Skip to main content
Log in

Gröbner–Shirshov basis and minimal projective resolution of \(U_q^+(A_{n})\)

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

In this paper, by using Anick’s resolution and Gröbner–Shirshov basis for quantum group of type \(A_n\), we compute the first three steps of a minimal projective resolution of the trivial module of \(U_q^+(A_n)\) and as an application we compute the global dimension of \(U_q^+(A_n)\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Faddeev, L.D.: Integravble models in \((1+1)\)-dimensional quantum field theory. In: Zuber, J.-B., Stora, R. (eds.) Recent Advances in F ield Theory and Statistical Mechanics, pp. 561-608. North-Holland, Amsterdam (1984)

  2. Sklyanin, E.K.: Some algebraic structures connected with the Yang–Baxter equation. Representations of quantum algebras. Funct. Anal. Appl. 17, 273–284 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. Drinfeld, V.G.: Hopf algebras and quantum Yang–Baxter equations. Soviet Math. Dokl. 32, 254–258 (1985)

    MathSciNet  Google Scholar 

  4. Jimbo, M.: A q-analogue of \(U({g})\) and the Yang–Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buchberger, B.: An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal. Ph.D. Thesis, University of Innsbruck (1965)

  6. Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Shirshov, A.I.: Some algorithmic problems for Lie algebras. Siberian Math. J. 3, 292–296 (1962)

    MATH  Google Scholar 

  8. Bokut, L.A.: Imbeddings into simple associative algebras. Algebra Logic 15, 117–142 (1976)

    Article  MathSciNet  Google Scholar 

  9. Anick, D.J.: On the homology of associative algebras. Trans. Am. Math. Soc. 296(2), 641–659 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bokut, L.A., Malcolmson, P.: Gröbner–Shirshov basis for quantum enveloping algebras. Israel J. Math. 96, 97–113 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ringel, C.M.: Hall algebras and quantum groups. Invent. math. 101, 583–592 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ringel, C.M.: PBW-bases of quantum groups. J. Reine Angew. Math. 470, 51–88 (1996)

    MathSciNet  MATH  Google Scholar 

  13. Yamane, H.: A Poincaré–Birkhoff–Witt theorem for quantized universal enveloping algebras of type \(A_n\). Publ. RIMS. Kyoto Univ. 25, 503–520 (1989)

  14. Lorentz, M.E., Lorentz, M.: On crossed products of Hopf algebras. Proc. Am. Math. Soc. 123, 33–38 (1995)

    Article  MathSciNet  Google Scholar 

  15. Avramov, L.L., Foxby, H-B., Halperin, S.: Differential Graded Homological Algebra (2007). (in preparation)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Abdukadir Obul.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 11361056).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yunus, G., Obul, A. Gröbner–Shirshov basis and minimal projective resolution of \(U_q^+(A_{n})\) . Rend. Circ. Mat. Palermo, II. Ser 65, 283–296 (2016). https://doi.org/10.1007/s12215-016-0233-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-016-0233-2

Keywords

Mathematics Subject Classification

Navigation