Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Cartan matrix
Summary
In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.
A (symmetrizable) generalized Cartan matrix is a square matrix with integral entries such that
For diagonal entries, .
For non-diagonal entries, .
if and only if
can be written as , where is a diagonal matrix, and is a symmetric matrix.
For example, the Cartan matrix for G2 can be decomposed as such:
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).
An matrix A is decomposable if there exists a nonempty proper subset such that whenever and . 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.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related concepts (8)
Semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra , if nonzero, the following conditions are equivalent: is semisimple; the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate; has no non-zero abelian ideals; has no non-zero solvable ideals; the radical (maximal solvable ideal) of is zero.
Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising (if for all , then ). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic . In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., ), a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements x such that the adjoint endomorphism is semisimple (i.
E8 (mathematics)
DISPLAYTITLE:E8 (mathematics) In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases.