The notion of commutator width of a group, defined as the smallest number of commutators needed to represent each element of the derived group as their product, has been extensively studied over the past decades. In particular, in 1992 Barge and Ghys discovered the first example of a simple group of commutator width greater than one among groups of diffeomorphisms of smooth manifolds. We consider a parallel notion of bracket width of a Lie algebra and present the first examples of simple Lie algebras of bracket width greater than one.
Bibliographical noteFunding Information:
We thank Joseph Bernstein, Yuly Billig, Mikhail Borovoi, Zhihua Chang, Be’eri Greenfeld, Hanspeter Kraft, Leonid Makar-Limanov, Olivier Mathieu, Anatoliy Petravchuk, Vladimir Popov, Oksana Yakimova, and Efim Zelmanov for useful discussions regarding various aspects of this work. We are grateful to the anonymous referee for helpful comments. The first author was partially supported by the French ANR grant FIBALGA ANR-18-CE40-0003-01 and the EIPHI Graduate School ANR-17-EURE-0002. Research of the second author was supported by the ISF grants 1623/16 and 1994/20. This research started during the visit of the second author to the MPIM (Bonn) and continued during the visit of the third author to Bar-Ilan University. The support of these institutions is gratefully acknowledged.
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- Danielewski surfaces
- Lie algebras of algebraic
- Locally nilpotent derivations
- Simple lie algebras
- Smooth affine curves
- Symplectic and hamiltonian vector fields