Shifts and characterization of the elements of A
4
pages
English
Documents
2010
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe en t'inscrivant gratuitement
Découvre YouScribe en t'inscrivant gratuitement
4
pages
English
Ebook
2010
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Shifts and characterization of the elements of A? Matthieu Deneufchatel 21/04/2010 We take the following notations : if A be an algebra and f ? A?, – fx : y 7? f(xy) (left shift) ; – xf : y 7? f(yx) (right shift) ; – xfy : z 7? f(yzx) ; – A? denotes the finite dual (also called Sweedler's dual). Theorem 0.1. Theorem by Abe (extended by Schutzenberger's condition (property (vi) be- low)) - Characterization of the elements of A? : Let A be an algebra and f ? A?. The following properties are equivalent : – (i) tµ(f) ? A? ?A? ; – (ii) The family (fx)x?A is of finite rank ; – (iii) The family (xf)x?A is of finite rank ; – (iv) The family (xfy)x,y?A is of finite rank ; – (v) f(xy) = n∑ i=1 fi(x)gi(y) ; – (vi) ? ? : A ? kn?n and (?, ?) ? k1?n ? kn?1 such that ?x ? A, f(x) = ??(x) ?; – (vii) Ker(f) contains an ideal of finite codimension (i.
Voir
- therefore tµ
- argument allows
- now
- argument also applies
- take ?
- finite rank
- among all
- ?fi ?
- therefore
◦ Shifts and characterization of the elements ofA
MatthieuDeneufchˆatel 21/04/2010
∗ We take the following notations : ifAbe an algebra andf∈ A, –fx:y7→f(xy) (left shift); –f:y7→f(yx;) (right shift) x –fy:z7→f(yzx) ; x ◦ –Adenotes the finite dual (also called Sweedler’s dual). Theorem 0.1.u¨hcneztgrebs’rebeyAxt(edeenySdbytcmobnrdeietoiTohn(proper(vi)be-◦ low))-Characterization of the elements ofA: ∗ LetAbe an algebra andf∈ A. The following properties are equivalent : t∗ ∗ –(i)µ(f)∈ A⊗ A; –(ii)The family(fx)is of finite rank; x∈A –(iii)The family(f);is of finite rank x x∈A f;is of finite rank –(iv)The family(x y)y∈A x, n X –(v)f(xy) =fi(x)gi(y); i=1 n×n1×n n×1 –(vi)∃α:A →kand(λ, γ)∈k×ksuch that ∀x∈ A, f(x) =λ α(x)γ; –(vii) Ker(f)contains an ideal of finite codimension (i.e. there exists an idealIsuch that dim(Ker(f)/I)<∞).(1) Proof :We begin by two lemmas : n X Lemma 0.2.Lettbe an element ofV⊗W. If we decomposetasfi⊗giwithnminimal, then the i=1 fi’s and thegi’s form free families. n−1 X Proof :Indeed, if that was not the case, one could write thatyn=αiyiand i=1 n n−1 X X t=xn⊗yn+xi⊗yi= (αixn+xi)⊗yi. i=1i=1 But this equation shows thatnis not minimal and it is impossible by hypothesis.✷ Lemma 0.3.LetU, VandWbe vector spaces andφ:U×V→Wa bilinear map. Then ifz∈Im(φ) P n and ifz=φ(xi, yi)withnminimal among all decompositions of this form, the families(xi)1≤i≤n i=1 and(yi)1≤i≤nare free in their respective spaces. Note that the first lemma is a consequence of the second one : takeφ=⊗.
1
