M.Sc Student M.Sc Thesis David Ofir On Regular Gradings of Algebras Department of Mathematics PROF. Eli Aljadeff

Abstract

Let G be a finite abelian group and A a G-graded algebra over a field F. The G-grading is called regular with commutation function θ: G?G→F?, if for any sequence (g1,?,gn) in Gn  there are ai homogeneous of degree gi such that a1*a2*?*an ≠0 , and for any ag, bh of degrees g and h respectively we have ag*bh = θ(g,h) bh*ag.

Each such function θ must also be a skew symmetric bicharacter, meaning that for any g, h in G we have θ(g,h) = θ(h,g)-1, and for fixed h in G the maps g→θ(h,g) and g→θ(g,h) are characters on G.

A regular grading is called minimal (and θ is called non-degenerate) if for any e≠g in G there is some h in G such that θ(g,h)≠1.

A prime example for a regular graded algebra is the Grassmann algebra with its standard Z2-grading. In addition, any matrix algebra Mn(F) admits regular gradings.

For each regular graded algebra A we show that A has the same polynomial identities as Mn(F); Mn(E) or M2n;n(E), where E is the Grassmann algebra. We use this presentation of regularly graded algebras to prove that if A has a minimal regular G-grading, then the cardinality |G| is an invariant of the algebra A. Moreover, if (H, η) is another minimal regular grading then the matrices defined by (MG)g1,g2= θ(g1,g2) and (MH)h1,h2= η(h1,h2)  are conjugate.

If θ is a skew symmetric bicharacter on G, then we construct a regular G-graded algebra A with commutation function θ. For any group G we classify all the non-degenerate skew symmetric bicharacters on G.

Finally, we define a twisted tensor product of two regular graded algebras, and prove some properties of this operation.