Tables Fin and Aff

Chapter 4 of the book Infinite dimensional Lie algebras by Kac classifies the Cartan matrices of finite and affine type. Each Cartan matrix of finite type defines a crystallographic root system and finite Weyl group, and each is listed in Table Fin. Tables Aff 1, Aff 2, and Aff 3 list the Cartan matrices of affine type.

See also: how to read Dynkin diagrams

Table Fin

Table Fin lists the finite-type Dynkin diagrams, along with various data about their associated Weyl groups and root systems.

The columns in this table are:

  • The Dynkin diagram is a compact representation of the Cartan matrix. The arrow points from long roots to short roots; for example C_3 has two short simple roots. When taking the untwisted affinisation, the affine vertex will attach at the red vertex. (These can be determined by looking at the highest root \widetilde{\alpha} in the fundamental weight basis)
  • |S|: The rank of the simple Lie algebra, or Coxeter group.
  • |R|: the number of roots. The number of positive roots is half the number of roots.
  • |Q/P|: the index of connection, the index of the root lattice in the weight lattice. Alternatively, the determinant of the Cartan matrix.
  • |W| the order of the Weyl group, calculated using the index of connection and the coefficients of the highest root: |W| = |S|! \times [Q : P] \times a_1 \cdots a_{|S|}.
  • l(w_0) is the length of the longest element of W. This is equal to the number of positive roots.
  • h the Coxeter number, the order of a Coxeter element. A Coxeter element is a product of reflections incident on a single chamber, and all such elements are conjugate in W. Alternatively, h = 1 + \sum_i a_i, or one more than the height of the highest root.
  • a_i the coefficients of the highest root in the root basis: \tilde{\alpha} = \sum_{i} a_i \alpha_i.
  • h^\vee the dual Coxeter number: h^\vee = 1 + \sum_i a_i^\vee.
  • a_i^\vee the coefficients of the dual of the highest root (which is the highest short coroot): (\tilde{\alpha})^\vee = \sum_i a_i^\vee \alpha_i^\vee.
  • m_i: the exponents of the group. These describe the eigenvalues (with multiplicity) of a Coxeter element, where each eigenvalue is written as \exp\left(2 \pi i m_i / h\right). These are not indexed by simple roots, are are instead listed in weakly increasing order.
  • d_i (not shown): the degrees of the fundamental invariants of W acting on \Sym(V^*) with V an irreducible representation of W. These can be derived from the exponents: d_i = m_i + 1.

The Cartan types B_n and C_n are dual under the switching of their root and coroot systems (alternatively, transposing the Cartan matrix). However, looking at the special vertex in their Dynkin diagrams, as well as the coefficients a_i^\vee, one sees that this data is not swapped under duality. A consequence of this is that taking duals does not commute with affinisation.

Table Aff 1

Table Aff 1 contains the Dynkin diagrams of the untwisted affine algebras: these are the affine Dynkin diagrams that can be obtained by an untwisted affinisation of the finite type Dynkin diagrams in Table Fin. The Cartan entries are obtained as follows: if \alpha_0 is the new affine root, then the new column of the Cartan matrix is defined by a_{i0} = \innprod{\alpha_i^\vee, \alpha_0} = \innprod{\alpha_i^\vee, -\widetilde{\alpha}}, where \widetilde{\alpha} is the highest root in the finite root system. Similarly, a_{0j} = \innprod{\alpha_0^\vee, \alpha_j} = \innprod{-\widetilde{\alpha}^\vee, \alpha_j}, where \widetilde{\alpha}^\vee is the coroot associated to the highest root (equivalently, the highest short coroot).

In Kac’ notation, A_{n}^{(1)} means the untwisted affinisation starting from A_n; in particular A_{n}^{(1)} has n + 1 nodes.

Some columns of this table need to be interpreted differently:

  • The red vertex in the Dynkin diagram is indicating the affine vertex that was attached during the construction of the affine algebra from the finite one. This is not a canonical property of the diagram, but is useful to know when studying the algebra. We have a_0 = 1 in all but the case of A_{2n}^{(2)} (shown in a table below), and a_0^\vee = 1 always.
  • The marks a_i are the coefficients of a linear dependence in the columns of the Coxeter matrix. They define a null root \delta = \sum_i a_i \alpha_i (not a root, but rather a distinguished element of the root lattice), so that in any realisation of the root system we have \innprod{\alpha_i^\vee, \delta} = 0 for all coroots \alpha_i^\vee.
  • The Coxeter number h = \sum_i a_i.
  • The comarks a_i^\vee are the coefficients of a linear dependence in the rows of the Coxeter matrix. They define the canonical central element K = \sum_i a_i^\vee \alpha_i^\vee (commonly also written as c). In any realisation of the root system we have \innprod{K, \alpha_i} = 0 for all i. The canonical central element defines the level of a representation.
  • The dual Coxeter number h^\vee = \sum_i a_i^\vee.

Note that now due to the symmetry in the definitions of a_i and a_i^\vee, we do have for affine Dynkin diagrams that a_i in the dual diagram is equal to a_i^\vee. There is a twist (ha-ha) however: some of the dual diagrams do not appear in Table Aff 1, rather they only appear in tables Aff 2 and 3, meaning that their corresponding algebras are more complicated to construct when starting from a finite-dimensional algebra.

Tables Aff 2 and Aff 3

The remaining affine type Dynkin diagrams arise through twisted affinisation, which is a more complicated process. Roughly how this is done is we first isolate a nontrivial diagram automorphism of one of the finite type diagrams. There are not so many of these: we can flip A_n for n \geq 2 (an order r = 2 automorphism), we can flip E_6 along its longest path (order r = 2), we can interchange the two end vertices of D_n for n \geq 4 (order r = 2), and we can rotate the leaves of D_4 (order r = 3). One then performs the loop algebra construction on the fixed-point subalgebra defined by the automorphism, using the automorphism to twist the Lie algebra structure somehow. We get several new types of diagrams:

The only entry in table Aff 3 is D_4^{(3)}. Also observe that if we were only interested in the Coxeter matrices associated to each Dynkin diagram, we would have already found all of the affine ones in table Aff 1.

Every indecomposable Dynkin diagram of finite or affine type appears in the tables above. Note that A_3^{(2)} = C_2^{(1)}, so by convention we remove A_2^{(3)} in order to get an irredundant list.