Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-28T16:01:10.769Z Has data issue: false hasContentIssue false
  • English
  • Français

The algebra of differential forms on a full matric bialgebra

Published online by Cambridge University Press:  24 October 2008

A. Sudbery
A. Sudbery
Affiliation:
Department of Mathematics, University of York, York, YO1 5DD
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct a non-commutative analogue of the algebra of differential forms on the space of endomorphisms of a vector space, given a non-commutative algebra of functions and differential forms on the vector space. The construction yields a differential bialgebra which is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. We give necessary and sufficient conditions for such an algebra to exist, show that it is uniquely determined by the differential algebra on the vector space, and show that it is a non-commutative superpolynomial algebra in the matrix elements and their differentials (i.e. that it has the same dimensions of homogeneous components as in the classical case).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 111 - 130
Copyright
Copyright © Cambridge Philosophical Society 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Cabow-Watamura, U., Schlieker, M., Watamuba, S. and Weich, W.. Bioovariant differential calculus on quantum groups SUq(N) and SOq(N). Commun. Math. Phys. 142 (1991), 605–641.CrossRefGoogle Scholar
[2]Bergman, G. M.. The diamond lemma for ring theory. Adv. Math. 29 (1978), 178–218.CrossRefGoogle Scholar
[3]Demidov, E. E., Manin, Yu. I., Mukhin, E. E. and Zhdanovich, D. V.. Non-standard quantum deformations of GL(n) and constant solutions of the Yang–Baxter equation. Prog. Theor. Phys. Supp. 102 (1990), 203–218.CrossRefGoogle Scholar
[4]Faddeev, L. D., Reshetikhxn, N. Y. and Takhtajan, L. A.. Quantization of Lie groups and Lie algebras. Algebra and Analysis 1 (1987), 178–206.Google Scholar
[5]Maltsiniotis, G.. Calcul différentiel sur le groupe linéaire quantique, preprint.Google Scholar
[6]Manin, Yu. I.. Quantum groups and non-commutative geometry (C.R.M., 1988).Google Scholar
[7]Manin, Yu. I.. Notes on quantum groups and quantum de Rham complexes, preprint. Max-Planck-Institut, Bonn.Google Scholar
[8]Rosso, M.. Algèbres enveloppantes quantifées, groupes quantiques compacts de matrices et calcul différentiel non commutatif. Duke Math. J. 61 (1990), 11–40.CrossRefGoogle Scholar
[9]Schmidke, W. B., Vokos, S. P. and Zumino, B.. Differential geometry of the quantum supergroup GLq(1|1). Z. Phys. C 48 (1990), 249–255.CrossRefGoogle Scholar
[10]Schwenk, J.. Differential calculus for the n-dimensional quantum hyperplane. In Quantum Groups (ed. Curtright, T., Fairlie, D. and Zachos, C.) (World Scientific, 1991), pp. 53–61.Google Scholar
[11]Sudbery, A.. Matrix-element bialgebras determined by quadratic coordinate algebras. J. Algebra, to appear.Google Scholar
[12]Takeuchi, M.. Matric bialgebras and quantum groups. Israel J. Math. 72 (1990), 232–252.CrossRefGoogle Scholar
[13]Woronowicz, S. L.. Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys. 122 (1989), 125–170.CrossRefGoogle Scholar
[14]Wess, J. and Zumino, B.. Covariant differential calculus on the quantum hyperplane. Nucl. Phys. B (Proc. Suppl.) 18B (1990), 302.Google Scholar