Abstract
Let H be a Hopf algebra and A an H-bimodule algebra. This paper investigates Gorenstein global dimensions and representation dimensions of L-R smash products \(A\natural H\). Several well-known results are generalized.
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Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Springer, Berlin (1974)
Assem, I., Platzeck, M.I., Trepode, S.: On the representation dimension of tilted and laura algebras. J. Algebra 296, 426–439 (2006)
Auslander, M.: Representation dimension of artin algebras. Math. Notes, Queen Mary College, London (1971)
Bennis, D., Mahdou, N.: Global Gorenstein dimensions. Proc. Am. Math. Soc. 138, 461–465 (2010)
Bennis, D., Mahdou, N., Ouarghi, K.: Rings over which all modules are strongly Gorenstein projective. Rocky Mountain J. Math. 40, 749–759 (2010)
Bieliavsky, P., Bonneau, P., Maeda, Y.: Universal deformation formulae for three-dimensional solvable Lie groups. In: Quantum Field Theory and Noncommutative Geometry. Lecture Notes in Phys., vol. 662, pp. 127–141. Springer, Berlin (2005)
Bonneau, P., Gerstenhaber, M., Giaquinto, A., Sternheimer, D.: Quantum groups and deformation quantization: explicit approaches and implicit aspects. J. Math. Phys. 45, 3703–3741 (2004)
Bonneauand, P., Sternheimer, D.: Topological Hopf algebras, quantum groups and deformation quantization. In: Hopf Algebras in Noncommutative Geometry and Physics. Lecture Notes in Pure and Appl. Math., vol. 239, pp. 55–70. Marcel Dekker, New York (2005)
Enochs, E.E., Jenda, O.M.G.: Relative Homological Algebra. Walter de Gruyter, Berlin, New York (2000)
Erdmann, K., Holm, T., Iyama, O., Schr\(\ddot{o}\)er, J.: Radical embeddings and representation dimension. Adv. Math. 185, 159–177 (2004)
Holm, H.: Gorensrein homological dimensions. J. Pure Appl. Algebra 189, 167–193 (2004)
Iyama, O.: Finiteness of representation dimension. Proc. Am. Math. Soc. 131, 1011–1014 (2002)
Li, F., Zhang, M.M.: Invariant properties of representations under cleft extensions. Sci. China Ser. 50, 121–131 (2007)
Mahdou, N., Tamekkante, M.: On (strongly) Gorenstein (semi)hereditary rings. Arab. J. Sci. Eng. 36, 431–440 (2011)
Montgomery, S.: Hopf Algebras and Their Actions on Rings. CBMS, Rhode Island (1993)
Panaite, F., Van Oystaeyen, F.: L-R-smash product for (quasi-)Hopf algebras. J. Algebra 309, 168–191 (2007)
Pan, Q.X.: On L-R-smash products of Hopf algebras. Comm. Algebra 40, 3955–3973 (2012)
Radford, D.E.: Minimal quasitriangular Hopf algebras. J. Algebra 157, 285–315 (1993)
Radford, D.E.: The trace and Hopf algebras. J. Algebra 163, 583–622 (1994)
Rotman, J.J.: An Introduction to Homological Algebra. Academic Press, New York (1979)
Sweedler, M.E.: Hopf Algebras. Benjamin, New York (1969)
Xi, C.C.: Representation dimension and quasi-hereditary algebras. Adv. Math. 168, 193–212 (2002)
Zhang, L.Y., Tong, W.T.: Quantum Yang-Baxter H-module algebras and their braided products. Comm. Algebra 28, 2471–2495 (2003)
Zhang, L.Y.: L-R-smash products for bimodule algebras. Prog. Nat. Sci. 16, 580–587 (2006)
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Pan, Q., Cai, F. Gorenstein Global Dimensions and Representation Dimensions for L-R Smash Products. Algebr Represent Theor 17, 1349–1358 (2014). https://doi.org/10.1007/s10468-013-9451-x
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DOI: https://doi.org/10.1007/s10468-013-9451-x