An Introduction to Lie Groups and Lie Algebras by Alexander Kirillov Jr Jr

By Alexander Kirillov Jr Jr

It is a wickedly solid booklet. it really is concise (yeah!) and it really is good written. it misses out on plenty of stuff (spin representations, etc..). yet when you learn this ebook you've got the formalism down pat, after which every little thing else turns into easy.

if you install the hours to learn this ebook hide to hide -- like sitting down for three days directly eight hours an afternoon, then will research the stuff. in the event you do not persevere and get crushed with the stuff that's not transparent first and foremost, then you definately will most likely chuck it out the window.

lie teams and lie algebras in 2 hundred pages performed in a chic method that does not appear like lecture notes cobbled jointly is beautiful amazing.

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In this case, g = gl(n, K) is the space of all matrices. SL(n, K): Suppose X ∈ SL(n, K) is close enough to identity. Then X = exp(x) for some x ∈ gl(n, K). The condition that X ∈ SL(n, K) is equivalent to det X = 1, or det exp(x) = 1. But it is well-known that det exp(x) = exp(tr(x)) (which is easy to see by finding a basis in which x is upper-triangular), so exp(x) ∈ SL(n, K) if and only if tr(x) = 0. Thus, in this case the statement also holds, with g = {x ∈ gl(n, K) | tr x = 0}. 7. Classical groups 19 O(n, K), SO(n, K): The group O(n, K) is defined by XX t = I .

7) 34 Lie groups and Lie algebras This may not be defined globally, but for the moment, let us ignore this problem. What is the commutator [ξ , η]? 3), we need to consider tξ sη t−ξ s−η . It is well-known that this might not be the identity (if a plane flies 500 miles north, then 500 miles west, then 500 miles south, then 500 miles east, then it does not necessarily lands at the same spot it started – because Earth is not flat). 3), we expect that this expression can be written in the form 1 + ts[ξ , η] + · · · for some vector field [ξ , η].

Writing X = exp(x), X t = exp(xt ) (since the exponential map agrees with transposition), we see that x, xt also commute, and thus exp(x) ∈ O(n, K) implies exp(x) exp(xt ) = exp(x + xt ) = 1, so x + xt = 0; conversely, if x + xt = 0, then x, xt commute, so we can reverse the argument to get exp(x) ∈ O(n, K). Thus, in this case the theorem also holds, with g = {x | x + xt = 0} – the space of skew-symmetric matrices. What about SO(n, K)? In this case, we should add to the condition XX t = 1 (which gives x + xt = 0) also the condition det X = 1, which gives tr(x) = 0.

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