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Poincaré Lie algebra

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Contents

Idea

The Poincaré Lie algebra 𝔦𝔰𝔬(d1,1)\mathfrak{iso}(d-1,1) is the semidirect product of the special orthogonal Lie algebra 𝔰𝔬(d1,1)\mathfrak{so}(d-1,1) and the abelian translation Lie algebra d1,1\mathbb{R}^{d-1,1}.

The corresponding Lie group is the Poincaré group.

Definition

The CE-algebra

The Chevalley-Eilenberg algebra CE(𝔦𝔰𝔬(d1,1))CE(\mathfrak{iso}(d-1,1)) is generated from d,1\mathbb{R}^{d,1} and 2 d,1\wedge^2 \mathbb{R}^{d,1}. For {t a}\{t_a\} the standard basis of d1,1\mathbb{R}^{d-1,1} we write {ω ab}\{\omega^{a b}\} and {e a}\{e^a\} for these generators. With (η ab)(\eta_{a b}) the components of the Minkowski metric we write

ω a b:=ω acη cb. \omega^{a}{}_b := \omega^{a c}\eta_{c b} \,.

In terms of this the CE-differential that defines the Lie algebra structure is

d CE:ω ab=ω a cω cb d_{CE} : \omega^{a b} = \omega^a{}_c \wedge \omega^{c b}
d CE:e aω a bt b d_{CE} : e^a \mapsto \omega^{a}{}_b \wedge t^b

Properties

Cohomology

We discuss some elements in the Lie algebra cohomology of 𝔦𝔰𝔬(d1,1)\mathfrak{iso}(d-1,1).

The canonical degree-3 𝔰𝔬(d1,1)\mathfrak{so}(d-1,1)-cocycle is

ω a bω b cω c aCE(𝔦𝔰𝔬(d1,1)). \omega^a{}_b \wedge \omega^b{}_c \wedge \omega^c{}_a \in CE(\mathfrak{iso}(d-1,1)) \,.

The volume cocycle is the volume form

vol=ϵ a 1a de a 1e a dCE(𝔦𝔰𝔬(d1,1)). vol = \epsilon_{a_1 \cdots a_{d}} e^{a_1} \wedge \cdots \wedge e^{a_d} \in CE(\mathfrak{iso}(d-1,1)) \,.

Invariant polynomials and Chern-Simons elements

With the basis elements (e a,ω ab)(e^a, \omega^{a b}) as above, denote the shifted generators of the Weil algebra W(𝔦𝔰𝔬(d1,1))W(\mathfrak{iso}(d-1,1)) by θ a\theta^a and r abr^{a b}, respectively.

We have the Bianchi identity

d W:r abω acR c dR acω c b d_W : r^{a b} \mapsto \omega^{a c} \wedge R_c{}^d - R^{a c} \wedge \omega_c{}^b

and

d W:θ aω a bθ bR a be b. d_W : \theta^a \mapsto \omega^a{}_b \theta^b - R^{a}{}_b e^b \,.

The element η abθ aθ bW(𝔦𝔰𝔬(d1,1))\eta_{a b} \theta^a \wedge \theta^b \in W(\mathfrak{iso}(d-1,1)) is an invariant polynomial. A Chern-Simons element for it is cs=η abe aθ bcs = \eta_{a b} e^a \wedge \theta^b. So this transgresses to the trivial cocycle.

Another invariant polynomial is r abr abr^{a b} \wedge r_{a b}. This is the Killing form of 𝔰𝔬(d1,1)\mathfrak{so}(d-1,1). Accordingly, it transgresses to a multiple of ω a bω b cω c a\omega^a{}_b \wedge \omega^b{}_c \wedge \omega^c{}_a.

This is the first in an infinite series of Pontryagin invariant polynomials

P n:=r a 1 a 2r a 2 a 3r a n a 1. P_n := r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_n}{}_{a_1} \,.

There is also an infinite series of mixed invariant polynomials

C 2n+2:=θ a 1r a 1 a 2r a 2 a 3r a n1 a nθ a n. C_{2n + 2} := \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge \theta^{a_n} \,.

Chern-Simons elements for these are

B 2n+1:=θ a 1r a 1 a 2r a 2 a 3r a n1 a ne a n. B_{2n + 1} := \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge e^{a_n} \,.

Lie algebra valued forms

A Lie algebra-valued form with values in 𝔦𝔰𝔬(d1,1)\mathfrak{iso}(d-1,1)

Ω (X)W(𝔦𝔰𝔬(d1,1)):(E,Ω) \Omega^\bullet(X) \leftarrow W(\mathfrak{iso}(d-1,1)) : (E,\Omega)

is

The curvature 2-form (T,R)(T, R) consists of

If the torsion vanishes, then Ω\Omega is a Levi-Civita connection for the metric E aE bη abE^a \otimes E^b \eta_{a b} defined by EE.

The volume form is the image of the volume cocycle

Ω (X)(E,Ω)W(𝔦𝔰𝔬(d1,1))volW(b d1):vol(E). \Omega^\bullet(X) \stackrel{(E,\Omega)}{\leftarrow} W(\mathfrak{iso}(d-1,1)) \stackrel{vol}{\leftarrow} W(b^{d-1} \mathbb{R}) : vol(E) \,.

We have

vol(E)=ϵ a 1a dE a 1E a d. vol(E) = \epsilon_{a_1 \cdots a_d} E^{a_1} \wedge \cdots \wedge E^{a_d} \,.

If the torsion vanishes, this is indeed a closed form.

Revised on August 22, 2011 16:43:39 by Urs Schreiber (82.113.99.25)