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Yang-mills and Some Related Algebras Dedicated to Jacques Bros

After a short introduction on the theory of homogeneous algebras we describe the application of this theory to the analysis of the cubic Yang-Mills algebra , the quadratic self-duality algebras, their " super " versions as well as to some generalization. Consider the classical Yang-Mills equations in (s + 1)-dimensional pseudo Euclidean space R s+1 with pseudo metric denoted by g µν in the canonical basis of R s+1 corresponding to coordinates x λ. For the moment the signature plays no role so g µν is simply a real nondegenerate symmetric matrix with inverse denoted by g µν. In terms of the covariant derivatives ∇ µ = ∂ µ + A µ (∂ µ = ∂/∂x µ) the Yang-Mills equations read g λµ [∇ λ , [∇ µ , ∇ ν ]] = 0 (1.1) for ν ∈ {0,. .. , s}. By forgetting the detailed origin of these equations, it is natural to consider the abstract unital associative algebra A generated by (s + 1) elements ∇ λ with the (s + 1) cubic relations (1.1). This algebra will be refered to as the Yang-Mills algebra. It is worth noticing here that Equations (1.1) only involve the product through commutators so that, by its very definition the Yang-Mills algebra A is a universal enveloping algebra. Our aim here is to present the analysis of the Yang-Mills algebra and of some related algebras based on the recent development of the theory of homogeneous algebras [2], [4]. This analysis is only partly published in [10].