Classification of Left-Covariant Differential Calculi on the Quantum Group $\SLq 2$

Abstract

For transcendental values of q the quantum tangent spaces of all left-covariant first order differential calculi of dimension less than four on the quantum group $\SLq 2$ are given. All such differential calculi $\Gamma$ are determined and investigated for which the left-invariant differential one-forms $\omega (u_2^1)$, $\omega (u_1^2)$ and $\omega (u_1^1-u_2^2)$ generate $\Gamma$ as a bimodule and the universal higher order differential calculus has the same dimension as in the classical case. Important properties (cohomology spaces, *-structures, braidings, generalized Lie brackets) of these calculi are examined as well. Keywords: quantum groups, noncommutative differential calculus, quantum tangent space

arXiv commentLaTeX2e, 43 pages, to appear in Journal of Algebra

References in corpus (4)

• On Quantum Lie Algebras and Quantum Root Systems
• On the Construction of Covariant Differential Calculi on Quantum Homogeneous Spaces
• Bicovariant Differential Geometry of the Quantum Group $S{L}_h(2)$
• Classification of bicovariant differential calculi on the Jordanian quantum groups GL_{g,h}(2) and SL_{h}(2) and quantum Lie algebras

Cited by in corpus (5)

• Twisted Homology of Quantum SL(2) - Part II
• Warped Products and Yang-Mills equations on non commutative spaces
• A Hodge - De Rham Dirac operator on the quantum $\mathrm{SU}(2)$
• Examples of twisted cyclic cocycles from covariant differential calculi
• Non-commutative integration, zeta functions and the Haar state for $S{U}_q(2)$