Construction and classification of complex simple Lie algebras via projective geometry
From the text: We first present an algorithm that constructs the minuscule varieties using elementary algebraic geometry. The minuscule varieties are a preferred class of homogeneous varieties. They are essentially the homogeneous projective varieties that admit an irreducible Hermitian symmetric metric. \(\ldots\) Next we present an algorithm that constructs the fundamental adjoint varieties using the ideals of the tangential and secant varieties of certain minuscule varieties. By an adjoint variety, we mean the unique closed orbit in the projectivization \(P\mathfrak g\) of a simple complex Lie algebra \(\mathfrak g\). We say that an adjoint variety is fundamental if the adjoint representation is fundamental. In particular, we construct all complex simple Lie algebras without any reference to Lie theory. Complex simple Lie algebras were first classified by Cartan and Killing 100 years ago. Their classification proof proceeds by reducing the question to a combinatorial problem: the classification of irreducible root systems, and then classifying root systems. We present a new proof of the classification of minuscule varieties and complex simple Lie algebras by showing our algorithms produce all minuscule (resp. fundamental adjoint) varieties without using the classification of root systems, although we do use some properties of root systems. We also provide a proof that the only non-fundamental adjoint varieties are the adjoint varieties of \(A_m\) and \(C_m\), and thus we obtain a new proof of the classification of complex simple Lie algebras. Our proof can be translated into a combinatorial argument: the construction consists of two sets of rules for adding new nodes to marked Dynkin diagrams. Our constructions have applications that go well beyond the classification proof presented in this article. This is the second paper in a series. In [(*) J. Algebra 239, No. 2, 477--512 (2001; Zbl 1064.14053)], [(**) Series of Lie algebras, after Deligne, Freudenthal and Zak. In preparation] and [(***) Adv. Math. 171, No. 1, 59--85 (2002; Zbl 1035.17016)] we present geometric and representation-theoretic applications of our algorithms. In [(*)] we study the geometry of the exceptional homogeneous spaces using the constructions of this paper. In [(**)] and [(***)] we apply the results of this paper, especially our observations about the Casimir in Section 5, to obtain decomposition and dimension formulas for tensor powers of some preferred representations.
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