On the properties of branching coefficients for affine Lie groups

It is demonstrated that the decompositions of integrable highest weight modules of a simple Lie algebra (classical or affine) with respect to its reductive subalgebra obey a set of algebraic relations leading to recursive properties for the corresponding branching coefficients. These properties are encoded in a special element $\Gamma _{\mathfrak{g}\supset\mathfrak{a}}$ of the formal algebra $\mathcal{E}_{\mathfrak{a}}$ that describes the injections $\mathfrak{a}\to \mathfrak{g}$ and is called a fan. In the simplest case where $\mathfrak{a}=\mathfrak{h}\left(\mathfrak{g}\right)$, the recursion procedure generates the weight diagram of a module $L_{\mathfrak{g}}$. When the recursion described by a fan is applied to highest weight modules, it provides a highly efficient tool for explicit calculations of branching coefficients.