Operatoren in Lieschen Ringen.

Wever, Franz

doi:10.1515/crll.1950.187.44

Cited by 15 publications

(17 citation statements)

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“…We recover in particular the theorem of Dynkin [4], Specht [19], Wever [20] (case f = Id) and obtain an extension thereof to the case f = δ x i . We let the reader derive similar results for other families of Lie derivations.…”

Section: Twisted Dynkin Operators On Free Lie Algebrasmentioning

confidence: 72%

Logarithmic derivatives and generalized Dynkin operators

Menous¹,

Patras²

2013

J Algebr Comb

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Motivated by a recent surge of interest for Dynkin operators in mathematical physics and by problems in the combinatorial theory of dynamical systems, we propose here a systematic study of logarithmic derivatives in various contexts. In particular, we introduce and investigate generalizations of the Dynkin operator for which we obtain Magnus-type formulas.Keywords Dynkin operator · Free Lie algebras · Logarithmic derivative · Rota-Baxter algebra IntroductionDynkin operators are usually defined as iterated bracketings. They are particularly popular in the framework of linear differential equations and the so-called continuous Baker-Campbell-Hausdorff problem (to compute the logarithm of an evolution operator). We refer to [17] for details and an historical survey of the field. Dynkin operators can also be expressed as a particular type of logarithmic derivatives (see Corollary 3 below). They have received increasingly more and more interest during recent years, for various reasons.(1) In the theory of free Lie algebras and noncommutative symmetric functions, it was shown that Dynkin operators generate the descent algebra (the direct sum of Solomon's algebras of type A, see [9,17]) and play a crucial role in the theory of Lie idempotents.

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Section: Twisted Dynkin Operators On Free Lie Algebrasmentioning

confidence: 72%

Logarithmic derivatives and generalized Dynkin operators

Menous¹,

Patras²

2013

J Algebr Comb

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“…This process defines a standard way, the "left to right" or "left normed" bracketing, to associate a Lie bracket to each monomial. The following important result, proved successively by Dynkin [31], Specht [64] and Wever [73] states that, if a polynomial is a Lie element, then it is equal to its left normed bracketing.…”

Section: Preliminary Algebraic Resultsmentioning

confidence: 99%

On expansions for nonlinear systems, error estimates and convergence issues

Beauchard,

Borgne,

Marbach

2020

Preprint

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Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied.First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann's infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation.Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input.Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann's infinite product expansion.Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.

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“…, x n ∈ X. The Dynkin-Specht-Wever theorem [Dyn47, Spe48,Wev49] says that ω 2 n = nω n or, equivalently, that left action of 1 n ω n yields a projection from A n (X) onto L n (X). Any such idempotent e in F S n is a Lie idempotent.…”

Section: Lie Idempotentsmentioning

confidence: 99%

The module structure of the Solomon–Tits algebra of the symmetric group

Schocker¹

2006

Journal of Algebra

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Let (W, S) be a finite Coxeter system. Tits defined an associative product on the set Σ of simplices of the associated Coxeter complex. The corresponding semigroup algebra is the Solomon-Tits algebra of W . It contains the Solomon algebra of W as the algebra of invariants with respect to the natural action of W on Σ. For the symmetric group S n , there is a 1-1 correspondence between Σ and the set of all set compositions (or ordered set partitions) of {1, . . . , n}. The product on Σ has a simple combinatorial description in terms of set compositions. We study in detail the representation theory of the Solomon-Tits algebra of S n over an arbitrary field, and show how our results relate to the corresponding results on the Solomon algebra of S n . This includes the construction of irreducible and principal indecomposable modules, a description of the Cartan invariants, of the Ext-quiver, and of the descending Loewy series. Our approach builds on a (twisted) Hopf algebra structure on the direct sum of all Solomon-Tits algebras.

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Operatoren in Lieschen Ringen.

Cited by 15 publications

References 0 publications

Logarithmic derivatives and generalized Dynkin operators

Logarithmic derivatives and generalized Dynkin operators

On expansions for nonlinear systems, error estimates and convergence issues

The module structure of the Solomon–Tits algebra of the symmetric group

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