# Skoruppa N.-P.'s A crash course in Lie algebras PDF

By Skoruppa N.-P.

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

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Similarly, if a = 1, one deduces the contradiction β − α ∈ R. Proposition. The Cartan matrix uniquely determines R up to isomorphism. Proof. Let R and R two root systems in E and E with basis αi and αi such that the associated Cartan matrices coincide. Let φ be the isomorphism of E with E defined by αi → αi (for all i). Then σφ(αi ) = φ ◦ σαi ◦ φ−1 . Since R is the set σαi (αj ) (all i and j) (and similarly for R ) we see that φ maps R bijectively to R , and that σφ(αi ) = φ ◦ σαi ◦ φ−1 for any root α.

Otherwise choose a maximal weight λ ≥ µ of V , a nonzero v ∈ Vλ , set U = U (L)v and let U a maximal proper submodule of U (which exists by the diagonalization lemma). Then 0, U , U, V is a filtration such that U/U ∼ = L(λ). But n(U ), n(V /U ) < n(V ), and applying the induction hypothesis to V /U and U yields than the desired filtration. Step 2: Since A is symmetrizable, there is a regular diagonal matrix D such that AD = (bα,β )α,β is symmetric. Thus one can define a scalar product on the root lattice by setting (α, β) := bα,β for simple roots α, β.

E. AD is symmetric and positive definite for a suitable diagonal matrix D). Note that these GCMs have nonzero determinant, and that any minor of such a matrix is again a direct sum of Cartan matrices of simple root systems, as is immediate from the definition of FIN. By minor we mean a matrix of the form (ai,j )i,j∈S , where S ⊆ {1, 2, . . , n}, and where A = (ai,j )1≤i,j≤n . The associated Kac-Moody algebras are the classical simple Lie algebras discussed in the preceding sections. The second class is the class of simple GCMs A such that det(A) = 0, but any proper minor of A is a direct sum of Cartan matrices of simple root systems.

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

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