Introduction to Lie Algebras

In my last post I’ve described the structure called Algebra. Although this structure has some other uses, I wanted at first to define a new structure, called Lie Algebras.

Lie Algebras is a very useful structure, and it is being repeatedly used in Quantum mechanics and Analytical mechanics.

Definition:Lie algebra \mathfrak{g} is an algebra with the operation (usually denoted by [-,-]) that satisfying the following two axioms:

  1. Anti-commutativity: \forall a\in\mathfrak{g},\; [a,a]=0.
  2. Jacobi identity: \forall a,b,c\in\mathfrak{g},\; [a,[b,c]]+[b,[c,a]]+[c,[a,b]]=0.

One can easily show that a subalgebra or a quotient algebra of a Lie algebra is a Lie algebra.

Using the definition we can prove a quick lemma:

Lemma: Let \mathfrak{g} be a Lie Algebra (over a field \mathbb{F}, and let a,b\in\mathfrak{g}. Then: [a,b]=-[b,a]

Proof: From the bi-linearity of the operation [-,-] we have:

    \[ [a+b,a+b]=[a,a]+[b,b]+[a,b]+[b,a] \]

If the characteristics of the field \mathbb{F} is not 2 (we will always assume that), from the first axiom of Lie Algebras we know that:

    \[ [a+b,a+b]=[a,a]=[b,b]=0 \]

Thus:

    \[ [a,b]=-[b,a] \]

as required. ■

We can also define the center of an Lie algebra (also of a general algebra, but that’s not the case here) as follows:

Definition: The center of a Lie Algebra \mathfrak{g} is the set:

    \[ \mathcal{Z}(\mathfrak{g})=\{z\in\mathfrak{g}\mid\;[a,z]=[z,a]\} \]

Note that in the case of Lie Algebra, because of the lemma we’ve proven, we can conclude that:

    \[ \mathcal{Z}(\mathfrak{g})=\{z\in\mathfrak{g}\mid\;[a,z]=0\} \]

Examples

  1. Take any vector space V, and define that operation (usually called the bracket) to be: [-,-]=0. This is a commutative (also called abelian) Lie algebra. And of course the center of \mathfrak{g} is \mathfrak{g} itself.
  2. \mathfrak{g}=\mathbb{R}^3, [a,b]=a\times b (vector product). In this case the center of \mathfrak{g} is zero.
  3. Let A be an associative algebra with the product ab(For example the algebra of n\times n matrices, The the space A with the bracket [a,b]:=ab-ba is a Lie algebra denoted by A_- or by \mathrm{Lie}A

Moreover, In Analytical mechanics, for the physicists readers, you’ve met the Poisson Brackets which maintains the Jacobi identity. And in quantum mechanics, physicist always talking about the commutator of two operators (For example the Hamiltonian and the momentum). And theorems from Lie algebra can easily be applied to these subject, and it’s another way of showing how we can learn facts about the universe simply from playing with math.

I hope I will have time to add a little bit more posts about the subject (such as main theorems and such).

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