By Gerstenhaber M., Schack D.

This paper is an improved model of comments introduced via the authors in lectures on the June, 1990 Amherst convention on Quantum teams. There we have been requested to explain, in as far as attainable, the elemental rules and effects, in addition to the current nation, of algebraic deformation thought. So this paper incorporates a mix of the previous and the recent. we have now tried to supply a clean standpoint even at the extra "ancient" issues, highlighting difficulties and conjectures of common curiosity all through. We hint a course from the seminal case of associative algebras to the quantum teams that are now riding deformation thought in new instructions. certainly, one of many delights of the topic is that the research of btalgebra deformations has resulted in clean insights within the classical case of associative algebra - even polynomial algebra! - deformations.

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**Extra resources for Algebras, bialgebras, quantum groups, and algebraic deformation**

**Example text**

Stream(X). will systematically produce all ﬁnite streams one by one, starting from the [] stream. Suppose now we remove the base case and obtain the program P2: stream([H|T]) :- number(H), stream(T). - stream(X). fails, since the model of P2 does not contain any instances of stream/1. The problems are two-fold: (i) the Herbrand universe does not contain inﬁnite terms; (ii) the least Herbrand model does not allow for inﬁnite proofs, such as the proof of stream(X); yet these concepts are commonplace in computer science, and a sound mathematical foundation exists for them in the ﬁeld of hyperset theory [2].

N ≤n ≤n ≤n ≤n ≤n ≤n ≤n Fig. 2. An example of the structure of a bamboo: it consists of a stem of unbounded length from which subtrees of height at most n sprout; on the right it is shown with its stem straightened Figure 2 illustrates the definition. The definition of “bamboo” directly leads to an unfolding rule for G: in every rule we limit the recursion depth of all but one terminal to n in the same way as we did in the case of the Kleene approximation. Notice that, since V (G) = V (L) by idempotence, we do not need to ensure that each derivation tree of the unfolded grammars uniquely corresponds to a derivation tree of G.

Let us “unfold” the grammar G of (4) by augmenting the nonterminal X with a counter keeping track of the height of a derivation: 24 J. Esparza and M. Luttenberger X 1 X [1] X 2 X [2] X 3 X [3] →c →X 1 → aX 1 X 1 | bX 1 → X 2 | X [1] → aX 2 X 2 | aX [1] X → X 3 | X [2] .. 2 2 | aX X h → aX h−1 X h−1 | aX [h−2] X X [h] → X h | X [h−1] .. X [1] | bX h−1 | aX 2 h−1 X [h−2] | bX h−1 Let G[h] (G h ) be the grammar consisting of those “unfolded” rules whose left-hand side is given by one of the variables of X [h] = {X 0 , X [0] , .