Cartan matrix

The Cartan matrix of a finite-dimensional semi-simple Lie algebra over an algebraically closed field of characteristic is a matrix

where is some system of simple roots of with respect to a fixed Cartan subalgebra and is the scalar product on the dual space of defined by the Killing form on . (For the Cartan matrix of an arbitrary system of roots see Root system.) Up to a transformation induced by a permutation of the indices , a Cartan matrix is an invariant of , that is, it does not depend on the choice of or the system of simple roots. This invariant completely determines : Two semi-simple Lie algebras are isomorphic if and only if their Cartan matrices are the same, up to a transformation induced by permutation of the indices. A semi-simple Lie algebra is simple if and only if its Cartan matrix is indecomposable, that is, not expressible as a block-diagonal matrix after some permutation of indices.

Let be a decomposition of into a direct sum of simple subalgebras and let be the Cartan matrix of the simple Lie algebra . Then the block-diagonal matrix

is the Cartan matrix of . (For the explicit form of the Cartan matrix of simple Lie algebras, see Lie algebra, semi-simple.)

The entries of a Cartan matrix have the following properties:

(1)

The Cartan matrix is closely related to the description of in terms of generators and relations. Namely, there exist in linearly independent generators , , , (the so-called canonical generators), connected by the following relations:

(2)

Any two systems of canonical generators can be transformed one into the other by automorphism of . In addition to (2), the canonical generators satisfy the relations

(3)

where, by definition, . For the chosen system of generators , , , , relations (2) and (3) are defining for (see [2]).

For any matrix satisfying (1), the Lie algebra over the field with generators , , , , and defining relations (2) and (3) is finite-dimensional if and only if is the Cartan matrix of a semi-simple Lie algebra [3].

References

[1]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[2]	J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966)
[3]	V.G. [V.G. Kats] Kac, "Simple irreducible graded Lie algebras of finite growth" Math. USSR Izv. , 2 : 6 (1968) pp. 1271–1311 Ivz. Akad. Nauk SSSR Ser. Mat. , 32 : 6 (1968) pp. 1323–1367

Comments

A matrix satisfying (1) defines a finite-dimensional Lie algebra if and only if it is positive definite; in other cases, notably the semi-positive definite case, other interesting algebras arise, cf. Kac–Moody algebra, [a2].

Let be a semi-simple Lie algebra over an algebraically closed field of characteristic zero. Then a set of generators , , such that (2) holds is also called a set of Chevalley generators or a Chevalley basis. That such a set of generators exists is Chevalley's theorem. The result that the relations (2) and (3) together define the Lie algebra is often called Serre's theorem.

References

[a1]	J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972)
[a2]	V.G. Kac, "Infinite dimensional Lie algebras" , Cambridge Univ. Press (1985)

The Cartan matrix of a finite-dimensional associative algebra with a unit over a field is the matrix , , defined by a complete set of finite-dimensional irreducible left -modules. Specifically, is the multiplicity of occurrence of in a composition series of an indecomposable projective left -module for which . Such a module exists for each and is uniquely defined up to an isomorphism.

In certain cases the Cartan matrix turns out to be symmetric, positive definite, and even , where is an integral, not necessarily square, matrix (and denotes taking the transpose matrix). Such is the case for the Cartan matrix of the group algebra of a finite group over a field of characteristic (see [1]); in this case the form a complete set of non-isomorphic principal indecomposable left -modules, that is, indecomposable -modules into which the left -modules is decomposed as a direct sum. Another example when such an equality holds for a Cartan matrix: is the restricted universal enveloping algebra of a Lie algebra over an algebraically closed field of characteristic , obtained from a semi-simple complex Lie algebra by reduction to characteristic (see [2]).

References

[1]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)
[2]	J.E. Humphreys, "Modular representations of classical Lie algebras and semi-simple groups" J. of Algebra , 19 (1971) pp. 51–79