A crash course in Lie algebras by Skoruppa N.-P.

By Skoruppa N.-P.

Those are the notes of a path on Lie algebras which I gave on the collage of Bordeaux in spring 1997. The path used to be a so-called "Cours PostDEA", and as such needed to be held inside 12 hours. much more hard, no prior wisdom approximately Lie algebras can be assumed. however, I had the aim to arrive as top of the direction the nature formulation for Kac-Moody algebras, and, even as, to offer entire proofs so far as attainable.

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As it turns out, all matrices in this resulting list are Cartan matrices. Hence FIN coincides with the set of simple Cartan matrices of root systems. This leads to a classification of all simple root systems. See the table section for the complete list. The first (simple) specimen in FIN can immediately be written down: A1 = 2 , A2 = 2 −1 , −1 2 B2 = 2 −1 , −2 2 G2 = 2 −1 . −3 2 28 CHAPTER 3. SEMISIMPLE LIE ALGEBRAS (We put the classical names in front). 3 Serre’s Theorem and consequences In the preceding two sections we have constructed a map {semisimple Lie algebras}/ ∼ = → {Cartan matrices of root systems}/ ∼ =, where the second ∼ = means “up to permutation of indices”.

Let the notations as in the first theorem of this paragraph, and let A the Cartan matrix of L. Let L(A) be the Lie algebra from Serre’s theorem. From the definition of the latter (cf. next section for details) it will be clear then that there is a unique Lie algebra morphism L(A) → L mapping the Xα , Yα , Hα to xα , yα , hα , respectively. This map is clearly surjective. On the other hand, both Lie algebras have equivalent root systems (since the have the same Cartan matrices). Thus if r is the number of roots and l the number of lines of the Cartan matrix, then both Lie algebras have dimension r + l.

It is convenient to use a slightly different description of the loop space, namely L(L) := C[t, t−1 ] ⊗ L. Here a P ⊗ x may be viewed as the map z → P (z) z. 3. AFFINE KAC-MOODY ALGEBRAS given by [P ⊗ x, Q ⊗ y]L(L) = P Q [x, y]. In the next step we consider a certain central extension of L(L). Let C be a symbol and set [f, C] := 0, L(L) := L(L) ⊕ C · C, d [f, g] := [f, g]L(L) + Res B( f, g) · C dt (f, g ∈ L(L)). Here B is the Killing form on L, extended linearly to L(L) with values in the ring of Laurent polynomials C[t, t−1 ].

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