Download e-book for kindle: An Introduction to Lie Groups and Lie Algebras by Alexander Kirillov Jr Jr

By Alexander Kirillov Jr Jr

ISBN-10: 0511424183

ISBN-13: 9780511424182

ISBN-10: 0521889693

ISBN-13: 9780521889698

It is a wickedly strong booklet. it is concise (yeah!) and it is good written. it misses out on plenty of stuff (spin representations, etc..). yet when you learn this e-book you might have the formalism down pat, after which every little thing else turns into easy.

if you install the hours to learn this booklet disguise to hide -- like sitting down for three days instantly eight hours an afternoon, then will examine the stuff. if you happen to do not persevere and get crushed with the stuff that's not transparent at the start, then you definitely will most likely chuck it out the window.

lie teams and lie algebras in 2 hundred pages performed in a sublime means that does not seem like lecture notes cobbled jointly is beautiful notable.

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Extra resources for An Introduction to Lie Groups and Lie Algebras

Example text

The map γx will be called the one-parameter subgroup corresponding to x. Proof. Let us first consider the case of a real Lie group. We begin with uniqueness. The usual argument, used to compute the derivative of ex in calculus, shows that if γ (t) is a one-parameter subgroup, then γ˙ (t) = γ (t) · γ˙ (0) = γ˙ (0) · γ (t). 6, we interpret γ (t) · γ˙ (0) as (Lγ (t) )∗ γ˙ (0) and similarly for the right action. This gives us a differential equation for γ : if vx is a left-invariant vector field on G such that vx (1) = x, then γ is an integral curve for v.

25). Show that Fn (C) = GL(n, C)/B(n, C) = U(n)/T (n) where B(n, C) is the group of invertible complex upper triangular matrices, and T (n) is the group of diagonal unitary matrices (which is easily shown to be the n-dimensional torus (R/Z)n ). Deduce from this that Fn (C) is a compact complex manifold and find its dimension over C. 5. Let Gn,k be the set of all dimension k subspaces in Rn (usually called the Grassmanian). Show that Gn,k is a homogeneous space for the group O(n, R) and thus can be identified with coset space O(n, R)/H for appropriate H .

8. 3. Complex classical groups. G GL(n, C) SL(n, C) O(n, C) SO(n, C) π0 (G) π1 (G) {1} Z {1} {1} Z2 Z2 {1} Z2 Note that some of the classical groups are not simply-connected. 7, in this case the universal cover has a canonical structure of a Lie group. Of special importance is the universal cover of SO(n, R) which is called the spin group and is denoted Spin(n); since π1 (SO(n, R)) = Z2 , this is a twofold cover, so Spin(n) is a compact Lie group. 8. 1. Let G be a Lie group and H – a closed Lie subgroup.

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


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