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Poincaré Lie algebra

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\infty-Lie theory

∞-Lie theory (higher geometry)

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Contents

Idea

The Poincaré Lie algebra 𝔦𝔰𝔬( d1,1)\mathfrak{iso}(\mathbb{R}^{d-1,1}) is the Lie algebra of the isometry group of Minkowski spacetime: the Poincaré group. This happens to be the semidirect product of the special orthogonal Lie algebra 𝔰𝔬(d1,1)\mathfrak{so}(d-1,1) with the the abelian translation Lie algebra d1,1\mathbb{R}^{d-1,1}.

Definition

Definition

For dd \in \mathbb{N}, write d1,1\mathbb{R}^{d-1,1} for Minkowski spacetime, regarded as the inner product space whose underlying vector space is d\mathbb{R}^d and equipped with the bilinear form given in the canonical linear basis of d\mathbb{R}^d by

ηdiag(1,+1,+1,,+1). \eta \coloneqq diag(-1,+1,+1, \cdots, +1) \,.

The Poincaré group Iso( d1,1)Iso(\mathbb{R}^{d-1,1}) is the isometry group of this inner product space. The Poincaré Lie algebra 𝔦𝔰𝔬( d1,1)\mathfrak{iso}(\mathbb{R}^{d-1,1}) is the Lie algebra of this Lie group (its Lie differentiation)

𝔦𝔰𝔬( d1,1)Lie(Iso( d1,1)). \mathfrak{iso}(\mathbb{R}^{d-1,1}) \coloneqq Lie(Iso(\mathbb{R}^{d-1,1})) \,.
Remark

The Poincaré group is the semidirect product group

Iso( d1,1) d1,1O(d1,1) Iso(\mathbb{R}^{d-1,1}) \simeq \mathbb{R}^{d-1,1} \rtimes O(d-1,1)

of the Lorentz group O(d1,1)O(d-1,1) (the group of linear isometries of Minkowski spacetime) with the d\mathbb{R}^d regarded as the translation group along itself, via the defining action.

Accordingly, the Poincaré Lie algebra is the semidirect product Lie algebra

𝔦𝔰𝔬( d1,1) d1,1𝔰𝔬 +(d1,1) \mathfrak{iso}(\mathbb{R}^{d-1,1}) \simeq \mathbb{R}^{d-1,1} \rtimes \mathfrak{so}^+(d-1,1)

of the abelian Lie algebra on d\mathbb{R}^d with the (orthochronous) special orthogonal Lie algebra 𝔰𝔬(d1,1)\mathfrak{so}(d-1,1).

Proposition

For {P a}\{P_a\} the canonical linear basis of d\mathbb{R}^d, and for {L ab=L ba}\{L_{a b} = - L_{b a}\} the corresponding canonical basis of 𝔰𝔬(d1,1)\mathfrak{so}(d-1,1), then the Lie bracket in 𝔦𝔰𝔬( d1,1)\mathfrak{iso}(\mathbb{R}^{d-1,1}) is given as follows:

[P a,P b] =0 [L ab,L cd] =η daL bcη bcL ad+η acL bdη dbL ac [L ab,P c] =η acP bη bcP a \begin{aligned} [P_a, P_b] & = 0 \\ [L_{a b}, L_{c d}] & = \eta_{d a} L_{b c} -\eta_{b c} L_{a d} +\eta_{a c} L_{b d} -\eta_{d b} L_{a c} \\ [L_{a b}, P_c] & = \eta_{a c} P_b -\eta_{bc} P_a \end{aligned}
Proof

Since Lie differentiation sees only the connected component of a Lie group, and does not distinguish betwee a Lie group and any of its discrete covering spaces, we may equivalently consider the Lie algebra of the spin group Spin(d1,1)SO +(d1,1)Spin(d-1,1) \to SO^+(d-1,1) (the double cover of the proper orthochronous Lorentz group) and its action on d1,1\mathbb{R}^{d-1,1}.

By the discussion at spin group, the Lie algebra of Spin(d1,1)Spin(d-1,1) is the Lie algebra spanned by the Clifford algebra bivectors

L abΓ aΓ b L_{a b} \leftrightarrow \Gamma_a \Gamma_b

and its action on itself as well as on the vectors, identified with single Clifford generators

P aΓ a P_a \leftrightarrow \Gamma_a

is given by forming commutators in the Clifford algebra:

[L ab,P c]12[Γ ab,Γ c] [L_{a b}, P_c] \leftrightarrow \tfrac{1}{2}[\Gamma_{a b}, \Gamma_c ]
[L ab,L cd]12[Γ ab,Γ cd]. [L_{a b}, L_{c d}] \leftrightarrow \tfrac{1}{2}[\Gamma_{a b}, \Gamma_{c d} ] \,.

Via the Clifford relation

Γ aΓ b+Γ bΓ a=2η ab \Gamma_a \Gamma_b + \Gamma_b \Gamma_a = -2 \eta_{a b}

this yields the claim.

Remark

Dually, the Chevalley-Eilenberg algebra CE(𝔦𝔰𝔬( d1)CE(\mathfrak{iso}(\mathbb{R}^{d-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 bω acη cb. \omega^{a}{}_b \coloneqq \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} \colon \omega^{a b} = \omega^a{}_c \wedge \omega^{c b}
d CE:e aω a bt b d_{CE} \colon 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 January 4, 2017 16:56:37 by Urs Schreiber (87.191.41.20)