My question is regarding a vector space and Lie algebra. Why is it that whenever I read advanced physics texts I always hear about Lie algebra? What does it mean to "endow a vector space with a lie algebra"?
I'm assuming it is the same Lie from the Lie groups? My current knowledge is that the Lie groups are "to do with rotations of molecules".
I'm not after lots of detail but would like a basic understanding of what this means and why it is such a prevalent idea.
Answer
It's an enormous subject, but briefly:
Lie groups are smooth groups. Technically, Lie groups are sets that are both a smooth manifold, like a sphere for instance, and also have a group structure (multiplication operator, inverses, and an identity). The group multiplication and inverse must be smooth (differentiable) functions on the manifold.
As you mentioned, the group of rotations in 3-dimensional space, called SO(3), is an example of a Lie group. Rotations have a group structure because you can compose or invert rotations and get other rotations, and they are also a smooth manifold because you can smoothly vary the axis or angle and so move continuously from one rotation to another.
There are many other examples of Lie groups. Many types of geometric transformations on different spaces form Lie groups. They show up in physics in transformations of spacetime (the Poincaré group, most generally), and in so-called "internal symmetries" that transform different quantum fields into each other (often special unitary groups of various dimensions). Another example is diffeomorphism groups, which show up in general relativity and string theory and are also Lie groups.
As for Lie algebras, they are closely related to Lie groups. A Lie algebra basically consists of the "infinitesimal elements" of a Lie group, i.e. the "elements infinitesimally close to the identity". (I put that in scare quotes because in standard analysis, infinitesimal elements don't really exist—technically, a Lie algebra is defined on the tangent space of the Lie group at the identity. Still, the picture of infinitesimal elements is a useful and intuitive way of thinking about this.)
For example, in the case of rotations, we would be talking about rotations about any axis by infinitesimal angles.
When you multiply two group elements that are very close to the identity, the group multiplication looks like a vector sum—basically the same way that $(1+\delta)(1+\epsilon) \approx 1 + (\delta + \epsilon)$ when $\delta$ and $\epsilon$ are small. Similarly, $(1+\epsilon)^{-1} \approx 1 - \epsilon$ and so group inversion looks like vector negation. So the Lie algebra inherits its operations from those of the underlying Lie group, but it doesn't itself look like a group—instead, it looks like a vector space.
The Lie algebra also has, in addition to the standard vector space operations, a bilinear operation called the Lie bracket, denoted $[x, y]$ (where $x, y$ are two vectors in the Lie algebra and $[x, y]$ generates another vector). This operation measures "how noncommutative" the Lie group is; roughly speaking, it corresponds to the commutator $[A, B] = AB - BA$ of the Lie group.
Now, the funny and interesting thing about a Lie algebra is that even though it's derived from just an infinitesimal slice of a Lie group, it contains, encoded within it, almost everything there is to know about the group that it came from! You can actually reconstruct the entire Lie group, starting from just the Lie algebra, by using the exponential map—a generalization of the ordinary exponential function.
(It's almost because there are some cases where different Lie groups have the same Lie algebra, but different global structures—for example, SO(3) and SU(2).)
And since Lie algebras are vector spaces, and the Lie bracket is a bilinear operation, all you really need to have is a set of basis vectors for the Lie algebra and know what the Lie bracket does to each pair of basis vectors.
Such a set of basis vectors is called a set of generators of the group. If you apply the Lie bracket to each pair of generators and write down the resulting vectors as coordinates in the same basis, the set of numbers you obtain are called structure constants.
From the generators and the structure constants, you can generate the Lie algebra and thence the entire Lie group (except for ambiguities of global structure as mentioned above)! This makes Lie algebras a very powerful tool for understanding the Lie groups that show up in physics. For example, in particle physics, the gauge bosons (photon, W, Z, gluons) are closely related to the generators of internal symmetry groups; momentum and angular momentum are related to the generators of the Poincaré group, and so on.
There's much more that could be said about this—I haven't even mentioned representations!—but this is probably enough for one answer, so I'll stop here. :)
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