one-parameter group texp(tξ)t \mapsto \exp(-t\xi) of transformations on MM is nowhere vanishing.

If the action is locally free, we can find linear differential forms θ1,,θn\theta^{1},\ldots, \theta^{n} on MM which are everywhere dial to our basis ξ1,,ξn\xi_{1},\ldots, \xi_{n} in the sense that

ıaθb=δab. \imath_{a} \theta^{b} = \delta_{a}^{b}.

Conversely, if we have a GG-action on a manifold on which there exist forms θa\theta^{a} satisfying the above equation, then the action is locally free.

A linear differential form ω\w is called horizontal if it satisfies

ıaω=0,a=1,,n. \imath_{a}\w = 0,\qquad a = 1,\ldots, n.