Ph.D Thesis | |

Ph.D Student | David Ofir |
---|---|

Subject | The Center of the Generic G-Crossed Product |

Department | Department of Mathematics |

Supervisor | PROF. Eli Aljadeff |

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: GxG→F^{*},
if for any sequence (g_{1},?,g_{n}) in G^{n } there
are a_{i} homogeneous of degree g_{i} such that a_{1}*a_{2}*?*a_{n} ≠0
, and for any a_{g}, b_{h} of degrees g and h respectively
we have a_{g}*b_{h} = θ(g,h) b_{h}*a_{g}.

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 M_{n}(F) admits regular
gradings.

For
each regular graded algebra A we show that A has the same polynomial identities
as M_{n}(F); M_{n}(E) or M_{2n;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 (M_{G})_{g1,g2}= θ(g_{1},g_{2})
and (M_{H})_{h1,h2}= η(h_{1},h_{2})
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.