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Hopf Algebra Forms of the Multiplicative Group and Other Groups 121 Hayashi, N.: Classical Solutions of Nonlinear Schrödlinger Equations 171 Heath, P manuscripta mathematictl a Volume 55 1986 Editores M. Barner, Freiburg S. Hildebrandt, Bonn A. Prestel, Konstanz H. Br6zis, Paris E. Hlawka, Wien P. Roquette, Heidelberg P. M. Cohn, London T. Kato, Berkeley J. Tits, Paris A. Dold, Heidelberg H. Kraft, Basel Copyright. 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Springer-Verlag Berlin Heidelberg New York Tokyo Printers: Brühlsche Universitätsdruckerei, Giessen Printed in Germany — © Springer-Verlag Berlin Heidelberg 1986 Springer-Verlag GmbH & Co. KG, 1000 Berlin 33 Springer-Verlag Berlin Heidelberg New York Tokyo Index Afflerbach, L.: The Sub-Lattice Structure of Linear Congruential Random Number Generators. 455 Bear, D.: Wild Hereditary Artin Algebras and Linear Methods 69 Baldes, A.: Degenerate Elliptic Operators Diagonal Systems and Variational Integrals 467 Balibrea, F., Vera, G.: On the Sublinear Functional Associated to a Family of Invariant Means.. 101 Bauer, H.: A Class of Means and Related Inequalities 199 Bracho, J.: Cyclic Crystallizations of Spheres 213 Brandt, R., Stichtenoth, H.: Die Automorphismengruppen hyperelliptischer Kurven 83 Brion, M.: Quelques proprietes des espaces homogenes spheriques 191 Büch, J.: Bifurkation von Minimalflächen und elementare Katastrophen 269 Defant, A., Govaerts, W.: Tensor Products and Spaces of Vector-Valued Continuous Functions . 433 Degen, W.: Die zweifachen Bluteischen Kegelschnittflächen 9 Elias, J.: A Sharp Bound for the Minimal Number of Generators of Perfect Height Two Ideals .. 93 Forstneric, F.: Some Totally Real Embeddings of Three-Manifolds 1 Frey, G., Rück, H.-G.: The Strong Lefschetz Principle in Algebraic Geometry 385 Garcia, A.: Weights of Weierstrass Points in Double Coverings of Curves of Genus One or Two . 419 Govaerts, W., see Defant, A 433 Grüter, M.: Eine Bemerkung zur Regularität stationärer Punkte von konform invarianten Varia• tionsintegralen 451 Haggenmüller, R., Pareigis, B.: Hopf Algebra Forms of the Multiplicative Group and Other Groups 121 Hayashi, N.: Classical Solutions of Nonlinear Schrödlinger Equations 171 Heath, P. R.: A Pullback Theorem for Locally-Equiconnected Spaces 233 Herzog, J., Waldi, R.: Cotangent Functors of Curve Singularities 307 Klein, C.: Arzela-Ascoli's Theorem for Riemann-Integrable Functions on Compact Spaces 403 Koltsaki, P., Stamatakis, S.: Über eine von J. Hoschek erzeugte Klasse von Strahlensystemen... 359 Kuhlmann, F.-V., Pank, M., Roquette, P.: Immediate and Purely Wild Extensions of Valued Fields 39 Manolache, N.: On the Normal Bündle to Abelian Surfaces Embedded in P4 (€) 111 Michel, J.: Randregularität des 6 -Problems für die Halbkugel in Cn 239 Pälfy, P. P.: On Partial Ordering of Chief Factors in Solvable Groups 219 Pank, M., see Kuhlmann, F.-V., et al 39 Pareigis, B., see Haggenmüller, R 121 Roquette, P., see Kuhlmann, F.-V., et al 39 Rück, H.-G., see Frey, G 385 Sander, W.: Weighted-Additive Deviations with the Sum Property 373 Sanders, H.: Cohen-Macaulay Properties of the KoszufHomology 343 Stamatakis, S., see Koltsaki, P 359 Stichtenoth, H., see Brandt, R 83 Takase, K.: On the Trace Formula of the Hecke Operators and the Special Values of the Second L-Functions Attached to the Hilbert Modular Forms 137 Vera, G., see Balibrea, F 101 Waldi, R., see Herzog, J 307 Covered by Zentralblatt für Mathematik and Current Mathematical Publications manuscripta math. 55, 121 - 136 (1986) manuscripta mathematica © Springer-Verlag 1986 HOPF ALGEBRA FORMS OF THE MULTIPLICATIVE GROUP AND OTHER GROUPS Rudolf Haggenmüller and Bodo Pareigis The multiplicative group functor, which associates with each k-algebra its group of units, is affine with Hopf algebra k[x,x~ ]. The purpose of this gaper is to determine explicitly all Hopf alge• bra forms of k[x,x ] with only minor restrictions on k ( 2 not a zero-divisor and Pic(2)(k) = 0 ). We also describe explicitly (by generators and relations) the Hopf algebra forms of kCß , kC4 and kC$ , where Cn is the cyclic group of order n . Some of our results could be drawn from [l,III §5.3.3] where a similar result as ours is indicated (and left as an exercise). We prefer however a less technical approach, in particular we do not use the extended theory of algebraic groups and functor sheaves. The principal tool of this note is the theory of faithfully flat descent which is used to prove that the Hopf algebra forms of kG with finitely generated group G (with finite automorphism group F) are in one-to-one correspondence with the F-Galois extensions of k. The progress in recent years in describing the quadratic extensions of k and the explicit construction of the correspondence allow us to compute the forms of kG for all groups with Aut(G) = in terms of generators and relations. Consider the functor C: k-AZg^ —>• Gl , the CA.KcZt $uncto>i9 defined by C(A) = {(a,b)<?AxA | a2+b2=l} . The group structure is given by (a,b)*(c,d) = (ac-bd,ad+bc) . The neutral element is (1,0) and the inverse of (a,b) is (a,-b). To understand this multiplication observe that C is represented 121 HAGGENMÜLLER - PAREIGIS by the k-algebra H = k[c,s]/(s2+c2-1) . This must be a Hopf alge• bra and is called the tK^igOYiomQÄHA,c aJLgohKOL* The coalgebra structure of H is given by A(c) = c»c-s«s A(s) =c«s+s«c e(c) = 1 e(s) = 0 . The antipode is S(c) = c , S(s) = -s . Obviously c and s play the role of cos and sin resp. and the diagonal map reflects the summation formulas for cos and sin: cos(x + y) = cos(x)cos(y) - sin(x)sin(y) sin(x + y) - cos(x)sin(y) + sin(x)cos(y) . e gives the value at 0° and S is the reflection on the x-axis. The geometric meaning of the group structure on C(A) is the addi- tion of the corresponding angles with the x-axis for the points (a,b) resp. (c,d) . Let us now ask for a QKOULp-ZJJlQ, QJtWQ.vit e f 1 in H , i.e. an element e with A(e) = e e e , e(e) = 1 . A little calculation shows that such an element exists if and only if there is ick with i2 = -1 and all the group-like elements are then of the form (c + is)n , n € TL . Observe that e * = c - is , if e = c + is . The diagonal map on e reflects the summation formula for the exponential function exp(x + y) = exp(x)exp(y) . If furthermore 2 is invertible in k then we get an isomorphism of Hopf algebras k[c,s]/(s2 + c2 - 1) ^ kS (where U2L corresponds to e = c + is) because of c = -j(e + e *) , s = "^"(e - e *) . The affine k-group represented by k2Z - k[x,x is the mLUL&pLLccLtive, gKOUp or the group of units. Hence we have the following: If A is a k-algebra over a commutative ring k with 2 ^€ k and iek , then the circle group C(A) is isomorphic to the multiplicative group U(A) .
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