By José Bueso, José Gómez-Torrecillas, Alain Verschoren (auth.)
The already vast diversity of functions of ring concept has been greater within the eighties through the expanding curiosity in algebraic buildings of substantial complexity, the so-called classification of quantum teams. one of many basic homes of quantum teams is they are modelled by way of associative coordinate earrings owning a canonical foundation, which permits for using algorithmic buildings in keeping with Groebner bases to review them. This booklet develops those equipment in a self-contained method, targeting an in-depth research of the idea of an enormous classification of non-commutative earrings (encompassing so much quantum groups), the so-called Poincaré-Birkhoff-Witt jewelry. We contain algorithms which deal with crucial facets like beliefs and (bi)modules, the calculation of homological size and of the Gelfand-Kirillov measurement, the Hilbert-Samuel polynomial, primality exams for top beliefs, etc.
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Extra info for Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups
B of x resp. y in k, the former satisfying ba = qab. Of course, it then follows that a = O or b = O, as q =1= 1, so, looking at kemels, obviously the ideals ofthe form (x - a,y) and (x,y - b) are prime (and even maximal). With a little more effort, one may show that if q is not a primitive root of unity, then these are the only prime ideals of kq [x, y], except for the zero ideal and the ideals (x) and (y). It follows that the prime spectrum of the quantum plane thus consists of the x -axis and y -axis as well as the points on it.
This ring should be the coordinate ring of the "quantum 49 6. QUANTUM GROUPS plane" (whatever this geometric object may be), but is usualIy itself referred to as the quantum plane. At this moment, there is a whole bunch of rings which alI deserve the name quantum group - we refer to the literature, ef. [46, 71,87,88], and the rest of this section for examples. 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.
If R is commutative and if r, sER, then the following assertions are equivalent: PROPOSITION (1) (2) r r ~ ~ s; s. is obvious that the first statement implies the second one. Conversely, if r ~ s, then there exists some isomorphism of left Rmodules cI>: R/Rr --=-. R/Rs: x + Rr xa + Rs, for some aER. , that Rs ~ Rr. The other inclusion Rr ~ Rs follows by the same argument, applied to the isomorphism ci> -1. , that r ~ s, indeed. O PROOF. It 1--+ 31 4. FACTORIZATION As we will see beIow, this result shows that the non-commutative theory generalizes the "usual" theory in the commutative case.
Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups by José Bueso, José Gómez-Torrecillas, Alain Verschoren (auth.)