Lie Algebras
| Lie groups |
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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) |
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Simple Lie groups
List of simple Lie groups Classical: An, Bn, Cn, Dn Exceptional: G2, F4, E6, E7, E8 |
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Other Lie groups
Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group |
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Lie algebras
Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Lie point symmetry |
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Semi-simple Lie groups
Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra |
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Representation theory
Representation of a Lie group Representation of a Lie algebra |
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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, aii = 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 ].](http://upload.wikimedia.org/math/8/8/7/8874c62f4b110ab076b6bdd53a036447.png)
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 ri 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 ri and rj with a positive coefficient for rj and so, the coefficient for ri 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
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