Abstract
We introduce a large class of bicovariant differential calculi on any quantum group A, associated to Ad-invariant elements. For example, the deformed trace element on SLq (2) recovers Woronowicz's 4D ± calculus. More generally, we obtain a class of differential calculi on each quantum group A(R), based on the theory of the corresponding braided groups B(R). Here R is any regular solution of the QYBE.
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References
Abe, E., Hopf Algebras, Cambridge Univ. Press, 1980.
Brzeziński, T., Remarks on bicovariant differential calculi and exterior Hopf algebras, Preprint DAMTP/92-11, 1992.
BrzezińskiT., DabrowskiH., and RembielińskiJ., On the quantum differential calculus and the quantum holomorphicity, J. Math. Phys. 33, 19 (1992).
Brzeziński, T. and Majid, S., Quantum group gauge theory on quantum spaces, Preprint DAMTP/92-27, 1992.
CoquerauxR. and KastlerD., Remarks on the differential envelopes of associative algebras, Pacific J. Math. 137, 245 (1989).
Fadeev, L. D., Reshetikhin, N. Yu., and Takhtajan, L. A., Quantization of Lie groups and Lie algebras, Algebra i Analiz 1, 1989.
JurcoB., Differential calculus on quantized simple Lie groups, Lett. Math. Phys. 22, 177 (1991); U. Carow-Watamura et al., Bicovariant differential calculus on quantum groups SU q (N) and SO q (N), Comm. Math. Phys. 142, 605 (1991).
KastlerD., Cyclic Cohomology within Differential Envelope, Hermann, Paris, 1988.
MajidS., Examples of braided groups and braided matrices, J. Math. Phys. 32, 3246 (1991); Rank of quantum groups and braided groups in dual form, in Lecture Notes in Math. 1510, Springer-Verlag, New York, 1992, pp. 79–88.
Majid, S., Quantum and braided linear algebra, J. Math. Phys. (in press).
MaltsiniotisG., Groupes quantiques et structures différentielles, C. R. Acad. Sci. Paris, Serie I311, 831 (1990).
Manin, Yu. I., Notes on quantum groups and quantum de Rham complexes, Preprint MPI, 1991.
Sudbery, A., The algebra of differential forms on a full metric bialgebra, Yourk University preprint, 1991.
Sudbery, A., Canonical differential calculus on quantum linear groups and super-groups, York University preprint, 1992.
SweedlerM. E., Hopf Algebras, Benjamin, New York, 1969.
WoronowiczS. L., Twisted su2 group. An example of a non-commutative differential calculus, Publ. RIMS Kyoto 23, 117 (1987).
WoronowiczS. L., Differential calculus on compact matric pseudogroups (quantum groups), Comm. Math. Phys. 122, 125 (1989).
Zumino, B., Introduction to the differential geometry of quantum groups, in K. Schmüdgen (ed), Proc. Mathematical Physics X, Leipzig, Germany, 1991, Springer-Verlag, 1992, p. 20.
SchuppP., WattsP., and ZuminoB., Differential geometry on linear quantum groups, Lett. Math. Phys. 25, 139 (1992).
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Supported by St John's College, Cambridge and KBN grant 2 0218 91 01.
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Brzeziński, T., Majid, S. A class of bicovariant differential calculi on hopf algebras. Lett Math Phys 26, 67–78 (1992). https://doi.org/10.1007/BF00420519
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DOI: https://doi.org/10.1007/BF00420519