Cartan matrix

Lie algebras

A (symmetrizable) generalized Cartan matrix is a square matrix ${\displaystyle A=(a_{ij})}$ with integer entries such that

1. For diagonal entries, ${\displaystyle a_{ii}=2}$.
2. For non-diagonal entries, ${\displaystyle a_{ij}\le 0}$.
3. ${\displaystyle a_{ij}=0}$ if and only if ${\displaystyle a_{ji}=0}$
4. ${\displaystyle A}$ can be written as ${\displaystyle DS}$, where ${\displaystyle D}$ is a diagonal matrix, and ${\displaystyle S}$ is a symmetric matrix.

For example, the Cartan matrix for G₂ can be decomposed as such:

${\displaystyle \left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}3& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}\frac{2}{3}& -1\\ -1& 2\end{array}\right].}$

The third condition is not independent but is really a consequence of the first and fourth conditions.

We can always choose a D with positive diagonal entries. In that case, if S in the above decomposition is positive definite, then A is said to be a Cartan matrix.

The Cartan matrix of a simple Lie algebra is the matrix whose elements are the scalar products

${\displaystyle a_{ji}=2\frac{(r_i,r_j)}{(r_j,r_j)}}$

(sometimes called the Cartan integers) where r_i are the simple roots of the algebra. The entries are integral from one of the properties of roots. The first condition follows from the definition, the second from the fact that for ${\displaystyle i\ne j,r_j-\frac{2(r_i,r_j)}{(r_i,r_i)}{r}_i}$ is a root which is a linear combination of the simple roots r_i and r_j with a positive coefficient for r_j and so, the coefficient for r_i has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let ${\displaystyle D_{ij}=\frac{{\delta }_{ij}}{(r_i,r_i)}}$ and ${\displaystyle S_{ij}=2(r_i,r_j)}$. Because the simple roots span a Euclidean space, S is positive definite.

Conversely, given a generalized Cartan matrix, one can recover its corresponding Lie algebra. (See Kac–Moody algebra for more details).

Classification

An ${\displaystyle n\times n}$ matrix A is decomposable if there exists a nonempty proper subset ${\displaystyle I\subset \{1,\dots ,n\}}$ such that ${\displaystyle a_{ij}=0}$ whenever ${\displaystyle i\in I}$ and ${\displaystyle j\notin I}$. A is indecomposable if it is not decomposable.

Let A be an indecomposable generalized Cartan matrix. We say that A is of finite type if all of its principal minors are positive, that A is of affine type if its proper principal minors are positive and A has determinant 0, and that A is of indefinite type otherwise.

Finite type indecomposable matrices classify the finite dimensional simple Lie algebras (of types ${\displaystyle A_n,B_n,C_n,D_n,E_6,E_7,E_8,F_4,G_2}$), while affine type indecomposable matrices classify the affine Lie algebras (say over some algebraically closed field of characteristic 0).

Determinants of the Cartan matrices of the simple Lie algebras

The determinants of the Cartan matrices of the simple Lie algebras are given in the following table (along with A₁=B₁=C₁, B₂=C₂, D₃=A₃, D₂=A₁A₁, E₅=D₅, E₄=A₄, and E₃=A₂A₁).

Another property of this determinant is that it is equal to the index of the associated root system, i.e. it is equal to ${\displaystyle |P/Q|}$ where P, Q denote the weight lattice and root lattice, respectively.