Calculating Lie Algebra 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?
 A: 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.
A: I think about Lie algebras by building upon other concepts, and specific examples of them:  


*

*Group

*Lie Group

*Lie Algebra


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. 
A: 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!
