Frobenius extensions and weak Hopf algebras. (English) Zbl 1018.16020

Let \(M/N\) be a Frobenius extension of \(k\)-algebras with Frobenius homomorphism \(E\) and dual bases \(\{x_i\}\), \(\{y_i\}\). Let \(U=C_M(N)\). The extension is called symmetric if \(E\) commutes with every \(u\in U\), and Markov if the extension is strongly separable (i.e. \(E(1)=1\) and \(\sum_ix_iy_i=\lambda^{-1}1\)) and there is a (Markov) trace \(T\colon N\to k\) such that \(T(1)=1_k\) and \(T_0=T\circ E\colon M\to k\) is a trace. The basic construction theorem says that if \(N\subseteq M\) is a symmetric Markov extension and \(M_1=M\otimes_NM=\text{End}(M_N)\) then \(M_1/M\) is a symmetric Markov extension; the Frobenius endomorphism \(E_M\) and the dual bases are described, and the Markov trace is \(T_0\). If in addition \(U\) is Kanzaki separable, \(T_0|_U\) is non-degenerate and \(\sum_ix_iy_i=\sum_iy_ix_i\) then \(V=C_{M_1}(M)\) is Kansaki separable and the restriction of \(T_1=T_0\circ E_M\) to \(V\) is non-degenerate. The construction can be iterated to obtain the Jones tower \(N\subseteq M\subseteq M_1\subseteq M_2\). Let \(A=C_{M_1}(N)\) and \(B=C_{M_2}(M)\). Assuming the existence of dual bases for \(E_M\) resp. \(E_{M_1}\) in \(A\) resp. \(B\) (depth 2 condition) the authors prove some properties of algebra extensions involving \(A\), \(B\), \(U\) and \(V\). Examples of extensions with depth 2 are provided in the final appendix. Then semisimple weak Hopf algebra structures are defined in \(A\) and \(B\), and also \(A\) resp. \(B\)-module algebra structures on \(M\) resp. \(M_1\). Two isomorphisms \(M_1\simeq M\#A\) and \(M_2\simeq M_1\#B\) are also provided. Finally, in the absence of a trace, the basic construction is iterated to the right, extending a result of M. Pimsner and S. Popa [Trans. Am. Math. Soc. 310, No. 1, 127-133 (1988; Zbl 0706.46047)].


16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S40 Smash products of general Hopf actions
16S50 Endomorphism rings; matrix rings
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16L60 Quasi-Frobenius rings


Zbl 0706.46047
Full Text: DOI arXiv


[1] Blattner, R.; Montgomery, S., A duality theorem for Hopf module algebras, J. Algebra, 95, 153-172 (1985) · Zbl 0589.16010
[2] Böhm, G.; Nill, F.; Szlachányi, K., Weak Hopf algebras, I. Integral theory and \(C\)*-structure, J. Algebra, 221, 385-438 (1999) · Zbl 0949.16037
[3] Böhm, G.; Szlachányi, K., A coassociative \(C\)*-quantum group with nonintegral dimensions, Lett. Math. Phys., 35, 437-456 (1996) · Zbl 0872.16022
[4] Etingof, P.; Nikshych, D., Dynamical quantum groups at roots 1, Duke Math. J., 108, 135-168 (2001) · Zbl 1023.17007
[5] Goodman, F.; de la Harpe, P.; Jones, V. F.R., Coxeter Graphs and Towers of Algebras. Coxeter Graphs and Towers of Algebras, M. S. R. I. Publ. 14 (1989), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0698.46050
[6] Hirata, K.; Sugano, K., On semisimple extensions and separable extensions over non commutative rings, J. Math. Soc. Japan, 18, 360-373 (1966) · Zbl 0178.36802
[7] Jones, V. F.R., Index for subfactors, Invent. Math, 72, 1-25 (1983) · Zbl 0508.46040
[8] Jones, V. F.R., Index for subrings of rings, Contemp. Math., 43, 181-190 (1985) · Zbl 0607.46033
[9] Kadison, L., The Jones polynomial and certain separable Frobenius extensions, J. Algebra, 186, 461-475 (1996) · Zbl 0880.16011
[10] Kadison, L., New Examples of Frobenius Extensions. New Examples of Frobenius Extensions, University Lecture Series, 14 (1999), Am. Math. Soc: Am. Math. Soc Providence · Zbl 0929.16036
[11] Kadison, L.; Nikshych, D., Outer actions of centralizer Hopf algebras on separable extensions, Comm. Algebra, 29 (2001) · Zbl 1010.16037
[12] Kanzaki, T., Special type of separable algebra over commutative ring, Proc. Japan Acad., 40, 781-786 (1964) · Zbl 0143.05603
[13] Kasch, F., Projektive Frobenius Erweiterungen, Sitzungsber. Heidelberg. Akad. Wiss. Math.-Natur. Kl., 89-109 (1960/1961) · Zbl 0104.26201
[14] Kasch, F., Dualitätseigenschaften von Frobenius-Erweiterungen, Math. Z., 77, 219-227 (1961) · Zbl 0112.26502
[15] Montgomery, S., Hopf Algebras and Their Actions on Rings. Hopf Algebras and Their Actions on Rings, CBMS Regional Conf. Series in Math., 82 (1993), A. M. S: A. M. S Providence · Zbl 0793.16029
[16] Nikshych, D., A duality theorem for quantum groupoids, (Andruskiewitsch, N.; Santos, F.; Schneider, H.-J., New Trends in Hopf Algebra Theory. New Trends in Hopf Algebra Theory, Contemp. Math., 267 (2000)), 237-243 · Zbl 0978.16032
[17] D. Nikshych, V. Turaev, and, L. Vainerman, Quantum groupoids and invariants of knots and 3-manifolds, J. Topology Appl, to appear.; D. Nikshych, V. Turaev, and, L. Vainerman, Quantum groupoids and invariants of knots and 3-manifolds, J. Topology Appl, to appear. · Zbl 1021.16026
[18] Nikshych, D.; Vainerman, L., A characterization of depth 2 subfactors of \(II_1\) factors, J. Funct. Anal., 171, 278-307 (2000) · Zbl 1010.46063
[19] Nikshych, D.; Vainerman, L., A Galois correspondence for actions of quantum groupoids on \(II_1\)-factors, J. Funct. Anal., 178, 113-142 (2000) · Zbl 0995.46041
[20] D. Nikshych, and, L. Vainerman, Finite dimensional quantum groupoids and their applications, in; D. Nikshych, and, L. Vainerman, Finite dimensional quantum groupoids and their applications, in · Zbl 1026.17017
[21] Ocneanu, A., Quantized groups, string algebras and Galois theory for algebras, Operator Algebras and Applications. Operator Algebras and Applications, London Math. Soc. Lecture Notes Series 135, 2 (1988), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0696.46048
[22] Onodera, T., Some studies on projective Frobenius extensions, J. Fac. Sci. Hokkaido Univ. Ser. I, 18, 89-107 (1964) · Zbl 0127.25801
[23] Pimsner, M.; Popa, S., Iterating the basic construction, Trans. Amer. Math. Soc., 310, 127-133 (1988) · Zbl 0706.46047
[24] Stolin, A. A.; Kadison, L., Separability and Hopf algebras, (Huynh; Jain; Lopez-Permouth, Algebra and Its Applications. Algebra and Its Applications, Contemp. Math., 259 (2000)), 279-298 · Zbl 0974.16033
[25] K. Szlachányi, Weak Hopf Algebra Symmetries of \(C\); K. Szlachányi, Weak Hopf Algebra Symmetries of \(C\)
[26] Szymański, W., Finite index subfactors and Hopf algebra crossed products, Proc. Amer. Math. Soc., 120, 519-528 (1994) · Zbl 0802.46076
[27] Watatani, Y., Index of \(C\)*-subalgebras, Mem. Amer. Math. Soc., 83 (1990) · Zbl 0697.46024
[28] Xiaolong, J.; Yongchua, X., \(H\)-separable rings and their Hopf-Galois extensions, Chinese Ann. Math. Ser. B., 19, 311-320 (1998) · Zbl 0910.16021
[29] Yamagata, K., Frobenius algebras, (Hazewinkel, M., Handbook of Algebra (1996), Elsevier: Elsevier Amsterdam), 841-887 · Zbl 0879.16008
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