Hopf algebra actions on strongly separable extensions of depth two. (English) Zbl 1029.46098
Summary: We bring together ideas in analysis on Hopf *-algebra actions on \(\text{II}_1\) subfactors of finite Jones index [F. M. Goodman, P. de la Harpe and V. F. R. Jones, Coxeter graphs and towers of algebras, Springer, New York (1989; Zbl 0698.46050); W. Szymański, Proc. Am. Math. Soc. 120, No. 2, 519-528 (1994; Zbl 0802.46076)] and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions [Y. Doi and M. Takeuchi, Commun. Algebra 14, 801-817 (1986; Zbl 0589.16011); F. Kasch, Sitzungsber. Heidelberger Akad. Wiss., Math.-Nat. Kl. 1960/61, 89-109 (1961; Zbl 0104.26201); H. F. Kreimer and M. Takeuchi, Indiana Univ. Math. J. 30, 675-692 (1981; Zbl 0451.16005)] to prove a non-commutative algebraic analogue of the following classical theorem: a finite degree field extension is Galois iff it is separable and normal. Suppose \(N\hookrightarrow M\) is a separable Frobenius extension of \(k\)-algebras with trivial centralizer \(C_M(N)\) and split as \(N\)-bimodules. Let \(M_1:=\text{End}(M_N)\) and \(M_2:=\text{End}(M_1)_M\) be the endomorphism algebras in the Jones tower \(N\hookrightarrow M\hookrightarrow M_1\hookrightarrow M_2\). We place depth 2 conditions on its second centralizers \(A:=C_{M_1}(N)\) and \(B:=C_{M_2}(M)\). We prove that \(A\) and \(B\) are semisimple Hopf algebras dual to one another, that \(M_1\) is a smash product of \(M\) and \(A\), and that \(M\) is a \(B\)-Galois extension of \(N\).
MSC:
| 46L37 | Subfactors and their classification |
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 46L35 | Classifications of \(C^*\)-algebras |
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