Lie Algebra

In this article, I will introduce Lie algebra, which essentially is related to the Poisson brackets we studied before.

Definition

Lie Algebra

For a vector space $V$ defined on field $F$ , a Lie algebra is tuple $⟨ V, [\cdot, \cdot] ⟩$ , where $[\cdot, \cdot]$ is a bilinear mapping: $[\cdot, \cdot] : V \times V \to V$ , s.t. all of the 3 following conditions are fulfilled:

Bilinearity:

[a x + b y, z] = a [x, z] + b [y, z], \forall x, y, z \in V .

Anti-Symmetry:

[x, x] = 0, \forall x \in V .

If the field $F$ does not have character $2$ , we also have

[x, y] = - [y, x], \forall x, y \in V .

Jacobi Identity:

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, \forall x, y, z \in V .

Comment 1.: The character of a field $F$ is defined as:

char (F) = {argmin}_{n > 0} (\underset{n times}{\underset{⏟}{1 + 1 + . . . + 1}} = 0),

if $n = \infty$ , $char (F) := 0$ .

Comment 2.: If we denote ${ad}_{x} (y) = [x, y]$ (derivation), Jacobi identity can be rewritten as:

{ad}_{x} ([y, z]) = [y, {ad}_{x} (z)] + [{ad}_{x} (y), z] .

Why Poisson brackets have Lie algebra structure

What is Poisson bracket

See the link.

Poisson Bracket

{Q_{1}, Q_{2}} = \frac{\partial Q_{1}}{\partial x} \frac{\partial Q_{2}}{\partial p} - \frac{\partial Q_{1}}{\partial p} \frac{\partial Q_{2}}{\partial x},

where $Q_{1}, Q_{2}$ are observables.

Easy to know that, Poisson brackets are bilinear and anti-symmetric. For the Jacobi identity, the proof is technical but not too difficult. This is an exercise for the reader.

Jacobi identity is an interesting property, because we know in Hamiltonian mechanics the time evolution of observables can be written:

\dot{f} = {f, H} .

If $f, g$ are conserved ( ${f, H} = {g, H} = 0$ ), from Jacobi identity:

{f, {g, H}} + {g, {f, H}} + {H, {f, g}} = 0 ⟹ {{f, g}, H} = 0.

${f, g}$ is also conserved! This implies that the conserved quantities form a Lie subalgebra.

Moreover, if we define the derivation operator $X_{f} (g) = {f, g}$ , from Jacobi identity we can write:

X_{{f, g}} = X_{f} X_{g} - X_{g} X_{f} .

Poisson algebra

Poisson’s bracket is as a matter of fact stronger than the bilinear mapping required by Lie algebra in the way that it satisfies the Leibniz rule in addition:

{f, g h} = {f, g} h + g {f, h},

or in the derivation notation

X_{f} (g h) = X_{f} (g) h + g X_{f} (h) .

This rule, which is a generalization of the product rule of derivative, makes $⟨ V, {\cdot, \cdot} ⟩$ a Poisson algebra.

本文采用署名-非商业性使用-相同方式共享 4.0 国际许可协议，转载请注明出处。