By Mugnai D.

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Example. We p r e s e n t example of a n o n c o m m u t a t i v e k-group of d i m e n s i o n (see below) Let to k-wound It was by S h i z u o Endo. a ~ k, a ~ k p, and 0 £ m £ n. We p r o d u c e 2. Let a nonzero G2 × G2 + G1 m x ax p and let y] assignments x t ,> connected discovered m, n be ~ m up ~) k[u, t], k[u, t] Q u first for integers m=n=l subject G 2 = ~(M(n,l+aF2m)). biadditive Writing the unipotent first antisymmetric U ~ = Spec k[x, is p r o b a b l y G 1 = ~=D(M(m, l - a F m ) ) , as f o l l o w s : a function what function m G 1 = Spec k [ x , y ] , yP = n 2m tp = u + au p , we d e f i n e k[u, m u ® up t] through the n-m and Y I that this > u @ tp n-m tp @ u.

From a k [F] -linear map f: X x y ÷ Z algebra C. D(L), B = O(Y), or e q u i v a l e n t l y in y) = F x ~ ® U(N). bi-additive and identify elements that F(x~ We view Conversely, X, Y C(~) B ÷ A<~)B, is a B-Hopf + D(M)x Every Let k[F]-modules. be a k [F] -linear U(L) X x y ÷ Z x y, Y-group Hence N by from a k [ F ] - l i n e a r and let morphism. be left k [F] -module f: L ÷ M ~ to an algebra N L ÷ M~ denote elements it follows we have Since D(M) N. the p r i m i t i v e of the B-Hopf from above f(P(C))C__ P(U(M)) × D(N) that A ( ~ P(B).

Ga Then, there exist exists ~ : ~i k-group of > G, a n o n c o n s t a n t a nonconstant k-homomorphism >G. Proof. , of = 0 G a × ... 3, we m a y c o n s i d e r factors) = an index p-polynomial; set), G defined where each thus, for each a~ . v, one can write ~v ~i + "'" + ~ n By a s s u m p t i o n , least one there exist is n o n c o n s t a n t ~ (f(r)) = for all yEN. , fn E k[T] of w h i c h at such that + ... as we may, with 1 < i < n b ~ 0. := 0 if no terms is p r e s e n t in ¢~l(gl(T)) + .....

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A note on an exponential semilinear equation of the fourth order by Mugnai D.


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