We give a classification of the Kac-Moody current algebras of all the possible massless fermion-gauge theories in two dimensions. It is shown that only Kac-Moody algebras based on AN, BN, CN, and DN in the Cartan classification with all possible central charge occur. The representation of local fermion fields and simply laced Kac-Moody algebras with minimal central charge in terms of free boson fields on a compactified space is discussed in detail, where stress is laid on the role played by the boundary conditions on the various collective modes. Fractional solitons and the possible soliton representation of certain nonsimply laced algebras is also analysed. We briefly discuss the relationship between the massless bound state sector of these two-dimensioned gauge theories and the critically coupled two-dimensional nonlinear sigma model, which share the same current algebra. Finally we briefly discuss the relevance of Sp(n) Kac-Moody algebras to the physics of monopole-fermion systems.

We illustrate through the examples of the osp(2,1) and spl(2,1) algebras the differences between the properties of the irreducible representations of simple graded Lie algebras and simple Lie algebras.

Hermitian representations play a fundamental role in the study of the representations of simple Lie algebras. We show how this concept generalizes for classical simple graded Lie algebras. Star and grade star representations are defined through adjoint and grade adjoint operations. Each algebra admits at most two adjoint and two grade adjoint operations (we list the various possibilities for all classical simple graded Lie algebras). To each adjoint (grade adjoint) operation corresponds a class of star (grade star) representations. The tensor product of two star representations belonging to one class is completely reducible into irreducible representations belonging to the same class. This property is very useful since in general the finite-dimensional representations of classical simple graded Lie algebras are not completely reducible.

The exceptional simple graded Lie algebras whose existence is suggested by the results of the preceding paper are explicitly constructed. In this way the classification of all simple graded Lie algebras whose Lie algebra is reductive is completed.

By definition simple pseudo Lie algebras do not contain any nontrivial ideal. We show that ’’graded simplicity’’ implies ’’simplicity’’ and discuss the uniqueness of invariant bilinear forms on a simple pseudo Lie algebra. A lot of examples of simple pseudo Lie algebras is given together with their invariant bilinear forms. Under certain general assumptions we derive that the Lie algebra ℊ contained in a simple pseudo Lie algebra 𝒶 is reductive. Assuming that ℊ is reductive, we prove that the ’’adjoint representation of ℊ in the odd subspace of 𝒶’’ is completely reducible with at most two irreducible components. Finally we show that the pseudo Lie algebras with nondegenerate ’’generalized Killing form’’ are direct products of simple pseudo Lie algebras.

Strictly semi-simple graded Lie algebras are defined and are shown to have properties similar to semi-simple Lie algebras. Their classification is presented and it is shown that they cannot be used as a symmetry of know hadrons.