By Ladislav NebeskyÌ

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**Example text**

Notice that the G i] are ideals of G . Property Let G be a Lie algebra. { G is solvable if and only if the derivative G 0 is nilpotent. { If G is nilpotent, then G is solvable (but a solvable Lie algebra is not necessary nilpotent: as an example, one can consider the two-dimensional Lie algebra generated by a and b such that a b] = b). { If G is solvable (resp. nilpotent), then any subalgebra H of G is a solvable (resp. nilpotent) Lie algebra. De nition Let G be a Lie algebra. The maximal solvable ideal of G is called the radical of G .

It is maximal as Abelian subalgebra of G , that is any Abelian subalgebra of G is in H (up to a conjugation). Because of its uniqueness, one can say that H is \the" maximal Abelian subalgebra of G . e. for any X 2 G , one can choose H such that X 2 H). P For any element H = ri=1 i Hi 2 H, one can write h i H E = (H ) E where is a linearP functional on H, that is an element of the dual H of H, such that (H ) = ri=1 i i . Since there exists a unique (up to a multiplicative factor) non-degenerate symmetric bilinear form B (!

In order to determine which irreducible representations of (su(p) su(q)) are contained in the irreducible representation ], we have to compute all the possible irreducible representations of su(q) and su(p), with n boxes, such that the \inner" product of such Young tableaux, considered as irreducible representations of the symmetric group Sn , contains the Young tableau ], with the appropriate multiplicity. Rules for performing such \inner" product have been given by several authors, see ref. 38] where references to the original papers can be found.

### Algebraic properties of trees (Acta Universitatis Carolinae : Philologica monographia) by Ladislav NebeskyÌ

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