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Knit products of graded Lie algebras and groups

Michor, Peter W. — 1990

Proceedings of the Winter School "Geometry and Physics"

Let A = k A k and B = k B k be graded Lie algebras whose grading is in 𝒵 or 𝒵 2 , but only one of them. Suppose that ( α , β ) is a derivatively knitted pair of representations for ( A , B ) , i.e. α and β satisfy equations which look “derivatively knitted"; then A B : = k , l ( A k B l ) , endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra A ( α , β ) B . This graded Lie algebra is called the knit product of A and B . The author investigates the general situation for any graded Lie subalgebras A and B of a graded...

A note on n-ary Poisson brackets

Michor, Peter W.Vaisman, Izu — 2000

Proceedings of the 19th Winter School "Geometry and Physics"

An n -ary Poisson bracket (or generalized Poisson bracket) on the manifold M is a skew-symmetric n -linear bracket { , , } of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order n , i.e., σ S 2 n - 1 ( sign σ ) { { f σ 1 , , f σ n } , f σ n + 1 , , f σ 2 n - 1 } = 0 , S 2 n - 1 being the symmetric group. The notion of generalized Poisson bracket was introduced by et al. in [J. Phys. A, Math. Gen. 29, No. 7, L151–L157 (1996; Zbl 0912.53019) and J. Phys. A, Math. Gen. 30, No. 18, L607–L616 (1997; Zbl 0932.37056)]. They established...

Riemannian geometries on spaces of plane curves

Peter W. MichorDavid Mumford — 2006

Journal of the European Mathematical Society

We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from S 1 to the plane modulo the group of diffeomorphisms of S 1 , acting as reparametrizations. In particular we investigate the metric, for a constant A > 0 , G c A ( h , k ) : = S 1 ( 1 + A κ c ( θ ) 2 ) h ( θ ) , k ( θ ) | c ' ( θ ) | d θ where κ c is the curvature of the curve c and h , k are normal vector fields to c . The term A κ 2 is a sort of geometric Tikhonov regularization because, for A = 0 , the geodesic distance between any two distinct curves is 0, while for A > 0 the...

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