In the present paper we analyze algebraic structures arising in Yang-Mills theory. The paper should be considered as a part of a project started with [15] and devoted to maximally supersymmetric Yang-Mills theories. In this paper we collected those of our results which are correct without assumption of supersymmetry and used them to give rigorous proofs of some results of [15]. We consider two different algebraic interpretations of Yang-Mills theory -in terms of A ∞ -algebras and in terms of representations of Lie algebras (or associative algebras). We analyse the relations between these two approaches and calculate some Hochschild (co)homology of algebras in question.

We consider ten-dimensional supersymmetric Yang-Mills theory (10D SUSY YM theory) and its dimensional reductions, in particular, BFSS and IKKT models. We formulate these theories using algebraic techniques based on application of differential graded Lie algebras and associative algebras as well as of more general objects, L∞-and A∞-algebras.We show that using pure spinor formulation of 10D SUSY YM theory equations of motion and isotwistor formalism one can interpret these equations as Maurer-Cartan equations for some differential Lie algebra. This statement can be used to write BV action functional of 10D SUSY YM theory in Chern-Simons form. The differential Lie algebra we constructed * The work of both authors was partially supported by NSF grant No. DMS 0204927 1 is closely related to differential associative algebra (Ω,∂) of (0, k)-forms on some supermanifold; the Lie algebra is tensor product of (Ω,∂) and matrix algebra . We construct several other algebras that are quasiisomorphic to (Ω,∂) and, therefore, also can be used to give BV formulation of 10D SUSY YM theory and its reductions. In particular, (Ω,∂) is quasi-

We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare Lie algebra. We give complete answers for (non-extended) supersymmetry in all dimensions ≤ 11. For dimensions D = 10, 11 we describe also the cohomology of reduction of supersymmetry Lie algebra to lower dimensions. Our methods can be applied to extended supersymmetry algebra.

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We study supersymmetric and super Poincaré invariant deformations of ten-dimensional super Yang-Mills theory and of its dimensional reductions. We describe all infinitesimal super Poincaré invariant deformations of equations of motion of ten-dimensional super Yang-Mills theory and its reduction to a point; we discuss the extension of them to formal deformations. Our methods are based on homological algebra, in particular, on the theory of L-infinity and A-infinity algebras. The exposition of this