nLab
Poincaré Lie algebra

Idea

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

The corresponding Lie group is the Poincaré group.

Definition

The CE-algebra

The Chevalley-Eilenberg algebra $\mathrm{CE}(\mathrm{𝔦𝔰𝔬}(d-1,1))$ is generated from ${ℝ}^{d,1}$ and ${\wedge }^2{ℝ}^{d,1}$. For $\{t_a\}$ the standard basis of ${ℝ}^{d-1,1}$ we write $\{{\omega }^{ab}\}$ and $\{e^a\}$ for these generators. With $({\eta }_{ab})$ the components of the Minkowski metric we write

$${\omega }^a{}_b:={\omega }^{ac}{\eta }_{cb}.$$\omega^{a}{}_b := \omega^{a c}\eta_{c b} \,.

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

$$d_{\mathrm{CE}}:{\omega }^{ab}={\omega }^a{}_c\wedge {\omega }^{cb}$$d_{CE} : \omega^{a b} = \omega^a{}_c \wedge \omega^{c b}

$$d_{\mathrm{CE}}:e^a\mapsto {\omega }^a{}_b\wedge t^b$$d_{CE} : e^a \mapsto \omega^{a}{}_b \wedge t^b

Properties

Cohomology

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

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

$${\omega }^a{}_b\wedge {\omega }^b{}_c\wedge {\omega }^c{}_a\in \mathrm{CE}(\mathrm{𝔦𝔰𝔬}(d-1,1)).$$\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

$$\mathrm{vol}={ϵ}_{a_1\cdots a_d}{e}^{a_1}\wedge \cdots \wedge e^{a_d}\in \mathrm{CE}(\mathrm{𝔦𝔰𝔬}(d-1,1)).$$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 $(e^a,{\omega }^{ab})$ as above, denote the shifted generators of the Weil algebra $W(\mathrm{𝔦𝔰𝔬}(d-1,1))$ by ${\theta }^a$ and $r^{ab}$, respectively.

We have the Bianchi identity

$$d_W:r^{ab}\mapsto {\omega }^{ac}\wedge R_c{}^d-R^{ac}\wedge {\omega }_c{}^b$$d_W : r^{a b} \mapsto \omega^{a c} \wedge R_c{}^d - R^{a c} \wedge \omega_c{}^b

and

$$d_W:{\theta }^a\mapsto {\omega }^a{}_b{\theta }^b-R^a{}_b{e}^b.$$d_W : \theta^a \mapsto \omega^a{}_b \theta^b - R^{a}{}_b e^b \,.

The element ${\eta }_{ab}{\theta }^a\wedge {\theta }^b\in W(\mathrm{𝔦𝔰𝔬}(d-1,1))$ is an invariant polynomial. A Chern-Simons element for it is $\mathrm{cs}={\eta }_{ab}{e}^a\wedge {\theta }^b$. So this transgresses to the trivial cocycle.

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

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

$$P_n:=r^{a_1}{}_{a_2}\wedge r^{a_2}{}_{a_3}\wedge \cdots \wedge r^{a_n}{}_{a_1}.$$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

$$C_{2n+2}:={\theta }_{a_1}\wedge r^{a_1}{}_{a_2}\wedge r^{a_2}{}_{a_3}\wedge \cdots \wedge r^{a_{n-1}}{}_{a_n}\wedge {\theta }^{a_n}.$$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

$$B_{2n+1}:={\theta }_{a_1}\wedge r^{a_1}{}_{a_2}\wedge r^{a_2}{}_{a_3}\wedge \cdots \wedge r^{a_{n-1}}{}_{a_n}\wedge e^{a_n}.$$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 $\mathrm{𝔦𝔰𝔬}(d-1,1)$

$${\Omega }^{•}(X)\leftarrow W(\mathrm{𝔦𝔰𝔬}(d-1,1)):(E,\Omega )$$\Omega^\bullet(X) \leftarrow W(\mathfrak{iso}(d-1,1)) : (E,\Omega)

is

• a vielbein $E$ on $X$;

• a “spin connection” $\Omega$ on $X$.

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

• the torsion $T=dE+[\Omega \wedge E]$;

• the Riemannian curvature $R=d\Omega +[\Omega \wedge \Omega ]$.

If the torsion vanishes, then $\Omega$ is a Levi-Civita connection for the metric $E^a\otimes E^b{\eta }_{ab}$ defined by $E$.

The volume form is the image of the volume cocycle

$${\Omega }^{•}(X)\stackrel{(E,\Omega )}{\leftarrow }W(\mathrm{𝔦𝔰𝔬}(d-1,1))\stackrel{\mathrm{vol}}{\leftarrow }W(b^{d-1}ℝ):\mathrm{vol}(E).$$\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

$$\mathrm{vol}(E)={ϵ}_{a_1\cdots a_d}{E}^{a_1}\wedge \cdots \wedge E^{a_d}.$$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.

Related structures

• special orthogonal Lie algebra

• super Poincaré Lie algebra