THE STRUCTURE THEOREM OF HOM-HOPF BIMODULES AND ITS APPLICATIONS
Yıl 2021,
Cilt: 30 Sayı: 30, 78 - 98, 17.07.2021
Huihui Zheng
Yuanyuan Chen
Liangyun Zhang
Öz
In this paper, we give the structure theorem of Hom-Hopf bimodules. Furthermore, we give the structure theorem of Hom-comodule algebras.
Finally, we consider and study the structure theorems of Hom-Hopf bicomodules and Hom-module coalgebras.
Kaynakça
- Y. Bespalov and B. Drabant, Hopf (bi-)modules and crossed modules in braided
monoidal categories, J. Pure Appl. Algebra, 123(1-3) (1998), 105-129.
- T. Brzezinski and R. Wisbauer, Corings and Comodules, London Mathematical
Society Lecture Note Series, 309, Cambridge University Press, Cambridge,
2003.
- S. Caenepeel and I. Goyvaerts, Monoidal Hom-Hopf algebras, Comm. Algebra,
39(6) (2011), 2216-2240.
- Y. Y. Chen, Z. W. Wang and L. Y. Zhang, Integrals for monoidal Hom-Hopf
algebras and their applications, J. Math. Phys., 54(7) (2013), 073515 (22 pp).
- Y. Y. Chen, Z. W. Wang and L. Y. Zhang, The fundamental theorem and
Maschke's theorem in the category of relative Hom-Hopf modules, Colloq.
Math., 144(1) (2016), 55-71.
- Y. Y. Chen and L. Y. Zhang, The category of Yetter-Drinfel'd Hom-modules
and the quantum Hom-Yang-Baxter equation, J. Math. Phys., 55(3) (2014),
031702 (18 pp).
- Y. Y. Chen, H. H. Zheng and L. Y. Zhang, Double Hom-associative algebra and
double Hom-Lie bialgebra, Adv. Appl. Cliord Algebr., 30(1) (2020), Paper No.
8 (25 pp).
- S. Dascalescu, C. Nastasescu and S. Raianu, Hopf Algebras: An Introduction,
Monographs and Textbooks in Pure and Applied Mathematics, 235, Marcel-
Dekker, Inc., New York, 2001.
- J. Dello, F. Panaite, F. Van Oystaeyen and Y. Zhang, Structure theorems for
bicomodule algebras over quasi-Hopf algebras, weak Hopf algebras, and braided
Hopf algebras, Comm. Algebra, 44(11) (2016), 4609-4636.
- J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras
using $\sigma$-derivations, J. Algebra, 295(2) (2006), 314-361.
- F. Hausser and F. Nill, Integral theory for quasi-Hopf algebras, (1999),
arXiv:math/9904164.
- R. G. Larson and M. E. Sweedler, An associative orthogonal bilinear form for
Hopf algebras, Amer. J. Math., 91(1) (1969), 75-94.
- H. Li and T. Ma, A construction of the Hom-Yetter-Drinfeld category, Colloq.
Math., 137(1) (2014), 43-65.
- L. Liu and B. Shen, Radford's biproducts and Yetter-Drinfeld modules for
monoidal Hom-Hopf algebras, J. Math. Phys., 55(3) (2014), 031701 (16 pp).
- A. Makhlouf and F. Panaite, Yetter-Drinfeld modules for Hom-bialgebras, J.
Math. Phys., 55(1) (2014), 013501 (17 pp).
- G. Militaru and D. Stefan, Extending modules for Hopf Galois extensions,
Comm. Algebra, 22(14) (1994), 5657-5678.
- S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional
Conference Series in Mathematics, 82, Published for the Conference Board of
the Mathematical Sciences, Washington, DC; by the American Mathematical
Society, Providence, RI, 1993.
- R. F. Niu, Y. Wang and L. Y. Zhang, The structure theorem of endomorphism
algebras for weak Doi-Hopf modules, Acta Math. Hungar., 127(3) (2010), 273-
290.
- P. Saracco, On the structure theorem for quasi-Hopf bimodules, Appl. Categ.
Structures, 25(1) (2017), 3-28.
- P. Schauenburg, Bialgebras over noncommutative rings and a structure theorem
for Hopf bimodules, Appl. Categ. Structures, 6(2) (1998), 193-222.
- M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series W. A. Benjamin, Inc., New York, 1969.
- Y. Wang and L. Y. Zhang, The structure theorem for weak module coalgebras,
Math. Notes, 88(1-2) (2010), 3-15.
- D. Yau, Hom-quantum groups I: quasi-triangular Hom-bialgebras, J. Phys. A,
45(6) (2012), 065203 (23 pp).
- L. Y. Zhang, The Structure theorem of weak comodule algebras, Comm. Algebra, 38(1) (2010), 254-260.
Yıl 2021,
Cilt: 30 Sayı: 30, 78 - 98, 17.07.2021
Huihui Zheng
Yuanyuan Chen
Liangyun Zhang
Kaynakça
- Y. Bespalov and B. Drabant, Hopf (bi-)modules and crossed modules in braided
monoidal categories, J. Pure Appl. Algebra, 123(1-3) (1998), 105-129.
- T. Brzezinski and R. Wisbauer, Corings and Comodules, London Mathematical
Society Lecture Note Series, 309, Cambridge University Press, Cambridge,
2003.
- S. Caenepeel and I. Goyvaerts, Monoidal Hom-Hopf algebras, Comm. Algebra,
39(6) (2011), 2216-2240.
- Y. Y. Chen, Z. W. Wang and L. Y. Zhang, Integrals for monoidal Hom-Hopf
algebras and their applications, J. Math. Phys., 54(7) (2013), 073515 (22 pp).
- Y. Y. Chen, Z. W. Wang and L. Y. Zhang, The fundamental theorem and
Maschke's theorem in the category of relative Hom-Hopf modules, Colloq.
Math., 144(1) (2016), 55-71.
- Y. Y. Chen and L. Y. Zhang, The category of Yetter-Drinfel'd Hom-modules
and the quantum Hom-Yang-Baxter equation, J. Math. Phys., 55(3) (2014),
031702 (18 pp).
- Y. Y. Chen, H. H. Zheng and L. Y. Zhang, Double Hom-associative algebra and
double Hom-Lie bialgebra, Adv. Appl. Cliord Algebr., 30(1) (2020), Paper No.
8 (25 pp).
- S. Dascalescu, C. Nastasescu and S. Raianu, Hopf Algebras: An Introduction,
Monographs and Textbooks in Pure and Applied Mathematics, 235, Marcel-
Dekker, Inc., New York, 2001.
- J. Dello, F. Panaite, F. Van Oystaeyen and Y. Zhang, Structure theorems for
bicomodule algebras over quasi-Hopf algebras, weak Hopf algebras, and braided
Hopf algebras, Comm. Algebra, 44(11) (2016), 4609-4636.
- J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras
using $\sigma$-derivations, J. Algebra, 295(2) (2006), 314-361.
- F. Hausser and F. Nill, Integral theory for quasi-Hopf algebras, (1999),
arXiv:math/9904164.
- R. G. Larson and M. E. Sweedler, An associative orthogonal bilinear form for
Hopf algebras, Amer. J. Math., 91(1) (1969), 75-94.
- H. Li and T. Ma, A construction of the Hom-Yetter-Drinfeld category, Colloq.
Math., 137(1) (2014), 43-65.
- L. Liu and B. Shen, Radford's biproducts and Yetter-Drinfeld modules for
monoidal Hom-Hopf algebras, J. Math. Phys., 55(3) (2014), 031701 (16 pp).
- A. Makhlouf and F. Panaite, Yetter-Drinfeld modules for Hom-bialgebras, J.
Math. Phys., 55(1) (2014), 013501 (17 pp).
- G. Militaru and D. Stefan, Extending modules for Hopf Galois extensions,
Comm. Algebra, 22(14) (1994), 5657-5678.
- S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional
Conference Series in Mathematics, 82, Published for the Conference Board of
the Mathematical Sciences, Washington, DC; by the American Mathematical
Society, Providence, RI, 1993.
- R. F. Niu, Y. Wang and L. Y. Zhang, The structure theorem of endomorphism
algebras for weak Doi-Hopf modules, Acta Math. Hungar., 127(3) (2010), 273-
290.
- P. Saracco, On the structure theorem for quasi-Hopf bimodules, Appl. Categ.
Structures, 25(1) (2017), 3-28.
- P. Schauenburg, Bialgebras over noncommutative rings and a structure theorem
for Hopf bimodules, Appl. Categ. Structures, 6(2) (1998), 193-222.
- M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series W. A. Benjamin, Inc., New York, 1969.
- Y. Wang and L. Y. Zhang, The structure theorem for weak module coalgebras,
Math. Notes, 88(1-2) (2010), 3-15.
- D. Yau, Hom-quantum groups I: quasi-triangular Hom-bialgebras, J. Phys. A,
45(6) (2012), 065203 (23 pp).
- L. Y. Zhang, The Structure theorem of weak comodule algebras, Comm. Algebra, 38(1) (2010), 254-260.