Graded generalizations of Weyl- and Clifford algebras

Graded skew bilinear forms {,} on graded vector spaces V are defined such that their restrictions to the even resp. odd subspaces are skew resp. odd. Over such graded symplectic vector spaces a (universal) factor algebra of the tensor algebra of V is described which reduces to a Weyl- resp. Clifford algebra if only one even resp. odd subspace is nontrivial. Introducing the total graduation on this polynomial algebra and graded symmetrization it is shown that the elements up to second power are closed under graded commutation. If the graduation is of type Z₂ the elements of second power are a Lie-graded algebra and this is the only graduation for which this is true. The graded commutation relations of this algebra are calculated. It is isomorphic to the graded symplectic algebra of (V,{,}) which is contained in the graded derivation algebra of the graded Heisenberg algebra of elements up to first power.