Isomorphic Lie Research Articles

Characterization of Lie n-isomorphisms of certain algebras

In this paper, Lie [Formula: see text]-isomorphisms on nest algebras and von Neumann algebras are constructed, which can be viewed as a common generalization of Lie isomorphisms and Lie triple isomorphisms. Besides, several 2-local Lie [Formula: see text]-isomorphisms and 2-local Lie [Formula: see text]-∗-isomorphisms on certain von Neumann algebras are studied.

Physics of the Inverted Harmonic Oscillator: From the lowest Landau level to event horizons

In this work, we present the inverted harmonic oscillator (IHO) Hamiltonian as a paradigm to understand the quantum mechanics of scattering and time-decay in a diverse set of physical systems. As one of the generators of area preserving transformations, the IHO Hamiltonian can be studied as a dilatation generator, squeeze generator, a Lorentz boost generator, or a scattering potential. In establishing these different forms, we demonstrate the physics of the IHO that underlies phenomena as disparate as the Hawking–Unruh effect and scattering in the lowest Landau level (LLL) in quantum Hall systems. We derive the emergence of the IHO Hamiltonian in the LLL in a gauge invariant way and show its exact parallels with the Rindler Hamiltonian that describes quantum mechanics near event horizons. This approach of studying distinct physical systems with symmetries described by isomorphic Lie algebras through the emergent IHO Hamiltonian enables us to reinterpret geometric response in the lowest Landau level in terms of relativistic effects such as Wigner rotation. Further, the analytic scattering matrix of the IHO points to the existence of quasinormal modes (QNMs) in the spectrum, which have quantized time-decay rates. We present a way to access these QNMs through wave packet scattering, thus proposing a novel effect in quantum Hall point contact geometries that parallels those found in black holes.

The Characterization of 2-Local Lie Automorphisms of Some Operator Algebras

Let M ⊑ B(X) be an algebra with nontrivial idempotents or nontrivial projections if M is a *-algebra and ZM = ℂI. In this paper, the notion of (strong) 2-local Lie automorphism normalized property is introduced and it is proved that if M has 2-local Lie automorphism normalized property and Φ: M → M is an almost additive surjective 2-local Lie isomorphism with idempotent decomposition property, then → = Ψ + τ, where Ψ is an automorphism of M or the negative of an anti-automorphism of M and τ is a homogenous map from M into ℂI. Moreover, it is proved that nest algebras on a separable complex Hilbert space II with dimII >2 and factor von Neumann algebras on a separable complex Hilbert space H with dimH > 2 have strong 2-local Lie automorphism normalized property.

Towards Stochasticity through Joint Invariant Functions of Two Isomorphic Lie Algebras of SL(2R) Type

In the motion fractal theory, the scale relativity dynamics of any complex system are described through various Schrödinger or hydrodynamic type fractal “regimes”. In the one dimensional stationary case of Schrödinger type fractal “regimes”, synchronizations of complex system entities implies a joint invariant function with the simultaneous action of two isomorphic groups of the S L ( 2 R ) type as solutions of Stoka type equations. Among these joint invariant functions, Gaussians become in the Jeans’s sense, probability density (i.e., stochasticity) whenever the information on the complex system analyzed is fragmentary. In the two-dimensional case of hydrodynamic type fractal “regimes” at a non-differentiable scale, the soliton and soliton-kink of fractal type of the velocity field generate the minimal vortex of fractal type that becomes the source of all turbulences in the complex systems dynamics. Some correlations of our model to experimental data were also achieved.

The SO(3)-instanton moduli space and tensor products of ADHM data

Let MnK be the moduli space of framed K-instantons with instanton number n over the four-sphere S4 when K is a compact simple Lie group of classical type. Due to Donaldson's theorem [13], its scheme structure is given by the regular locus of a GIT quotient of μ−1(0) where μ is the moment map on the associated symplectic vector space of ADHM data.A main theorem of this paper asserts that μ is flat for K=SO(3,R) and any n≥0. Hence we complete the interpretation of the K-theoretic Nekrasov partition function for the classical groups [29] in term of Hilbert series of the instanton moduli spaces together with the author's previous results [10][11].We also write ADHM data for the second symmetric and exterior products of the associated vector bundle of an instanton. This gives an explicit quiver-theoretic description of the isomorphism MnK≅MnK′ for all the pairs K,K′ with isomorphic Lie algebras.

Algorithm to compute minimal matrix representation of nilpotent lie algebras

ABSTRACTAs it is well-known there exist matrix representations of any given finite-dimensional complex Lie algebra. More concretely, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. However, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any finite-dimensional nilpotent Lie algebra from its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representative for each nilpotent Lie algebra of dimensions 6 and 7 and we have also obtained the representation of some families with an arbitrary dimension.

On two isomorphic Lie algebroids associated with feedback linearization

We present two Lie algebroids linked to the construction of the linearizing output of an input affine nonlinear system. The algorithmic development of the linearizing output proceeds inductively, and each stage has two structures, namely a codimension one foliation defined through an integrable 1-form ω, and a transversal vectorfield g to the foliation. Each integral manifold of the vectorfield g defines an equivalence class of points. Due to transversality, a leaf of the foliation is chosen to represent these equivalence classes. A Lie groupoid is defined with its base given as the particular chosen leaf and with the product induced by the pseudogroup of diffeomorphisms that preserve equivalence classes generated by the integral manifolds of g. Two Lie algebroids associated with this groupoid are then defined. The theory is illustrated with an example using polynomial automorphisms as particular cases of diffeomorphisms and shows the relation with the Jacobian conjecture.

Complete classification of pseudo H-type algebras: II

We classify a class of 2-step nilpotent Lie algebras related to the representations of the Clifford algebras in the following way. Let $J\colon \Cl(\mathbb R^{r,s})\toU$ be a representation of the Clifford algebra $\Cl(\mathbb R^{r,s})$ generated by the pseudo Euclidean vector space $\mathbb R^{r,s}$. Assume that the Clifford module $U$ is endowed with a bilinear symmetric non-degenerate real form $\la\cdot\,,\cdot\ra_U$ making the linear map $J_z$ skew symmetric for any $z\in\mathbb R^{r,s}$. The Lie algebras and the Clifford algebras are related by $\la J_zv,w\ra_U=\la z,[v,w]\ra_{\mathbb R^{r,s}}$, $z\in \mathbb R^{r,s}$, $v,w\in U$. We detect the isomorphic and non-isomorphic Lie algebras according to the dimension of $U$ and the range of the non-negative integers~$r,s$.

Lie isomorphisms of reflexive algebras. II

By characterizing minimal non-central Lie ideals, we describe the structure of Lie isomorphisms of reflexive algebras with J-subspace lattices. This result can apply to reflexive algebras with atomic Boolean subspace lattices or with pentagon subspace lattice algebras.

Complete classification of pseudo H-type Lie algebras: I

Let \({\mathscr {N}}\) be a 2-step nilpotent Lie algebra endowed with a non-degenerate scalar product \(\langle .\,,.\rangle \), and let \({\mathscr {N}}=V\oplus _{\perp }Z\), where Z is the centre of the Lie algebra and V its orthogonal complement. We study classification of the Lie algebras for which the space V arises as a representation space of the Clifford algebra \({{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\), and the representation map \(J:{{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\rightarrow {{\mathrm{End}}}(V)\) is related to the Lie algebra structure by \(\langle J_zv,w\rangle =\langle z,[v,w]\rangle \) for all \(z\in {\mathbb {R}}^{r,s}\) and \(v,w\in V\). The classification depends on parameters r and s and is completed for the Clifford modules V having minimal possible dimension, that are not necessary irreducible. We find necessary conditions for the existence of a Lie algebra isomorphism according to the range of the integer parameters \(0\le r,s<\infty \). We present a constructive proof for the isomorphism maps for isomorphic Lie algebras and determine the class of non-isomorphic Lie algebras.

Minimal faithful upper-triangular matrix representations for solvable Lie algebras

The existence of matrix representations for any given finite-dimensional complex Lie algebra is a classic result on Lie Theory. In particular, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. Unfortunately, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any finite-dimensional solvable Lie algebra when inserting its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representatives not only for each solvable Lie algebra with dimension less than 6, but also for some solvable Lie algebras of arbitrary dimension.

2-local Lie isomorphisms of nest algebras

2-local Lie isomorphisms of nest algebras

Commuting (centralizing) traces and Lie (triple) isomorphisms on triangular algebras revisited

The aim of the paper is to give a description of commuting (centralizing) traces of bilinear maps and Lie (triple) isomorphisms on certain triangular algebras. As consequences, commuting (centralizing) traces of bilinear maps and Lie (triple) isomorphisms on upper triangular matrix algebras over a 2-torsion free commutative ring are completely determined.

2-local Lie isomorphisms of operator algebras on Banach spaces

2-local Lie isomorphisms of operator algebras on Banach spaces

Commuting traces and Lie isomorphisms on generalized matrix algebras

Let G be a generalized matrix algebra over a commutative ring R, q: G × G ! G be an R-bilinear mapping and Tq: : G ! G be a trace of q. We describe the form of Tq satisfying the condition Tq(G)G = GTq(G) for all G 2 G. The question of when Tq has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie isomorphism of G to be almost standard. As applications we characterize Lie isomorphisms of full matrix algebras, of triangular algebras and of certain uni- tal algebras with nontrivial idempotents. Some further research topics related to current work are proposed at the end of this article.

WZW models as mutual super Poisson-Lie T-dual sigma models

A WZW model on the Lie supergroup (C3+A) is constructed. It is shown that this model contains super Poisson-Lie symmetry with the dual Lie supergroup C3 + A1,1|.i. Furthermore, we show that the dual model is also equivalent to the WZW model on isomorphic Lie supergroup (C3+A).i. In this manner, because of isomorphism of the (C3+A) with a Manin supertriple, it is shown that the N=(2,2) structure is preserved under the super Poisson-Lie T-duality transformation.

Classification of complex simply connected homogeneous spaces of dimension not greater than 2

The paper is devoted to the classification of finite-dimensional complex Lie algebras of analytic vector fields on the complex plane and the corresponding actions of Lie groups on complex two-dimensional manifolds. These Lie algebras were specified by Sophus Lie. He specified vector fields which form bases of the Lie algebras. However the structure of the Lie algebras was not clarified, and isomorphic Lie algebras among listed were not established. Thus, the classification was far from complete, and the situation has not been essentially changed until now. This paper is devoted to the completion of the above mentioned classification. We consider the part of this classification which concerns transitive actions of Lie groups.

Multiplicative *-Lie isomorphisms between von Neumann algebras

Let ℳ, 𝒩 be von Neumann algebras with no central summands of type I1 or I2. If Φ : ℳ → 𝒩 is a multiplicative *-Lie isomorphism, then there is a central projection E ∈ 𝒩 such that modulo a centre-valued map which annihilates commutators, EΦ is a linear Lie *-homomorphism and (I − E)Φ is a conjugate linear Lie *-homomorphism.

Lie isomorphisms of nest algebras on Banach spaces

We show that a Lie isomorphism ψ between nest algebras on Banach spaces can be decomposed as ψ=ϕ+τ, where ϕ is an (algebraic) isomorphism or a negative of an (algebraic) anti-isomorphism, and τ is a linear map with image in the center vanishing on each commutator.

Categorified central extensions, étale Lie 2-groups and Lieʼs Third Theorem for locally exponential Lie algebras

Lieʼs Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration procedure for Lie algebra cocycles. This paper remedies the obstructions for integrating cocycles and central extensions from Lie algebras to Lie groups by generalising the integrating objects. Those objects obey the maximal coherence that one can expect. Moreover, we show that they are the universal ones for the integration problem. The main application of this result is that a Mackey-complete locally exponential Lie algebra (e.g., a Banach–Lie algebra) integrates to a Lie 2-group in the sense that there is a natural Lie functor from certain Lie 2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphic Lie algebra.