Editing Cartan matrix (section)

== Lie algebras ==
{{Lie groups}}

A (symmetrizable) '''generalized Cartan matrix''' is a [[square matrix]] <math>A = (a_{ij})</math> with [[integer]] entries such that

# For diagonal entries, <math>a_{ii} = 2 </math>.
# For non-diagonal entries, <math>a_{ij} \leq 0 </math>.
# <math>a_{ij} = 0</math> if and only if <math>a_{ji} = 0</math>
# <math>A</math> can be written as <math>DS</math>, where <math>D</math> is a [[diagonal matrix]], and <math>S</math> is a [[symmetric matrix]].

For example, the Cartan matrix for [[G2 (mathematics)#Dynkin diagram and Cartan matrix|''G''<sub>2</sub>]] can be decomposed as such:
:<math>
  \begin{bmatrix}
     2 & -3 \\
    -1 &  2
  \end{bmatrix} =
  \begin{bmatrix}
    3&0\\
    0&1
  \end{bmatrix}\begin{bmatrix}
     \frac{2}{3} & -1 \\
    -1           &  2
  \end{bmatrix}.
</math>

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 matrix|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 product]]s

:<math>a_{ji}=2 {(r_i,r_j)\over (r_j,r_j)}</math><ref>{{cite book |last1=Georgi |first1=Howard |title=Lie Algebras in Particle Physics |publisher=Westview Press |isbn=0-7382-0233-9 |page=115 |edition=2|date=1999-10-22 }}</ref>

(sometimes called the '''Cartan integers''') where ''r<sub>i</sub>'' are the [[root system|simple roots]] of the algebra. The entries are integral from one of the properties of [[root system|root]]s. The first condition follows from the definition, the second from the fact that for <math>i\neq j, r_j-{2(r_i,r_j)\over (r_i,r_i)}r_i</math> is a root which is a [[linear combination]] of the simple roots ''r<sub>i</sub>'' and ''r<sub>j</sub>'' with a positive coefficient for ''r<sub>j</sub>'' and so, the coefficient for ''r<sub>i</sub>'' has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let <math>D_{ij}={\delta_{ij}\over (r_i,r_i)}</math> and <math>S_{ij}=2(r_i,r_j)</math>. 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 <math>n \times n</math> matrix ''A'' is '''decomposable''' if there exists a nonempty proper subset <math>I \subset \{1,\dots,n\}</math> such that <math>a_{ij} = 0</math> whenever <math>i \in I</math> and <math>j \notin I</math>. ''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 minor]]s 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 algebra]]s (of types <math>A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2 </math>), while affine type indecomposable matrices classify the [[affine Lie algebra]]s (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 A<sub>1</sub>=B<sub>1</sub>=C<sub>1</sub>, B<sub>2</sub>=C<sub>2</sub>, D<sub>3</sub>=A<sub>3</sub>, D<sub>2</sub>=A<sub>1</sub>A<sub>1</sub>, E<sub>5</sub>=D<sub>5</sub>, E<sub>4</sub>=A<sub>4</sub>, and E<sub>3</sub>=A<sub>2</sub>A<sub>1</sub>).<ref>[https://deepblue.lib.umich.edu/bitstream/handle/2027.42/70011/JMAPAQ-23-11-2019-1.pdf Cartan-Gram determinants for the simple Lie Groups] Alfred C. T. Wu, J. Math. Phys. Vol. 23, No. 11, November 1982</ref>
{| class="wikitable" border="1"
|- style="vertical-align:top"
! A<sub>''n''</sub>
! B<sub>''n''</sub>
! C<sub>''n''</sub>
! D<sub>''n''</sub><br/>''n'' ≥ 3
! E<sub>''n''</sub><br/>3 &le; ''n'' &le; 8
! F<sub>4</sub>
! G<sub>2</sub>
|- align=center
| ''n'' + 1 || 2 || 2 || 4 || 9 − ''n'' || 1 || 1
|}
Another property of this determinant is that it is equal to the index of the associated root system, i.e. it is equal to <math>|P/Q| </math> where {{mvar|P, Q}} denote the weight lattice and root lattice, respectively.