# Left-covariant differential calculi on $S{L}_{q}\left(N\right)$

Konrad Schmüdgen; Axel Schüler

Banach Center Publications (1997)

- Volume: 40, Issue: 1, page 185-191
- ISSN: 0137-6934

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topSchmüdgen, Konrad, and Schüler, Axel. "Left-covariant differential calculi on $SL_{q}(N)$." Banach Center Publications 40.1 (1997): 185-191. <http://eudml.org/doc/252249>.

@article{Schmüdgen1997,

abstract = {We study $N^\{2\} - 1$ dimensional left-covariant differential calculi on the quantum group $SL_q(N)$. In this way we obtain four classes of differential calculi which are algebraically much simpler as the bicovariant calculi. The algebra generated by the left-invariant vector fields has only quadratic-linear relations and posesses a Poincaré-Birkhoff-Witt basis. We use the concept of universal (higher order) differential calculus associated with a given left-covariant first order differential calculus. It turns out that the space of left-invariant k-forms has the dimension $N^\{2\} - 1\atopwithdelims ()k$ as in the case of the corresponding classical Lie group SL(N).},

author = {Schmüdgen, Konrad, Schüler, Axel},

journal = {Banach Center Publications},

keywords = {quantum Lie algebra; noncommutative differential calculus; quantum groups; differential calculi; left-covariance},

language = {eng},

number = {1},

pages = {185-191},

title = {Left-covariant differential calculi on $SL_\{q\}(N)$},

url = {http://eudml.org/doc/252249},

volume = {40},

year = {1997},

}

TY - JOUR

AU - Schmüdgen, Konrad

AU - Schüler, Axel

TI - Left-covariant differential calculi on $SL_{q}(N)$

JO - Banach Center Publications

PY - 1997

VL - 40

IS - 1

SP - 185

EP - 191

AB - We study $N^{2} - 1$ dimensional left-covariant differential calculi on the quantum group $SL_q(N)$. In this way we obtain four classes of differential calculi which are algebraically much simpler as the bicovariant calculi. The algebra generated by the left-invariant vector fields has only quadratic-linear relations and posesses a Poincaré-Birkhoff-Witt basis. We use the concept of universal (higher order) differential calculus associated with a given left-covariant first order differential calculus. It turns out that the space of left-invariant k-forms has the dimension $N^{2} - 1\atopwithdelims ()k$ as in the case of the corresponding classical Lie group SL(N).

LA - eng

KW - quantum Lie algebra; noncommutative differential calculus; quantum groups; differential calculi; left-covariance

UR - http://eudml.org/doc/252249

ER -

## References

top- [1] A. Connes, Non-commutative differential geometry, Publ. Math. IHES 62, 44-144 (1986).
- [2] L. K. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan, Quantization of Lie groups and Lie algebras, Algebra and Analysis 1, 178-206 (1987).
- [3] K. Schmüdgen and A. Schüler, Classification of bicovariant differential calculi on quantum groups of type A, B, C and D, Commun. Math. Phys. 167, 635-670 (1995).
- [4] A. Sudbery, Non-commuting coordinates and differential operators, in: Quantum Groups, T. Curtright, D. Fairlie and C. Zachos (eds.), pp. 33-52, World Scientific, Singapore, 1991.
- [5] S. L. Woronowicz, Twisted SU(2) group. An example of a non-commutative differential calculus, Publ. RIMS Kyoto Univ. 23, 177-181 (1987).
- [6] S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Commun. Math. Phys. 122, 125-170 (1989). Zbl0751.58042

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