If F [right arrow] F is a homogeneous linear operator, we can extend it to the linear operator [L.sub.A] = L(F) [right arrow] L(F) on the

graded algebra of linear operators L(F) by means of the graded q-commutator as follows:

In this note we modify the definition of a Rota-Baxter system by including a curvature term and then derive the conditions that the curvature has to satisfy in order to yield a pre-Lie, associative or curved differential

graded algebra structures.

Definition 1 (Polynomial realization): Let A := [[direct sum].sub.n[greater than or equal to]0][A.sub.n] be a

graded algebra. A polynomial realization r of A is a map which associates to each alphabet U an injective

graded algebra morphism [r.sub.U] from A to the free non-commutative algebra K<U> such that, if U [subset] B, then for all x [member of] A one has [r.sub.U](x) = [r.sub.B](x)/U, where [r.sub.B](x)/U is the sub linear combination obtained from [r.sub.B](x) by keeping only those words in [U.sup.*].

If the symmetric algebra of I (G) in R is a integrity domain, then the canonical morphism of

graded algebrasWe write A for the zero- degree component of a

graded algebra [A.sup.*] = [[direct sum].sub.n[member of]N [A.sup.n], and [a, b] denotes the graded commutator, i.e.

In the previous section we used a combinatorial interpretation of the leading ideal of I = ([f.sub.2], [f.sub.3], ...) to compute the HP-series of the corresponding

graded algebra. There are commutative algebra methods to do this as well which yield a new approach to the Rogers-Ramanujan identity.

By the dimension formula (2.2), the cohomology algebra of such a space is isomorphic to either Q[[x.sub.1]]/([f.sub.1]) or Q[[x.sub.1], [x.sub.2]]/([f.sub.1], [f.sub.2]) as a

graded algebra, where ([f.sub.1], [f.sub.2]) is the ideal generated by a regular sequence {[f.sub.1], [f.sub.2]}, and hence the rational homotopy types of this kind are intrinsically formal, that is, two spaces with the isomorphic rational cohomology algebras are rationally homotopy equivalent.

Let R be a positively

graded algebra and G be a finite group of grading preserving automorphisms of R.

This conjecture was confirmed by results of Shan, Varagnolo and Vasserot in [SVV11], in which they introduce and use a family of

graded algebras, by showing that categories of modules over these algebras are equivalent to categories of modules over affine Hecke algebras of type D.

Simn, "Crossed products by twisted partial actions and

graded algebras", J.

Noncommutative

graded algebras and their Hilbert series.