Versal normal form for nonsemisimple singularities

Mokhtari, Fahimeh; Sanders, Jan A.

doi:10.1016/j.jde.2019.03.039

Cited by 5 publications

(8 citation statements)

References 18 publications

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“…In addition, these formulas may prove useful in the study of the convergence of the (first level) normal form where one needs to estimate the transformation through the Homological Equation. We refer the interested reader to [15], where parts of the proof show a striking similarity to the techniques employed in the present paper and we hope to improve on these results using the techniques described in [11].…”

Section: Discussionmentioning

confidence: 70%

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Equivariant decomposition of polynomial vector fields

Mokhtari¹,

Sanders²

2019

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To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the linear nilpotent. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the present paper is to formulate a systematic approach to this calculation.We propose to do this using a rational method for the inversion of the Clebsch-Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D case.

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Section: Discussionmentioning

confidence: 70%

“…Then let a be the linear form in x, y that is in ker M and has H-eigenvalue 1. Recall from [2,11] the sl 2 -representation for two dimensional vector field by…”

Section: Unique Normal For the A 1 -Familymentioning

confidence: 99%

Equivariant decomposition of polynomial vector fields

Mokhtari¹,

Sanders²

2019

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“…Kostant refers to this as an S-triple in [14], but this terminology seems not to have been taken up) containing a given nilpotent. This triple need not to be contained in net C,N , but still allows us to define the nilpotent normal form in the same fashion as described in [17]. To formulate the versal normal form theory will need more work in this case and will not be attempted here.…”

Section: Introductionmentioning

confidence: 99%

The Lie algebraic structure of colored networks

Mokhtari¹,

Sanders²

2021

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In the computation of the normal form of a colored network vector field, following the semigroup(oid) approach in [19], one would like to be able to say something about the structure of the Lie algebra of linear colored network vector fields. Unlike the purely abstract approach in [10], we describe here a concrete algorithm that gives us the Levi decomposition. If we apply this algorithm to a given subalgebra, it does put the elements in the subalgebra in the block form given by the Levi decomposition, but this need not be the Levi decomposition of the given subalgebra.We show that for N -dimensional vector fields with C colors (different functions describing different types of cells in the network) this Lie algebra netC,N is isomorphic to the semidirect sum of a semisimple part, consisting of two simple components slC and slB, with B = N −C, which we write as a block-matrix and a solvable part, consisting of two elements representing the identity C in c ≃ gl C and B in b ≃ gl B , and an abelian algebra a ≃ Gr(C, N ), the Grassmannian, consisting of the C-dimensional subspaces of R N . The methods in this paper can be immediately applied to study the linear maps of colored networks.

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“…The representation theory of sl 2 has also played a role in the normalisation of nilpotent vector fields depending on parameters. For instance, [14] has constructed the nilpotent normal form of versal deformations of nilpotent and nonsemisimple vector fields. The construction of the normal form for maps with nilpotent linear part has received little attention in the literature.…”

Section: Introductionmentioning

confidence: 99%

“…The Jacobson-Morozov theorem [19,14,Section 12.5] guarantees that any nilpotent element of a reductive Lie algebra can be embedded in an sl 2 -triple, which consists of three elements N, H, M that satisfy the commutator relations…”

Section: Introductionmentioning

confidence: 99%

Normal form for maps with nilpotent linear part

Mokhtari¹,

Roell²,

Sanders³

2020

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The normal form for an n-dimensional map with irreducible nilpotent linear part is determined using sl 2 -representation theory. We sketch by example how the reducible case can also be treated in an algorithmic manner. The construction (and proof) of the sl 2 -triple from the nilpotent linear part is more complicated than one would hope for, but once the abstract sl 2 theory is in place, both the description of the normal form and the computational splitting to compute the generator of the coordinate transformation can be handled explicitly in terms of the nilpotent linear part without the explicit knowledge of the triple. If one wishes one can compute the normal form such that it is guaranteed to lie in the kernel of an operator and one can be sure that this is really a normal form with respect to the nilpotent linear part; one can state that the normal form is in sl 2 -style.Although at first sight the normal form theory for maps is more complicated than for vector fields in the nilpotent case, it turns out that the final result is much better. Where in the vector field case one runs into invariant theoretical problems when the dimension gets larger if one wants to describe the general form of the normal form, for maps we obtain results without any restrictions on the dimension.In the literature only the 2-dimensional nilpotent case has been described sofar, as far as we know.

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Versal normal form for nonsemisimple singularities

Cited by 5 publications

References 18 publications

Equivariant decomposition of polynomial vector fields

Equivariant decomposition of polynomial vector fields

The Lie algebraic structure of colored networks

Normal form for maps with nilpotent linear part

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