M.Sc Thesis | |

M.Sc Student | David Ofir |
---|---|

Subject | On Regular Gradings of Algebras |

Department | Department of Mathematics |

Supervisor | PROF. Eli Aljadeff |

Full Thesis text |

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 (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.