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

The already extensive diversity of functions of ring conception has been improved within the eighties through the expanding curiosity in algebraic constructions of substantial complexity, the so-called classification of quantum teams. one of many primary houses of quantum teams is they are modelled via associative coordinate jewelry owning a canonical foundation, which permits for using algorithmic constructions according to Groebner bases to review them. This ebook develops those tools in a self-contained method, focusing on an in-depth research of the concept of an unlimited classification of non-commutative jewelry (encompassing so much quantum groups), the so-called Poincaré-Birkhoff-Witt jewelry. We contain algorithms which deal with crucial features like beliefs and (bi)modules, the calculation of homological size and of the Gelfand-Kirillov measurement, the Hilbert-Samuel polynomial, primality checks for top beliefs, etc.

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5. ) 'k-algebra U is said to be a (universal) enveloping algebra for g, if it is endowed with a representation u : 9 ..... U, satisfying the following universal property: for any 'k-algebra V and any representation v : 9 ..... V, there is a unique 'k-algebra morphism w : U ..... V, making the following diagram commutative: DEFINITION g~U ~lw V It is clear that an enveloping algebra for g, if it exists, is necessarily unique (up to isomorphism) - we will denote it by U(g). On the other hand, enveloping algebras actually always do exist.

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. 11. Let R be a domain and let O =/=. r,s E R. The following assertions are equivalent: (1) there is an isomorphism of left R -modules LEMMA R/Rr"":" R/Rs; (2) there is an isomorphism of right R -modules R/rR"":" R/sR. PROOF. Let us prove that the first assertion implies the second one. The other implication follows by symmetry.

For any pair of indices 1 ~ i, j < n, put aii = 2 resp. aij = -1 if li - j I = 1, and aij = O otherwise. It may then be proved that EXAMPLE 47 5. OTHER EXAMPLES modulo the relations XiYj - YjXi HiHj -HjHi =O Hi XiYi - YiXi HiXj - XjHi ajiXj HiYj - YjHi -ajiYj as well as (ad(Xi))l- aj i(Xj) = O (ad(Yi))l-aji (Yj) = O for any i *" }, where ad is defined by ad (X) (Y) = XY - Y x. 9. The previous example is actually a special case of a much more general result, due to Serre, describing the enveloping algebra U (g) of a complex semisimple Lie algebra 9 in terms of its associated Cartan matrix (aij).