Serre's theorem on a semisimple Lie algebra explained

\Phi , there exists a finite-dimensional semisimple Lie algebra whose root system is the given \Phi .

Statement

The theorem states that: given a root system

\Phi in a Euclidean space with an inner product ( ) , \langle \beta, \alpha \rangle = 2 (\alpha, \beta)/(\alpha, \alpha), \beta, \alpha \in E and a base \{ \alpha _{1,} ... , \alpha _{n} \} of \Phi , the Lie algebra ak g defined by (1) 3 n generators e _{i,} f _{i,} h _{i} and (2) the relations[h _{i,} h _{j]} = 0 , [e _{i,} f _{i]} = h _{i,} [e _{i,} f _{j]} = 0 , i \ne j ,[h _{i,} e _{j]} = \langle \alpha _{i,} \alpha _{j} \rangle e _{j,} [h _{i,} f _{j]} = - \langle \alpha _{i,} \alpha _{j} \rangle f _{j} ,\operatorname{ad}(e ^{- \langle \alpha _{i,} \alpha _{j} \rangle +1} _{i)} (e _{j)} = 0 , i \ne j ,\operatorname{ad}(f ^{- \langle \alpha _{i,} \alpha _{j} \rangle +1} _{i)} (f _{j)} = 0 , i \ne j .is a finite-dimensional semisimple Lie algebra with the Cartan subalgebra generated by h _{i} 's and with the root system \Phi .

The square matrix

[\langle \alpha _{i,} \alpha _{j} \rangle] _{1} is called the Cartan matrix. Thus, with this notion, the theorem states that, given a Cartan matrix A, there exists a unique (up to an isomorphism) finite-dimensional semisimple Lie algebra ak g(A) associated to A . The construction of a semisimple Lie algebra from a Cartan matrix can be generalized by weakening the definition of a Cartan matrix. The (generally infinite-dimensional) Lie algebra associated to a generalized Cartan matrix is called a Kac–Moody algebra.

Sketch of proof

The proof here is taken from and .Let

a _{ij} = \langle \alpha _{i,} \alpha _{j} \rangle and then let \widetilde{ ak g} be the Lie algebra generated by (1) the generators e _{i,} f _{i,} h _{i} and (2) the relations:[h _{i,} h _{j]} = 0 ,[e _{i,} f _{i]} = h _{i} , [e _{i,} f _{j]} = 0 , i \ne j ,[h _{i,} e _{j]} = a _{ij} e _{j,} [h _{i,} f _{j]} = - a _{ij} f _{j} .

Let

ak{h} be the free vector space spanned by h _{i} , V the free vector space with a basis v _{1,} ... , v _{n} and $T\; =\; \backslash bigoplus\_^\; V^$ the tensor algebra over it. Consider the following representation of a Lie algebra:\pi : \widetilde{ ak g} \to ak{gl}(T) given by: for a \in T, h \in ak{h}, λ \in ak{h} ^{*} ,\pi(f _{i)a} = v _{i} ⊗ a, \pi(h) 1 = \langle λ , h \rangle 1 , \pi(h)(v _{j} ⊗ a) = - \langle \alpha _{j,} h \rangle v _{j} ⊗ a + v _{j} ⊗ \pi(h)a , inductively,\pi(e _{i)1} = 0 , \pi(e _{i)(v} _{j} ⊗ a) = \delta _{ij} \alpha _{i(a)} + v _{j} ⊗ \pi(e _{i)a} , inductively.It is not trivial that this is indeed a well-defined representation and that has to be checked by hand. From this representation, one deduces the following properties: let \widetilde{ ak{n}} _{+} (resp. \widetilde{ ak{n}} _{-} ) the subalgebras of \widetilde{ ak g} generated by the e _{i} 's (resp. the f _{i} 's).\widetilde{ ak{n}} _{+} (resp. \widetilde{ ak{n}} _{-} ) is a free Lie algebra generated by the e _{i} 's (resp. the f _{i} 's).

• As a vector space,

\widetilde{ ak g} = \widetilde{ ak{n}} _{+} o plus ak{h} o plus \widetilde{ ak{n}} _{-} .\widetilde{ ak{n}} _{+} = o plus _{0 \ne \alpha \in Q _{+}} \widetilde{ ak g} _{\alpha} where \widetilde{ ak g} _{\alpha} = \{ x \in \widetilde{ ak g}|[h, x] = \alpha(h) x, h \in ak h \} and, similarly, \widetilde{ ak{n}} _{-} = o plus _{0 \ne \alpha \in Q _{+}} \widetilde{ ak g} _{-\alpha} .

• (root space decomposition)

\widetilde{ ak g} = \left( o plus _{0 \ne \alpha \in Q _{+}} \widetilde{ ak g} _{-\alpha} \right) o plus ak h o plus \left( o plus _{0 \ne \alpha \in Q _{+}} \widetilde{ ak g} _{\alpha} \right) .

For each ideal

ak i of \widetilde{ ak g} , one can easily show that ak i is homogeneous with respect to the grading given by the root space decomposition; i.e., ak i = o plus _{\alpha} (\widetilde{ ak g} _{\alpha} \cap ak i) . It follows that the sum of ideals intersecting ak h trivially, it itself intersects ak h trivially. Let ak r be the sum of all ideals intersecting ak h trivially. Then there is a vector space decomposition: ak r = ( ak r \cap \widetilde{ ak n} _{-)} ⊕ ( ak r \cap \widetilde{ ak n} _{+)} . In fact, it is a \widetilde{ ak g} -module decomposition. Letak g = \widetilde{ ak g}/ ak r .Then it contains a copy of ak h , which is identified with ak h andak g = ak{n} _{+} o plus ak{h} o plus ak{n} _{-} where ak{n} _{+} (resp. ak{n} _{-} ) are the subalgebras generated by the images of e _{i} 's (resp. the images of f _{i} 's).

One then shows: (1) the derived algebra

[ ak g, ak g] here is the same as ak g in the lead, (2) it is finite-dimensional and semisimple and (3) [ ak g, ak g] = ak g .

References

• Book: Kac, Victor. Victor Kac

. Victor Kac. Infinite dimensional Lie algebras. 3rd . . 1990. 0-521-46693-8.

• Book: Humphreys, James E. . James E. Humphreys

. James E. Humphreys. Introduction to Lie Algebras and Representation Theory . . Berlin, New York . 978-0-387-90053-7 . 1972 . registration .

• Book: Serre, Jean-Pierre. Jean-Pierre Serre

. Algèbres de Lie semi-simples complexes. Jean-Pierre Serre. 1966. Benjamin. Complex Semisimple Lie Algebras. 978-3-540-67827-4. en. Jones. G. A..