Coactions of Hopf Algebras on Algebras in Positive Characteristic

Marilena Crupi; Gaetana Restuccia

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 2, page 349-361
  • ISSN: 0392-4041

Abstract

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Let K be a field of positive characteristic p > 0 . We study the coactions of the Hopf algebra of the multiplicative group H m with underlying algebra H = K ⁢ [ X 1 , ⋯ , X n ] / ( X 1 p s 1 , ⋯ , X n p s n ) , n ≥ 1 , s 1 ≥ ⋯ ≥ s n ≥ 1 on a K -algebra A . We give the rule for the set of additive endomorphism of A , that define a coaction of H m on A commutative. For s 1 = ⋯ = s n = 1 , we obtain the explicit expression of such coactions in terms of n derivations of A .

How to cite

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Crupi, Marilena, and Restuccia, Gaetana. "Coactions of Hopf Algebras on Algebras in Positive Characteristic." Bollettino dell'Unione Matematica Italiana 3.2 (2010): 349-361. <http://eudml.org/doc/290666>.

@article{Crupi2010,
abstract = {Let $K$ be a field of positive characteristic $p > 0$. We study the coactions of the Hopf algebra of the multiplicative group $H_\{m\}$ with underlying algebra $H = K \left[ X_\{1\},\cdots,X_\{n\} \right] / (X_\{1\}^\{p^\{s_\{1\}\}\},\cdots,X_\{n\}^\{p^\{s_\{n\}\}\})$, $n \ge 1$, $s_\{1\}\ge \cdots \ge s_\{n\} \ge 1$ on a $K$-algebra $A$. We give the rule for the set of additive endomorphism of $A$, that define a coaction of $H_\{m\}$ on $A$ commutative. For $s_\{1\} = \cdots = s_\{n\} = 1$, we obtain the explicit expression of such coactions in terms of $n$ derivations of $A$.},
author = {Crupi, Marilena, Restuccia, Gaetana},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {349-361},
publisher = {Unione Matematica Italiana},
title = {Coactions of Hopf Algebras on Algebras in Positive Characteristic},
url = {http://eudml.org/doc/290666},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Crupi, Marilena
AU - Restuccia, Gaetana
TI - Coactions of Hopf Algebras on Algebras in Positive Characteristic
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/6//
PB - Unione Matematica Italiana
VL - 3
IS - 2
SP - 349
EP - 361
AB - Let $K$ be a field of positive characteristic $p > 0$. We study the coactions of the Hopf algebra of the multiplicative group $H_{m}$ with underlying algebra $H = K \left[ X_{1},\cdots,X_{n} \right] / (X_{1}^{p^{s_{1}}},\cdots,X_{n}^{p^{s_{n}}})$, $n \ge 1$, $s_{1}\ge \cdots \ge s_{n} \ge 1$ on a $K$-algebra $A$. We give the rule for the set of additive endomorphism of $A$, that define a coaction of $H_{m}$ on $A$ commutative. For $s_{1} = \cdots = s_{n} = 1$, we obtain the explicit expression of such coactions in terms of $n$ derivations of $A$.
LA - eng
UR - http://eudml.org/doc/290666
ER -

References

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  9. RESTUCCIA, G. - TYC, A., Regularity of the ring of invariants under certain actions of finite abelian Hopf algebras in characteristic p > 0 , J. of Algebra, 159 , No. 2 (1993), 347-357. Zbl0814.16037MR1231218DOI10.1006/jabr.1993.1161
  10. SCHNEIDER, H.-J., Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math., 72 (1990), 167-195. Zbl0731.16027MR1098988DOI10.1007/BF02764619
  11. TYC, A., p -basis and smoothness in characteristc p > 0 , Proc. Am. Math. Soc., 103 (1998), 389-394. MR943051DOI10.2307/2047146
  12. WATERHOUSE, W. C., Introduction to Affine Group Schemes, in: Grad Texts in Math., Vol 66 (Springer, 1979). MR547117

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