By José Bueso, José Gómez-Torrecillas, Alain Verschoren (auth.)

The already wide diversity of functions of ring thought has been better within the eighties through the expanding curiosity in algebraic constructions of substantial complexity, the so-called type of quantum teams. one of many primary homes of quantum teams is they are modelled via associative coordinate earrings owning a canonical foundation, which permits for using algorithmic constructions in keeping with Groebner bases to check them. This publication develops those tools in a self-contained approach, targeting an in-depth examine of the suggestion of an unlimited classification of non-commutative jewelry (encompassing such a lot quantum groups), the so-called Poincaré-Birkhoff-Witt jewelry. We contain algorithms which deal with crucial elements like beliefs and (bi)modules, the calculation of homological measurement and of the Gelfand-Kirillov measurement, the Hilbert-Samuel polynomial, primality exams for high beliefs, etc.

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Besides their obvious geometric importance, these rings also appear to be very interesting from a purely ring-theoretic point of view. ActualIy, alI the elementary examples of quantum groups are (iterated) Ore extensions, whereas many more complicated ones stiU possess similar properties. In particular, we will see that these rings thus also motivate the study of PBW algebras, to be introduced below. Throughout this section, let us fix an arbitrary field k As we already announced above, the fust example we will study of so-called quantum groups over 1< is the quantum plane.

The quantum plane kq[x,y] is a noetherian domain a basis of which (as a vectorspace over k) is given by the elements xiyi, where i and j are positive integers. 4. Let R be an arbitrary k-algebra. Then the k-algebra homomorphisms f: kq[x,y] - R are said to be the R -points of kq [x, y]. Somewhat more generally, for any pair ofk-algebras R and S, the k-algebra homomorphisms S -- R are said to be R -points of S (defined over k). It is easy to see that R-points of the quantum plane correspond bijectively to couples (r,s) E R 2 with the property that sr = qrs.

Consider a quasi-derivation (u,8) on R and let S = R[x; u, 8] be the associated Ore extension. (1) if u is injective and if R is a domain, then so is S; (2) if u is an automorphism and if R is prime, then so is S; (3) if u is an automorphism and if R is left (resp. right) noetherian, then so is S. PROOF. To prove the fust assertion, pick non-zero elements j, 9 E S, say with degrees n = deg(f) and m = deg(g) and corresponding leading coefficients r n resp. Sm. We then know that jg has degree n + m (with leading coefficient rnum(sm), which is non-zero, in view of the fact that R is a domain and the injectivity of u) so, in particular, jg *0.