The Ore Extension of Group-Cograded Hopf Coquasigroups
Abstract
:1. Introduction
2. Preliminaries
2.1. Hopf Coquasigroups and Ore Extensions
2.2. The Turaev Category
3. Group-Cograded Hopf Coquasigroups
- (1)
- Each is an unital associative k-algebra with multiplication and unit .whenever and , and .
- (2)
- Comultiplication Δ is a family of algebra homomorphisms , and counit is an algeba homomorphism in the sense that, for
- (3)
- Antipode S is an algebra anti-homomorphism with , and for any ,
- (1)
- In the following, we use the Sweelder notation for the comultiplication: For any and ,
- (2)
- (3)
- If is a group-cograded Hopf coquasigroup with each component is finite dimensional, then is a group-graded Hopf quasigroup introduced in [15], with Q being a group.
4. The Ore Extension
- (1)
- there is a character such that for any ,
- (2)
- the following relations hold:
- (3)
- the -derivation satisfies the relation
- 1.
- Comultiplication. Assume that the comultiplication can be extended to (6). The homomorphism Δ then preserves the relationfor any . The last equation coincides with (10).We now show that (13) and (14) imply (8) and (9). Let . It is clear that . One can regard χ as a mapping . If, for any , τ is an endomorphism, it follows thatFor by (13), we haveTherefore, by (4), we haveUsing the formula above, one can recover τ from χ:Hence,
- 2.
- For this section, from [9], we know that, as admits a comultiplication, there is a counit extending and satisfying . It follows that ϵ admits an extension to R if and only if
- 3.
- Let R be as in Step 1. Recall that , with being an antiautomorphism. If R admits an antipode S that can be extended from H to R by means of (7), then S satisfies (4) and (5) and preserves (11). This means that, for any ,We will now prove (19). We haveTherefore, (20) can be represented in an equivalent form asWe denote and .Thus,Therefore, for any ,On the other hand, for any , we have . The action by on both sides givesWe then haveWe therefore conclude that . This proves both the relation (21) and the existence of the antipode.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Ore, O. Theory of Non-Commutative Polynomials. Ann. Math. 1933, 34, 480–508. [Google Scholar] [CrossRef]
- Kassel, C. Quantum Groups; Springer: New York, NY, USA, 1995. [Google Scholar]
- Sweedler, M.E. Hopf Algebras; Benjamin: New York, NY, USA, 1969. [Google Scholar]
- Montgomery, S. Hopf Algebras and Their Actions on Rings; Conference Board of the Mathematical Sciences: Poughkeepsie, NY, USA; American Mathematical Society: Providence, RI, USA, 1993. [Google Scholar]
- Zhang, J. Double ore extensions. J. Pure Appl. Algebra 2008, 212, 2668–2690. [Google Scholar] [CrossRef]
- Panov, A.N. Ore extensions of Hopf algebras. Math. Notes 2003, 74, 401–410. [Google Scholar] [CrossRef]
- Zhao, L.; Lu, D. Ore Extensions of Multiplier Hopf Algebras. Commun. Algebra 2012, 40, 248–272. [Google Scholar] [CrossRef]
- Li, C.; Li, J. Ore extensions of Hopf π-coalgebras. Int. J. Algebra 2020, 14, 247–257. [Google Scholar] [CrossRef]
- Jiao, Z.; Meng, H. Ore extensions of Hopf coquasigroups. Math. Notes 2014, 95, 338–345. [Google Scholar] [CrossRef]
- Wang, D.; Lu, D. Ore extensions of Hopf group coalgebras. J. Korean Math. Soc. 2014, 51, 325–344. [Google Scholar] [CrossRef]
- Klim, J.; Majid, S. Hopf quasigroups and the algebraic 7-sphere. J. Algebra 2010, 323, 3067–3110. [Google Scholar] [CrossRef]
- Caenepeel, S.; De Lombaerde, M. A categorical approach to Turaev’s Hopf group-coalgebras. Commun. Algebra 2006, 34, 2631–2657. [Google Scholar] [CrossRef]
- Virelizier, A. Hopf group coalgebras. J. Pure Appl. Algebra 2002, 171, 75–122. [Google Scholar] [CrossRef]
- Zhang, S.; Wang, S. A New Approach to Braided T-Categories and Generalized Quantum Yang-Baxter Equations. Mathematics 2022, 10, 968. [Google Scholar] [CrossRef]
- Shi, G.; Wang, S. Q-graded Hopf quasigroups. arXiv 2018, arXiv:1812.07321. [Google Scholar]
- Liu, H.; Yang, T.; Zhu, L. Yetter–Drinfeld Modules for Group-Cograded Hopf Quasigroups. Mathematics 2022, 10, 1388. [Google Scholar] [CrossRef]
- Turaev, V. Homotopy field theory in dimension 3 and crossed group-categories. arXiv 2000, arXiv:math/0005291. [Google Scholar]
- Yang, T. Multiplier Hopf Coquasigroup: Motivation and Biduality. Mathematics 2022, 10, 4006. [Google Scholar] [CrossRef]
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Zhu, L.; Jin, B.; Liu, H.; Yang, T. The Ore Extension of Group-Cograded Hopf Coquasigroups. Mathematics 2023, 11, 3703. https://doi.org/10.3390/math11173703
Zhu L, Jin B, Liu H, Yang T. The Ore Extension of Group-Cograded Hopf Coquasigroups. Mathematics. 2023; 11(17):3703. https://doi.org/10.3390/math11173703
Chicago/Turabian StyleZhu, Lingli, Bingbing Jin, Huili Liu, and Tao Yang. 2023. "The Ore Extension of Group-Cograded Hopf Coquasigroups" Mathematics 11, no. 17: 3703. https://doi.org/10.3390/math11173703