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action Lie algebroid

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\infty-Lie theory

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Contents

Definition

For VV a space, GG a group and ρ:G×VV\rho : G\times V \to V a action of GG on VV, we have the corresponding action groupoid. If everything is sufficiently smooth, this is a Lie groupoid denoted V// ρGV//_\rho G.

The action Lie algebroid of ρ\rho is the Lie algebroid that corresponds to this Lie groupoid (under Lie integration).

The Chevalley-Eilenberg algebra of an action Lie algebroid is in physics known as a BRST complex.

Details

Let GG be a Lie group, VV a smooth manifold and ρ:G×VV\rho : G \times V \to V a smooth action. Write V//GV//G for the corresponding action groupoid, itself a Lie groupoid. The Lie algebroid Lie(V//G)Lie(V//G) corresponding to this is the action Lie algebroid.

The Chevalley-Eilenberg algebra of the action Lie algebroid is

CE(Lie(V//G))=( C (V) 𝔤 *,d ρ), CE(Lie(V//G)) = (\wedge^\bullet_{C^\infty(V)} \mathfrak{g}^*, d_{\rho}) \,,

where the differential acts on functions fC (V)f \in C^\infty(V) by

d ρ:fρ()() *fC (V)𝔤 *. d_\rho : f \mapsto \rho(-)(-)^* f \in C^\infty(V)\otimes \mathfrak{g}^* \,.

Explicitly, for t𝔤t \in \mathfrak{g} this sends ff to the function (d ρf)(t)(d_\rho f)(t) which is the derivative along tT eGt \in T_e G of the function G×VρVfG \times V \stackrel{\rho}{\to}V \stackrel{f}{\to} \mathbb{R}.

Even more explicitly, if we choose local coordinates {v k}: dimVV\{v^k\} : \mathbb{R}^{dim V} \to V on a patch, and choose a basis {t a}\{t^a\} of 𝔤 *\mathfrak{g}^* then we have that restricted to this patch the differential is on generators by

d ρ:fρ μ at a kf d_\rho : f \mapsto \rho^\mu{}_a t^a \wedge \partial_k f
d ρ:t a12C a bct bt c. d_\rho : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c \,.

Specifically for VV a finite dimensional vector space, ρ:G\rho : G a linear action, {v k}\{v^k\} a choice of basis of that vector space and ff a linear function f=f kv kf= f_k v^k , we have that (f k:= kf) dimV(f_k := \partial_k f) \in \mathbb{R}^{dim V} are the components vector of the dual vector given by VV in this basis, and the above gives the matrix multiplication form of the action

d ρ:v kt aρ a k lv l. d_\rho : v^k \mapsto t^a \rho_a{}^k{}_l v^l \,.

Notice for completeness that the equation (d ρ) 2=0(d_\rho)^2 = 0 is equivalent to the Jacobi identity of the Lie bracket and the action property of ρ\rho:

d ρd ρv k=(t at bρ a k rρ b r l12C a bct bt cρ a k l)v l. d_\rho d_\rho v^k = (t^a \wedge t^b \rho_a{}^k{}_r \rho_b{}^r{}_l - \frac{1}{2}C^a{}_{b c}t^b \wedge t^c \rho_a{}^k{}_l ) v^l \,.

These local formulas shall be useful below for recognizing from our general abstract definition of covariant derivative the formulas traditionally given in the literature. For that notice that in the above local coordinates further restricting attention to linear actions, the Weil algebra of the action Lie algebroid is given by

W(Lie(V//G))=( C ( dimV) (Γ(T * dimV)𝔤 *𝔤 *[1]),d W ρ) W(Lie(V//G)) = (\wedge^\bullet_{C^\infty(\mathbb{R}^{dim V})} ( \Gamma(T^* \mathbb{R}^{dim V}) \oplus \mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W_\rho})

where the differential is given on generators by

d W ρ:v kρ a k lt av l+d dRv k d_{W_\rho} : v^k \mapsto \rho_a{}^k{}_l t^a \wedge v^l + d_{dR} v^k
d W ρ:t a12C a bct bt c+r a d_{W_\rho} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a

and where the uniquely induced differential on the shifted generators – the one encoding Bianchi identities – is

d W ρ:d dRv kρ a k kr av lρ a k lt ad dRv l d_{W_\rho} : d_{dR} v^k \mapsto \rho_a{}^k{}_k r^a \wedge v^l - \rho_a{}^k{}_l t^a \wedge d_{dR} v^l

and

d W:r aC a bct br c. d_{W} : r^a \mapsto C^a{}_{b c} t^b \wedge r^c \,.

Notice that we may identify the delooping Lie groupoid BG\mathbf{B}G of GG with the action groupoid of the trivial action on the point, BG*//G\mathbf{B}G \simeq *//G. On Lie algebroids this morphism is dually the inclusion

CE(Lie(V//G))CE(𝔤) CE(Lie(V//G)) \leftarrow CE(\mathfrak{g})

that is the identity on 𝔤 *\mathfrak{g}^*.

Applications

Revised on May 27, 2013 22:14:24 by Urs Schreiber (82.113.99.197)