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By Gerstenhaber M., Schack D.

This paper is an extended model of feedback added by means of 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 fundamental ideas and effects, in addition to the current country, of algebraic deformation concept. So this paper features a mix of the outdated and the recent. we now have tried to supply a clean viewpoint even at the extra "ancient" subject matters, highlighting difficulties and conjectures of basic curiosity all through. We hint a course from the seminal case of associative algebras to the quantum teams that are now riding deformation idea in new instructions. certainly, one of many delights of the topic is that the examine of btalgebra deformations has resulted in clean insights within the classical case of associative algebra - even polynomial algebra! - deformations.

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2) Seien H1 , H2 Halbgruppen und e2 neutrales Element in H2 . Dann ist H1 → H2 , x → e2 f¨ ur alle x ∈ H1 , ein (trivialer) Homomorphismus. (3) Die Exponentialfunktion x → ex , exp : (IR, +) → (IR>0 , · ), ist ein Homomorphismus, da ex+y = ex ey . Auch der Logarithmus log : (IR>0 , · ) → (IR, +), x → log(x), ist ein Homomorphismus, denn log(xy) = log(x) + log(y). (4) Sei (H, · ) eine Halbgruppe. Dann ist die Linksmultiplikation mit einem a ∈ H eine Abbildung von H in sich, La : H → H, x → a · x.

20). Zeigen Sie, daß es zu jedem k ∈ I einen Homomorphismus ε : Hk → I Hi gibt mit π ◦ ε = idHk . Man folgere, daß jedes Hk isomorph ist zu einer Untergruppe von I Hi . 51 7. Ringe und K¨ orper 7 Ringe und Ko ¨rper In diesem Abschnitt untersuchen wir algebraische Strukturen mit zwei Verkn¨ upfungen. 1 Definition Eine Menge R mit zwei Verkn¨ upfungen +, · , also ein Tripel (R, +, · ), heißt ein Ring, wenn gilt: (i) (R, +) ist eine abelsche Gruppe, (ii) (R, · ) ist eine Halbgruppe mit neutralem Element, (iii) f¨ ur alle a, b, c ∈ A gelten die Distributivgesetze a · (b + c) = a · b + a · c, (a + b) · c = a · c + b · c.

Wir wollen hier haupts¨achlich jene Punkte herausarbeiten, die f¨ ur uns von Interesse sein werden. h. R ⊂ A × A). (1) R heißt antisymmetrisch, wenn R ∩ R−1 ⊂ ∆A (vgl. h. wenn f¨ ur ⇒ alle a, b ∈ A gilt: (a, b) ∈ R und (b, a) ∈ R a = b. (2) R heißt Ordnungsrelation, wenn R reflexiv, antisymmetrisch und transitiv ist. Man nennt dann A auch eine (durch R) (teilweise) geordnete Menge. Schreibweise: (a, b) ∈ R ⇔ a ≤R b oder auch a ≤ b. ¨ Eine Ordnungsrelation R, die zugleich Aquivalenzrelation ist, kann nur die Identit¨at sein, denn   symmetrisch R = R−1  antisymmetrisch R ∩ R−1 ⊂ ∆A ⇒ ∆A = R.

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