Definition 1 Acommutative algebra is a (real) vector space Aequipped with a product satisfying:
1. bilinearity:
(αF +βG)H=αF H +βGH, ∀α, β ∈R
F(αG +βH ) = αF G +β F H, F, G, H ∈A
2. commutativity:
F G =GF
3. associativity:
(F G)H=F(GH)
C∞(X) becomes a commutative algebra with the “obvious” addition, scalar multiplication, and
multiplication:
(αF )(x) = αF (x), x ∈X
(F+G)(x) = F(x) + G(x)
(F G)(x) = F(x)G(x)
But, in our example X=R2n, there’s another operation, the Poisson bracket:
{F, G}=
n
X
i=1
∂F
∂pi
∂G
∂qi
−∂F
∂qi
∂G
∂pi
This makes C∞(X) into a ‘Lie algebra’:
Definition 2 ALie algebra is a (real) vector space Aequipped with a “Lie bracket” {·,·}
1. bilinearity:
{αF +βG, H }=α{F, H}+β{G, H }
{F, αG +β H}=α{F , G}+β{G, H}
2. antisymmetry:
{F, G}=−{G, F }
3. Jacobi identity:
{F, {G, H }} ={{F, G}, H}+{G, {F, H }}
(Rule 3 looks like the product rule: dGH = (dG)H+g(dH).)
In fact for all classical mechanics problems, the algebra of observables C∞(X) is always both a
commutative algebra and a Lie algebra, but even better, they fit together to form a Poisson alge-
bra.
Definition 3 APoisson algebra is a vector space Awith a product making it into a commutative
algebra and a bracket {·,·} making it into a Lie algebra such that
{F, GH }={F, G}H+G{F, H}
2
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