nLab Poincaré Lie algebra

Idea

The Poincaré Lie algebra $\mathfrak{iso}(\mathbb{R}^{d-1,1})$ 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 $\mathfrak{so}(d-1,1)$ with the the abelian translation Lie algebra $\mathbb{R}^{d-1,1}$.

Definition

Definition

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

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

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

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

Remark

The Poincaré group is the semidirect product group

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

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

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

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

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

Proposition

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

$$\begin{aligned} [P_a, P_b] & = 0 \\ [L_{a b}, L_{c d}] & = \eta_{d a} L_{b c} -\eta_{b c} L_{a d} +\eta_{a c} L_{b d} -\eta_{d b} L_{a c} \\ [L_{a b}, P_c] & = \eta_{a c} P_b -\eta_{bc} P_a \end{aligned}$$

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 $Spin(d-1,1) \to SO^+(d-1,1)$ (the double cover of the proper orthochronous Lorentz group) and its action on $\mathbb{R}^{d-1,1}$.

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

$$L_{a b} \leftrightarrow \Gamma_a \Gamma_b$$

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

$$P_a \leftrightarrow \Gamma_a$$

is given by forming commutators in the Clifford algebra:

$$[L_{a b}, P_c] \leftrightarrow \tfrac{1}{2}[\Gamma_{a b}, \Gamma_c ]$$

$$[L_{a b}, L_{c d}] \leftrightarrow \tfrac{1}{2}[\Gamma_{a b}, \Gamma_{c d} ] \,.$$

Via the Clifford relation

$$\Gamma_a \Gamma_b + \Gamma_b \Gamma_a = -2 \eta_{a b}$$

this yields the claim.

Remark

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

$$\omega^{a}{}_b \coloneqq \omega^{a c}\eta_{c b} \,.$$

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

$$d_{CE} \colon \omega^{a b} = \omega^a{}_c \wedge \omega^{c b}$$

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

Properties

Cohomology

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

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

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

The volume cocycle is the volume form

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

Invariant polynomials and Chern-Simons elements

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

We have the Bianchi identity

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

and

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

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

Another invariant polynomial is $r^{a b} \wedge r_{a b}$. This is the Killing form of $\mathfrak{so}(d-1,1)$. 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

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

There is also an infinite series of mixed invariant polynomials

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

Chern-Simons elements for these are

$$B_{2n + 1} := \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge e^{a_n} \,.$$

Lie algebra valued forms

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

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

is

• a vielbein $E$ on $X$;

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

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

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

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

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

The volume form is the image of the volume cocycle

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

We have

$$vol(E) = \epsilon_{a_1 \cdots a_d} E^{a_1} \wedge \cdots \wedge E^{a_d} \,.$$

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

Related structures

• special orthogonal Lie algebra

• Poincaré-Weyl algebra

• super Poincaré Lie algebra