Nonlinear Lie Research Articles

A Few Comments on Ado’s Theorem and Non-Linear Lie Groups

There are claims in the literature on Clifford algebras that every Lie group can be represented as a spin group. In a letter, Pertti Lounesto emphasized, by explicit counter-examples, that this statement is false. This self-contained short paper intends to present a survey of publications of well-known scientists where explicitly developed counter-examples prove the importance of Lounesto’s letter.

Momentum Maps and Stochastic Clebsch Action Principles

We derive stochastic differential equations whose solutions follow the flow of a stochastic nonlinear Lie algebra operation on a configuration manifold. For this purpose, we develop a stochastic Clebsch action principle, in which the noise couples to the phase space variables through a momentum map. This special coupling simplifies the structure of the resulting stochastic Hamilton equations for the momentum map. In particular, these stochastic Hamilton equations collectivize for Hamiltonians that depend only on the momentum map variable. The Stratonovich equations are derived from the Clebsch variational principle and then converted into It\^o form. In comparing the Stratonovich and It\^o forms of the stochastic dynamical equations governing the components of the momentum map, we find that the It\^o contraction term turns out to be a double Poisson bracket. Finally, we present the stochastic Hamiltonian formulation of the collectivized momentum map dynamics and derive the corresponding Kolmogorov forward and backward equations.

Conditional Lie–Bäcklund symmetry, second-order differential constraint and direct reduction of diffusion systems

The equivalence relation between conditional Lie–Bäcklund symmetry, higher-order differential constraint and direct reduction is presented in the case of evolution systems. The second-order nonlinear conditional Lie–Bäcklund symmetries and corresponding induced differential constraints admitted by nonlinear diffusion systems are identified as an application of the conditional Lie–Bäcklund symmetry of evolution systems. Consequently, the corresponding symmetry reductions are obtained due to the compatibility of the resulting differential constraints and the governing systems.

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

This paper is the first part of a two part paper which introduces a method for determining the vanishing topology of nonisolated matrix singularities. A foundation for this is the introduction in this first part of a method for obtaining new classes of free divisors from representations V of connected solvable linear algebraic groups G. For equidimensional representations where dimG = dimV , with V having an open orbit, we give sufficient conditions that the complement E of this open orbit, the “exceptional orbit variety”, is a free divisor (or a slightly weaker free* divisor). We do so by introducing the notion of a “block representation”which is especially suited for both solvable groups and extensions of reductive groups by them. This is a representation for which the matrix representing a basis of associated vector fields on V defined by the representation can be expressed using a basis for V as a block triangular matrix, with the blocks satisfying certain nonsingularity conditions. We use the Lie algebra structure of G to identify the blocks, the singular structure, and a defining equation for E. This construction naturally fits within the framework of towers of Lie groups and representations yielding a tower of free divisors which allows us to inductively represent the variety of singular matrices as fitting between two free divisors. We specifically apply this to spaces of matrices including m×m symmetric, skew-symmetric or general matrices, where we prove that both the classical Cholesky factorization of matrices and a further “modified Cholesky factorization”which we introduce are given by block representations of solvable group actions. For skew-symmetric matrices, we further introduce an extension of the method valid for a representation of a nonlinear infinite dimensional solvable Lie algebras. In part II, we shall use these geometric decompositions and results for computing the vanishing topology for nonlinear sections of free divisors to compute the vanishing topology for matrix singularities for all of the classes.

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

We introduce a method for obtaining new classes of free divisors from representations $V$ of connected linear algebraic groups $G$ where $\dim(G)=\dim(V)$, with $V$ having an open orbit. We give sufficient conditions that the complement of this open orbit, the "exceptional orbit variety", is a free divisor (or a slightly weaker free* divisor) for "block representations" of both solvable groups and extensions of reductive groups by them. These are representations for which the matrix defined from a basis of associated "representation vector fields" on $V$ has block triangular form, with blocks satisfying certain nonsingularity conditions. For towers of Lie groups and representations this yields a tower of free divisors, successively obtained by adjoining varieties of singular matrices. This applies to solvable groups which give classical Cholesky-type factorization, and a modified form of it, on spaces of $m \times m$ symmetric, skew-symmetric or general matrices. For skew-symmetric matrices, it further extends to representations of nonlinear infinite dimensional solvable Lie algebras.

Nonlinear Lie Derivations on Prime Rings

Let M be a unital prime ring containing a nontrivial projection P.We prove that every nonlinear Lie derivation on prime rings M is of the form A→ω(A)+h(A)I,whereω:M→M is an additive derivation and h:M→C is a nonlinear map with h(AB—BA)=0for all A,B∈M.

Characterizations of Nonlinear Lie Derivations ofB(X)

LetXbe an infinite dimensional Banach space, andΦ:B(X)→B(X)is a nonlinear Lie derivation. Then Φ is the formδ+τwhereδis an additive derivation ofB(X)andτis a map fromB(X)into its centerZB(X), which maps commutators into the zero.

Nonlinear Lie triple derivations of triangular algebras

Let 𝒯 be a triangular algebra over a 2-torsion free commutative ring R. In this article, under some mild conditions on 𝒯, we prove that if δ: 𝒯 → 𝒯 is a nonlinear mapping satisfying for any x, y, z ∈ 𝒯, then δ = d + τ, where d is an additive derivation of 𝒯 and τ: 𝒯 → Z(𝒯) (where Z(𝒯) is the centre of 𝒯) is a map vanishing at Lie triple products [[x, y], z].

Nonlinear Lie higher derivations on triangular algebras

Let ℛ be a commutative ring with identity, A, B be unital algebras over ℛ and M be a unital (A, B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let be the triangular algebra consisting of A, B and M. Motivated by the powerful works of Brešar [M. Brešar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), pp. 525–546] and Yu et al. [W.-Y. Yu and J.-H. Zhang, Nonlinear Lie derivations of triangular algebras, Linear Algebra Appl. 432 (2010), pp. 2953–2960], we will study nonlinear Lie higher derivations on 𝒯 in this article. Let D = {L n } n∈ℕ be a Lie higher derivation on 𝒯 without additive condition. Under mild assumptions, it is shown that D = {L n } n∈ℕ is of standard form, i.e. each component L n (n ≥ 1) can be expressed through an additive higher derivation and a nonlinear functional vanishing on all commutators of 𝒯. As applications, nonlinear Lie higher derivations on some classical triangular algebras are characterized.

Nonlinear Lie triple derivations on parabolic subalgebras of finite-dimensional simple Lie algebras

A map ϕ on a Lie algebra is called a nonlinear Lie triple derivation if ϕ([x, [y, z]]) = [ϕ(x), [y, z]] + [x, [ϕ(y), z]] + [x, [y, ϕ(z)]] for all , where ϕ may not be linear. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a standard parabolic subalgebra of L. In this article, we prove that a map ϕ on P is a nonlinear Lie triple derivation if and only if ϕ is a sum of an inner derivation and an additive quasi-derivation.

On Equivalence of Two Realizations for a Nonlinear Lie Algebra

Recently, there are two independent approaches related to a class of nonlinear Lie algebras of three generators; and the realizations of these generators are achieved respectively in Schwinger-boson and position representation. However, by use of the representation transformation between these two representations, the equivalence of the two realizations is therefore proved.

Nonlinear lie-type derivations on full matrix algebras

Let \({\mathcal{R}}\) be a 2-torsion free commutative ring with identity and \({{\rm M}_n(\mathcal{R}) (n\geq 2)}\) be the full matrix algebra over \({\mathcal{R}}\). In this note, we prove that every nonlinear Lie triple derivation on \({{\rm M}_n(\mathcal{R})}\) is of the standard form, i.e. it can be expressed as a sum of an inner derivation, an additive induced derivation and a functional annihilating all second commutators of \({{\rm M}_n(\mathcal{R})}\). A open conjecture about Lie n-derivations is posed at the end of this note.

The structure of nonlinear Lie derivation on von Neumann algebras

Let M be a von Neumann algebra with no central summands of type I1. If Φ:M→M is a nonlinear Lie derivation, then Φ is of the form σ+τ, where σ is an additive derivation of M and τ is a mapping of M into its center ZM which maps commutators into zero.

Nonlinear Lie derivations of triangular algebras

In this paper we prove that every nonlinear Lie derivation of triangular algebras is the sum of an additive derivation and a map into its center sending commutators to zero.

Non-linear Lie conformal algebras with three generators

We classify certain non-linear Lie conformal algebras with three generators, which can be viewed as deformations of the current Lie conformal algebra of sl_2. In doing so we discover an interesting 1-parameter family of non-linear Lie conformal algebras R_{-1}^d and the corresponding freely generated vertex algebras V_{-1}^d, which includes for d=1 the affine vertex algebra of sl_2 at the critical level k=-2. We construct free-field realizations of the algebras V_{-1}^d extending the Wakimoto realization of the affine vertex algebra of sl_2 at the critical level, and we compute their Zhu algebras.

Nonlinear Lie derivations on upper triangular matrices

Let M n (𝔸) and T n (𝔸) be the algebra of all n × n matrices and the algebra of all n × n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T n (𝔸) into M n (𝔸) is of the form A → AT − TA + A ϕ + ξ(A)I n , where T ∈ M n (𝔸), ϕ : 𝔸 → 𝔸 is an additive derivation, ξ : T n (𝔸) → 𝔸 is a nonlinear map with ξ(AB − BA) = 0 for all A, B ∈ T n (𝔸) and A ϕ is the image of A under ϕ applied entrywise.

Finite vs affine W-algebras

In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra ZhuΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra Zhu H V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra Zhu H R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence of three definitions of a finite W-algebra is established.

Supplements on the theory of exponential Lie groups

A Lie group with surjective exponential function is called exponential. There are presented supplying results on the theory of exponential Lie groups. An additional criterion for (Mal'cev) splittable exponential Lie groups is presented as well as an additional necessary condition for solvable exponential Lie groups. Both the conditions have the advantage that they are well practicable. Moreover, it is shown that there exist simple non-linear exponential Lie groups. The general case is also considered: There are given some conditions concerning the Mal'cev splittable radical. At the end of the paper, one can find some counterexamples to some conjectures concerning exponential Lie groups.

Nonlinear delay systems, Lie algebras and Lyapunov transformations

Nonlinear delay systems, Lie algebras and Lyapunov transformations

Convergence of Normal Form Transformations: The Role of Symmetries

We discuss the convergence problem for coordinate transformations which take a given vector field into Poincare–Dulac normal form. We show that the presence of linear or nonlinear Lie point symmetries can guarantee convergence of these normalizing transformations in a number of scenarios. As an application, we consider a class of bifurcation problems.