Summary pages – explanation

There follow a number of summary pages, one for each Lie algebra of finite or affine type, giving basic properties of the Lie algebra in question. The information given differs to some extent between the Lie algebras of finite type and those of affine type.

In the case of the algebras of finite type we give the name of the algebra, the Dynkin diagram with the labelling we have chosen for its vertices, the Cartan matrix, the dimension of the Lie algebra, its Coxeter number, the order of its Weyl group W and the degrees of the basic polynomial invariants of W. We also give information about its root system. The roots are most conveniently described in terms of a basis β1, …, βm of mutually orthogonal basis vectors all of the same length. In several cases it is convenient to choose m greater than the rank l of the Lie algebra, so that the root system lies in a proper subspace of the vector space spanned by β1, …, βm In the cases when there are roots of two different lengths the long roots and short roots are both described. The extended Dynkin diagram is given and the root lattice described in terms of the above orthogonal basis. The fundamental weights are given, as is the index of the root lattice in the weight lattice. Finally the standard invariant forms on Hℝ and H*ℝ are described, and the constant is given which converts the standard invariant form on Hℝ into the Killing form.