Actions of Hopf algebras

A. A. Totok

Sbornik. Mathematics, Tome 189 (1998) no. 1, pp. 147-157

Résumé

We consider an action of a finite-dimensional Hopf algebra $H$ on a non-commutative associative algebra $A$. Properties of the invariant subalgebra $A^H$ in $A$ are studied. It is shown that if $A$ is integral over its centre $\mathrm Z(A)$ then in each of three cases $A$ will be integral over $\mathrm Z(A)^H$ (the invariant subalgebra in $\mathrm Z(A)$): 1) the coradical $H_0$ is cocommutative and char $\operatorname {char}k=p>0$, 2) $H$ is pointed, $A$ has no nilpotent elements, $\mathrm Z(A)$ is an affine algebra, and $\operatorname {char}k=0$, 3) $H$ is cocommutative. We also consider an action of a commutative Hopf algebra $H$ on an arbitrary associative algebra, in particular, the canonical action of $H$ on the tensor algebra $T(H)$. A structure theorem on Hopf algebras is proved by application of the technique developed. Namely, every commutative finite-dimensional Hopf algebra $H$ whose coradical $H_0$ is a sub-Hopf algebra or cocommutative, where $\operatorname {char}k=0$ or $\operatorname {char}k>\dim H$, is cosemisimple, that is, $H=H_0$. In particular, a commutative pointed Hopf algebra with $\operatorname {char}k=0$ or $\operatorname {char}k>\dim H$ will be a group Hopf algebra. An example is also constructed showing that the restrictions on $\operatorname {char}k$ are essential.

Bibliographie

[1] Artamonov V. A., “Stroenie algebr Khopfa”, Itogi nauki i tekhniki. Algebra. Topologiya. Geometriya, 29, VINITI, M., 1991, 3–63 | MR

[2] Sweedler M., Hopf algebras, Benjamin, New York, 1969 | MR | Zbl

[3] Montgomery S., “Hopf algebras and their actions on rings”, Expanded version of ten lectures given at the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University in Chicago (USA, August 10–14, 1992), Amer. Math. Soc., Providence, RI, 1993 | MR

[4] Ferrer Santos Walter R., “Finite generation of the invariants of finite dimensional Hopf algebras”, J. Algebra, 165:3 (1994), 543–549 | DOI | MR | Zbl

[5] Artamonov V. A., “Invarianty algebr Khopfa”, Vestn. MGU. Ser. 1, matem., mekh., 1996, no. 4, 45–49 ; Вестн. МГУ. Сер. 1, матем., мех., 1997, No 2, 64 | MR | Zbl | MR

[6] Zhu Shenglin, “Integrality of module algebras over its invariants”, J. Algebra, 180:1 (1996), 187–205 | DOI | MR | Zbl

[7] Totok A. A., “Ob invariantakh konechnomernykh tochechnykh algebr Khopfa”, Vestn. MGU. Ser. 1, matem., mekh., 1997, no. 3, 31–34 | MR

[8] Quinn D., “Integrality over fixed rings”, J. London Math. Soc., 40:2 (1989), 206–214 | DOI | MR | Zbl

[9] Passman D. S., “Fixed rings and integrality”, J. Algebra, 68:2 (1981), 510–519 | DOI | MR | Zbl

[10] Taft E., Wilson R., “On antipodes in pointed Hopf algebras”, J. Algebra, 29:1 (1974), 27–32 | DOI | MR | Zbl

[11] Burbaki N., Kommutativnaya algebra, Mir, M., 1971 | MR

[12] Bergen J., Cohen M., “Actions of commutative Hopf algebras”, Bull. London Math. Soc., 18:2 (1986), 159–164 | DOI | MR | Zbl

[13] Cohen M., “Smash products, inner actions, and quotient rings”, Pacific J. Math., 125:1 (1986), 46–65 | MR

[14] Nichols W. D., Zoeller M. B., “A Hopf algebra freeness theorem”, Amer. J. Math., 111:2 (1989), 381–385 | DOI | MR | Zbl

[15] Schneider H. J., “Normal basis and transitivity of crossed products for Hopf algebras”, J. Algebra, 152:2 (1992), 289–312 | DOI | MR | Zbl