174 research outputs found
Supertrace and superquadratic Lie structure on the Weyl algebra, with applications to formal inverse Weyl transform
Using the Moyal *-product and orthosymplectic supersymmetry, we construct a
natural non trivial supertrace and an associated non degenerate invariant
supersymmetric bilinear form for the Lie superalgebra structure of the Weyl
algebra. We decompose adjoint and twisted adjoint actions. We define a
renormalized supertrace and a formal inverse Weyl transform in a deformation
quantization framework and develop some examples.Comment: 26 pages; v1: added a reference, corrected typos; v2: changed title,
added a reference, typos fixe
Back to the Amitsur-Levitzki theorem: a super version for the orthosymplectic Lie superalgebra osp(1, 2n)
We prove an Amitsur-Levitzki type theorem for the Lie superalgebras osp(1,2n)
inspired by Kostant's cohomological interpretation of the classical theorem. We
show that the Lie superalgebras gl(p,q) cannot satisfy an Amitsur-Levitzki type
super identity if p, q are non zero and conjecture that neither can any other
classical simple Lie superalgebra with the exception of osp(1,2n).Comment: 11 pages, to be published in Letters in Mathematical Physics; added
references, corrected typo
Lie algebras generated by extremal elements
We study Lie algebras generated by extremal elements (i.e., elements spanning
inner ideals of L) over a field of characteristic distinct from 2. We prove
that any Lie algebra generated by a finite number of extremal elements is
finite dimensional. The minimal number of extremal generators for the Lie
algebras of type An, Bn (n>2), Cn (n>1), Dn (n>3), En (n=6,7,8), F4 and G2 are
shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results
are related to group theoretic ones for the corresponding Chevalley groups.Comment: 28 page
On the robust synchronization of Brockett oscillators
International audienceIn this article, motivated by a recent work of R. Brockett Brockett (2013), we study a robust synchronization problem for multistable Brockett oscillators within an Input-to-State Stability (ISS) framework. Based on a recent generalization of the classical ISS theory to multistable systems and its application to the synchronization of multistable systems, a synchronization protocol is designed with respect to compact invariant sets of the unperturbed Brockett oscillator. The invariant sets are assumed to admit a decomposition without cycles (i.e. with neither homoclinic nor heteroclinic orbits). Contrarily to the local analysis of Brockett (2013), the conditions obtained in our work are global and applicable for family of non-identical oscillators. Numerical simulation examples illustrate our theoretical results
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