Serre's generators and relations for a semisimple Lie algebra. Chevalley basis for a semisimple Lie algebra. Compact form.

6 dezembro 2017, 11:00 • Gustavo Granja

Given a semisimple Lie algebra L with root system \Phi and a base \Delta=\{\alpha_1,\ldots,\alpha_l\} a choice of nonzero x_i in L_{\alpha_i} and y_i in L_{-\alpha_i} give generators for L. Moreover they satisfy the Serre relations S1,S2,S3,S_{ij}^+ and S_{ij}^- (which depend only on the Cartan matrix).Definition and construction of the free Lie algebra on a vector space. Generators and relations.Serre's Theorem: Given a root system \Phi with base \Delta={\alpha_1,\ldots,\alpha_l} the Lie algebra with generators x_i, h_i, y_i for i=1,\ldots l satisfying Serre's relation is a semisimple Lie algebra with Cartan subalgebra <h_1,\ldots, h_l> and root system \PhiCorollary: 1. For each root system \Phi there is a semisimple Lie algebra with root system isomorphic to \Phi.2. If L and L' are semisimple Lie algebras with root systems \Phi and \Phi' and f:\Phi to \Phi' is an isomorphism then there is an isomorphism F:L \to L' inducing f on the root system.Example: The automorphism of L corresponding to the automorphism \alpha \mapsto -\alpha of the root system can be chosen as the automorphism that sends x_i to -y_i, h_i to -h_i and y_i to -x_i. 
Let L be a semisimple lie algebra with root system \Phi and Cartan decomposition h \oplus \oplus_{\alpha \in \Phi} L_\alphaLemma: Let \alpha,\beta in \Phi be independent roots and \beta-r\alpha, \ldots, \beta+q\alpha the \alpha string through \beta. Then
1. If \alpha + \beta \in \Phi then r+1=q<\alpha+\beta,\alpha+\beta>/<\beta,\beta>2. Given x_\alpha \in L_\alpha non zero let x_{-\alpha} \in L_{-\alpha} be the element such that x_\alpha, x_{-\alpha}, h_\alpha is a copy of sl(2;k). Then [x_{-\alpha},[x_\alpha, x_\beta]] = q(r+1)\beta
Proposition: It is possible to pick x_\alpha \in L_\alpha non-zero for each \alpha in \Phi such that1. [x_\alpha, x_{-\alpha}] = h_\alpha2. For \alpha+\beta \in \Phi and [x_\alpha,x_\beta]=c_{\alpha\beta}x_{\alpha+\beta} we have c_{\alpha,\beta}=-c_{-\alpha,-\beta}Moreover, for any such choice we have c^2_{\alpha\beta} = q(r+1)<\alpha+\beta,\alpha+\beta>/<\beta,\beta> and hence  c_{\alpha\beta} = \pm(r+1).Theorem: Let L be a semisimple Lie algebra with root system \Phi and base \Delta=\{\alpha_1,\ldots,\alpha_l}. Then the structure constants of the Lie algebra with respect to a Chevalley basis, i.e. {h_{\alpha_1},\ldots,h_{\alpha_l} together with x_\alpha for \alpha \in \Phi satisfying conditions 1. and 2. in the previous proposition are integers. Corollary: Let L be a semisimple Lie algebra over \C. There is a compact (i.e. real semisimple with negative definite Killing form) subalgebra U of L such that L = U\otimes_\R \C

Remark: A real Lie subalgebra M of L such that M \otimes_\R \C is called a real form of L. A Chevalley basis gives a standard real form called the split real form (just take the real lie subalgebra spanned by the Chevalley basis). All the real forms of L can be obtained from U. They are conjugate to p+in with p and n the +1, -1 eigenspaces of an involution of U.  Example: L=sl(l+1;\C). The split real form is sl(l+1;\R), the compact form is su(l+1) and there are "intermediate" real forms su(p,q) with p+q=l+1 (Lie algebra of automorphisms of a hermitean form on \C^{l+1} with signature (p,q)).