An explicit classification of the semisimple complex Lie subalgebras of the simple complex Lie algebras is given for algebras up to rank 6. The notion of defining vector, introduced by Dynkin and valid for subalgebras of rank 1, has been extended to the notion of defining matrix, valid for any semisimple subalgebra. All defining matrices have been determined explicitly, which is equivalent to the determination of the embeddings of the generators of the Cartan subalgebra of a semisimple subalgebra in the Cartan subalgebra K of the simple algebra containing this subalgebra. Moreover, the embedding of the root system of the subalgebras in the dual space K* of an algebra is given for all subalgebras. For the S-subalgebras of the simple algebras (up to rank 6), the embedding of the whole subalgebra in an algebra is given explicitly. In addition, the decomposition (branching) of the defining (fundamental) and adjoint representations of an algebra with respect to the restriction to its S-subalgebras has been determined. In the first part of this article a brief review of Dynkin's theory of the classification of the semisimple Lie subalgebras of the simple Lie algebras is given. No proofs are repeated, and at places where concepts have been extended and new results derived, merely an indication for their proof is given. This part of the article will serve as a prescription for a classification of semisimple subalgebras of the simple Lie algebras of rank exceeding 6. Later in the article, explicit expressions are given for the index of an embedding of a simple Lie subalgebra in a simple Lie algebra. These expressions are valid for the classical Lie algebras of arbitrary rank as well as for the exceptional Lie algebras.

Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight. Using the Infeld-Hull factorization method, we generate from the raising and lowering operators the second order self-adjoint differential/difference operator of hypergeometric type.

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The Kravchuk and Meixner polynomials of discrete variable are introduced for the discrete models of the harmonic oscillator and hydrogen atom. Starting from Rodrigues formula we construct raising and lowering operators, commutation and anticommutation relations. The physical properties of discrete models are figured out through the equivalence with the continuous models obtained by limit process. PACS: 02.20.+b, 03.65.Bz, 03.65.Fd