Cartan Matrix - Lie Algebras

Lie Algebras

Lie groups

Classical groups General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n)
Simple Lie groups List of simple Lie groups Classical: A_n, B_n, C_n, D_n Exceptional: G₂, F₄, E₆, E₇, E₈
Other Lie groups Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group
Lie algebras Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Lie point symmetry
Semi-simple Lie groups Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Representation of a Lie group Representation of a Lie algebra
Lie groups in Physics Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group Representation theory of the Galilean group

A generalized Cartan matrix is a square matrix with integral entries such that

For diagonal entries, a_ii = 2.
For non-diagonal entries, .
if and only if
A can be written as DS, where D is a diagonal matrix, and S is a symmetric matrix.

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

$\left [ \begin{smallmatrix} \;\,\, 2&-3\\ -1&\;\,\, 2 \end{smallmatrix}\right ] = \left [ \begin{smallmatrix} 3&0\\ 0&1 \end{smallmatrix}\right ] \left [ \begin{smallmatrix} 2/3&-1\\ -1&\;2 \end{smallmatrix}\right ].$

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

(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 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 and . 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).

Read more about this topic: Cartan Matrix

Famous quotes containing the word lie:

“‘Tis to yourself I speak; you cannot know
Him whom I call in speaking such a one,
For you beneath the earth lie buried low,
Which he alone as living walks upon:”
—Jones Very (1831–1880)