G-Algebra Ontology

An informal description of the Ontology of the G-Algebra Knowledge Base developed so far.

Metadata and Semantics

All G-Algebra instances belong to the owl:Class sd:GAlgebra with predicates

• sd:relatedXMLResource URI - the Resource file
• standard predicates sd:createdAt, sd:createdBy, rdfs:comment

The G-Algebra is the factor of the “almost” commutative polynomial rings, i.e. polynomial rings R=k[x_1,…,x_n] with commutator relations

x_j*x_i=c_ij*x_i*x_j+d_ij for i<j, c_ij in k, d_ij in R, with default c_ij=1, d_ij=0 (x_i and x_j commute in this case),

by the left(?) ideal generated by the given basis.

For a G-Algebra all polynomials in R can be uniquely rewritten as sum of standard monimials in ordered distributive form, the Poincaré standard form.

The base domain k is a field, either Q or Q(a1,…,ak) for (commuting) parameters a1,…,ak

XML-Resources

The XML-Resource contains mainly the following tags

• (optionally) - a list a1,...,ak of (commuting) parameters
• - a list x1,...,xn of (almost commuting) variables,
• - a sequence of non trivial commutator rules (only such commutator rules are recorded)

A commutator rule

1</D> 2\*x </commutator> represents the rule snd\*fst = C\*fst\*snd + D. - - a sequence of entries of non commutative polynomials For details see the XSchema description [GAlgebras.xsd](http://symbolicdata.org/XMLResources/GAlgebras.xsd).