By Gerstenhaber M., Schack D.

This paper is an multiplied model of comments introduced 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 elemental rules and effects, in addition to the current kingdom, of algebraic deformation thought. So this paper incorporates a mix of the outdated and the recent. we've got tried to supply a clean standpoint even at the extra "ancient" issues, highlighting difficulties and conjectures of normal curiosity all through. We hint a course from the seminal case of associative algebras to the quantum teams that are now riding deformation concept 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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**Additional info for Algebras, bialgebras, quantum groups, and algebraic deformation**

**Example text**

Show that Θ is a 2-ary closure operator. [Hint: replace each n-ary f of A by unary operations f (a1 , . . , ai−1 , x, ai+1 , . . , an ), a1 , . . , ai−1 , ai+1 , . . ] 8. If A is a unary algebra and B is a subuniverse deﬁne θ by a, b ∈ θ iﬀ a = b or {a, b} ⊆ B. Show that θ is a congruence on A. 9. Let S be a semilattice. Deﬁne a ≤ b for a, b ∈ S if a · b = a. Show that ≤ is a partial order on S. Next, given a ∈ S deﬁne θa = { b, c ∈ S × S : both or neither of a ≤ b, a ≤ c hold}. Show θa is a congruence on S.

2, LC is indeed an algebraic lattice. So suppose X = {a1 , . . , ak } and C(X) ⊆ C(Ai ) = C Ai i∈I i∈I For each aj ∈ X we have by (C4) a ﬁnite Xj ⊆ ﬁnitely many Ai ’s, say Aj1 , . . , Ajnj , such that i∈I Ai with aj ∈ C(Xj ). Since there are Xj ⊆ Aj1 ∪ · · · ∪ Ajnj , then aj ∈ C(Aj1 ∪ · · · ∪ Ajnj ). But then X⊆ C(Aj1 ∪ · · · ∪ Ajnj ), 1≤j≤k so . X ⊆C 1≤j≤k 1≤i≤nj Aji , 23 §5. Closure Operators and hence C(X) ⊆ C 1≤j≤k 1≤i≤nj Aji = C(Aji), 1≤j≤k 1≤i≤nj so C(X) is compact.

An ∈ A, we have (β ◦ α)f A(a1 , . . , an ) = β(αf A(a1 , . . , an )) = βf B (αa1 , . . , αan ) = f C (β(αa1 ), . . , β(αan )) = f C ((β ◦ α)a1 , . . , (β ◦ α)an ). ✷ The next result says that homomorphisms commute with subuniverse closure operators. 6. If α : A → B is a homomorphism and X is a subset of A then α Sg(X) = Sg(αX). Proof. From the deﬁnition of E (see §3) and the fact that α is a homomorphism we have αE(Y ) = E(αY ) for all Y ⊆ A. Thus, by induction on n, αE n (X) = E n (αX) for n ≥ 1; hence α Sg(X) = α(X ∪ E(X) ∪ E 2 (X) ∪ .

### Algebras, bialgebras, quantum groups, and algebraic deformation by Gerstenhaber M., Schack D.

by Ronald

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