Abstract
A two-parameter quantum group is obtained from the usual enveloping algebra by adding two commutative grouplike elements. In this paper, we generalize this procession further by adding commutative grouplike elements b ik , c ik , g ik , h ik (i ∈ I, k = 1,..., m i ) of a Hopf algebra H to the quantized enveloping algebra U q (G) of a Borcherds superalgebra G defined by a symmetrizable integral Borcherds–Cartan matrix A = (a ij ) i,j∈I . Therefore, we define an extended Hopf superalgebra H ⊗′ U q (G). We also discuss the basis and the grouplike elements of H ⊗′ U q (G).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aizawwa, N., Isaac, P. S.: Weak Hopf algebras corresponding to U q[sl n]. J. Math. Phys., 44, 5250–5267 (2003)
Benkart, G., Kang, S.-J., Melville, D. J.: Quantized enveloping algebras for Borcherds superalgebras. Trans. Amer. Math. Soc., 350, 3297–3319 (1998)
Hong, J.: Center and universal R-matrix for quantized Borcherds superalgebras. J. Math. Phys., 40(6), 3123–3145 (1999)
Hong, Y. Y., Wu, Z. X.: Extension of quantum enveloping algebras. Appl. Math. J. Chinese Univ., 27(3), 290–304 (2012)
Hu, N. H., Pei, Y. F.: Notes on 2-parameter quantum groups, (I). Sci. China Ser. A, 51(6), 1101–1110 (2008)
Hu, N. H., Pei, Y. F.: Notes on two-parameter quantum groups, (II). Comm. in Algebra, 40(9), 3202–3220 (2012)
Hu, N. H., Rosso, M., Zhang, H.: Two-parameter affine quantum group Ur,s(sln). Drinfeld realization and quantum affine Lyndon basis. Comm. Math. Phys., 278(2), 453–486 (2008)
Li, F.: Weak Hopf algebra and new solutions of the quantum Yang–Baxter equation. J. Algebra, 208, 72–100 (1998)
Li, F., Duplij, S.: Weak Hopf algebras and singular solutions of quantum Yang–Baxter equation. Commun. Math. Phys., 225, 191–217 (2002)
Pei, Y. F., Hu, N. H., Rosso, M.: Multi-parameter quantum groups and quantum shuffles, (I). Quantum affine algebras, extended affine Lie algebras, and their applications, Contemp. Math., Vol. 506, Amer. Math. Soc., Providence, RI, 2010, 145–171
Wu, Z. X.: Construct weak Hopf algebras by using Borcherds–Cartan matrix. Acta Math. Sin., Engl. Ser., 25(8), 1337–1352 (2009)
Wu, Z. X.: A class of weak Hopf algebras related to a Borcheds–Cartan matrix. J. Phys. A: Math. Gen., 39, 14611–14626 (2006)
Wu, Z. X.: Extended quantum enveloping algebra of sl(2). Galsgow Math. J., 51, 441–465 (2009)
Wu, Z. X.: Extension of a quantum enveloping algebra by a Hopf algebra. Sci. China Ser. A, 39(12), 1–9 (2009)
Yang, S. L.: Weak Hopf algebras corresponding to Cartan matrices. J. Math. Phys., 46(7), 073502, 18 pages (2005)
Ye, L. X., Wu, Z. X., Mei, X. F.: Weak Hopf algebras corresponding to Borcherds–Cartan matrices. Acta Math. Sin., Engl. Ser., 23(10), 1729–1744 (2007)
Ye, L. X., Li, F.: Weak Hopf superalgebras and their representations. J. Math. Phys., 48(2), 023502, 16 pages (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Natural Science Foundation of China (Grant No. 11171296), the Foundation of Zhejiang Provincial Educational Committee (Grant Nos. Y201327644 and FX2014082) and the Natural Science Foundation of Zhejiang Province (Grant Nos. Q13A01005 and LZ14A010001)
Rights and permissions
About this article
Cite this article
Ye, L.X., Wu, Z.X. The extension of a quantized Borcherds superalgebra by a Hopf algebra. Acta. Math. Sin.-English Ser. 32, 363–372 (2016). https://doi.org/10.1007/s10114-016-4771-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-016-4771-5