I am studying Quantum Field Theory and I am having some difficulties with the concept of a Lie Algebra.
My understanding is that a Lie Algebra is a vector space equipped with the commutator $[x,y] = xy - yx$. However, I've often come across questions such as: "calculate the Lie Algebra of $SU(2)$".
I'm not sure how one can calculate such things? Are there any good (and hopefully shortish) references that I can use to understand this?
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I think about Lie algebras by building upon other concepts, and specific examples of them:
Speaking a bit broadly:
A group is a set of discrete elements with an operation that takes any element in the group to a unique another element. There is an inverse operation (that takes you back), an identity element, and a few other technical requirements. A simple example is the group of transformations that yield a square: you can leave it alone (identity transformation), rotate by $+90^\circ$, or $180^\circ$ or flip along a vertical line through the center, or along a horizontal line through the center, a diagonal line through the center, and so on. You have discrete transformations and the operations are "closed".
A Lie group is a continuous group, where the transformations are continuous. Think of the transformations of a sphere. You can rotate by any real-valued angle (e.g., $26.984^\circ$) around an axis pointed in an arbitrary direction (e.g., a vector $\{ -.2, \pi, .8777... \}$). Any such rotation yields the sphere. And these rotations can be composited: Do transformation 1 then transformation 2 and it is equivalent to a single transformation 3. You can have infinitessimal transformations, and the operations are "closed".
A Lie algebra is the mathematics that governs such continuous transformations. Any Lie Group has an association Lie Algebra. Formally, you need a vector space and a non-associative "multiplication" operation. Think of the sphere. The transformations (transformation 1 * transformation 2) * transformation 3 need not be the same as transformation 1 * (transformation 2 * transformation 3). You can express this with a Lie Algebra (whose elements are the rotations). If you work with rotations of a 5-dimensional sphere (for instance), the Lie Algebra will differ from that for a 3-sphere or 2-sphere.
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As mentioned, there are plenty of books about Lie groups and Lie algebras - it's a mathematically mature subject with lots of different strands. May I suggest, if you want a quick overview from a perspective that may be more on your wavelength, that you try to pick up a copy of Physics from Symmetry by J. Schwintenberg. It is isn't a mathematics book; it isn't theorem, proof, corollary, proof etc. It is just full of ideas and examples of applying Lie theory to physics; it even gets on to discussing quantum field theory too!
Lie algebras of matrices have a Lie bracket given by $[x,y]=xy-yx$, but abstract Lie algebras need not consist of matrices. Then the term $xy$ has a priori no meaning. There is a (difficult) theorem by Ado and Iwasawa, that every finite-dimensional Lie algebra over a field $K$ can be realised by matrices. The references you will need, I suppose, are called "Lie groups and Lie algebras for physicists". There are plenty of such books.