Terms Of Lie Algebra Research Articles

Some properties of c-covers of a pair of Lie algebras

In [H. Safa and H. Arabyani, On c-nilpotent multiplier and c-covers of a pair of Lie algebras, Commun. Algebra 45(10) (2017), 4429–4434], we characterized the structure of the c-nilpotent multiplier of a pair of Lie algebras in terms of its c-covering pairs and discussed some results on the existence of c-covers of a pair of Lie algebras. In the present paper, it is shown under some conditions that a relative c-central extension of a pair of Lie algebras is a homomorphic image of a c-covering pair. Moreover, we prove that a c-cover of a pair of finite dimensional Lie algebras, under some assumptions, has a unique domain up to isomorphism and also that every perfect pair of Lie algebras admits at least one c-cover. Finally, we discuss a result concerning the isoclinism of c-covering pairs.

Representations of quantized coordinate algebras via PBW-type elements

Inspired by the work of Kuniba-Okado-Yamada, we study some tensor product representations of quantized coordinate algebras of symmetrizable Kac-Moody Lie algebras in terms of quantized enveloping algebras. As a consequence, we describe structures and properties of certain reducible representations of quantized coordinate algebras. This paper includes alternative proofs of Soibelman's tensor product theorem and Kuniba-Okado-Yamada's common structure theorem based on our direct calculation method using global bases.

Degeneration of Bethe subalgebras in the Yangian of $$\mathfrak {gl}_n$$ gl n

We study degenerations of Bethe subalgebras $B(C)$ in the Yangian $Y(\mathfrak{gl}_n)$, where $C$ is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parametrizes all possible degenerations, is the Deligne-Mumford moduli space of stable rational curves $\overline{M_{0,n+2}}$. All subalgebras corresponding to the points of $\overline{M_{0,n+2}}$ are free and maximal commutative. We describe explicitly the "simplest" degenerations and show that every degeneration is the composition of the simplest ones. The Deligne-Mumford space $\overline{M_{0,n+2}}$ generalizes to other root systems as some De Concini-Procesi resolution of some toric variety. We state a conjecture generalizing our results to Bethe subalgebras in the Yangian of arbitrary simple Lie algebra in terms of this De Concini-Procesi resolution.

Filtrations on Springer fiber cohomology and Kostka polynomials

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules.

Maurer–Cartan Elements in the Lie Models of Finite Simplicial Complexes

AbstractIn a previous work, we associated a complete diòerential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete diòerential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

Groups of automorphisms of local fields of period p and nilpotent class < p, II

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be the maximal quotient of period [Formula: see text] and nilpotent class [Formula: see text] of the Galois group of a maximal [Formula: see text]-extension of [Formula: see text]. We describe the ramification filtration [Formula: see text] and relate it to an explicit form of the Demushkin relation for [Formula: see text]. The results are given in terms of Lie algebras attached to the appropriate [Formula: see text]-groups by the classical equivalence of the categories of [Formula: see text]-groups and Lie algebras of nilpotent class [Formula: see text].

Nonlinear oblique projections

We construct nonlinear oblique projections along subalgebras of nilpotent Lie algebras in terms of the Baker–Campbell–Hausdorff multiplication. We prove that these nonlinear projections are real analytic on every Schubert cell of the Grassmann manifold whose points are the subalgebras of the nilpotent Lie algebra under consideration.

Universal formula for the Hilbert series of minimal nilpotent orbits

We show that the Hilbert series of the projective variety $X=P(O_{min}),$ corresponding to the minimal nilpotent orbit $\mathcal O_{min},$ is universal in the sense of Vogel: it is written uniformly for all simple Lie algebras in terms of Vogel's parameters $\alpha,\beta,\gamma$ and represents a special case of the generalized hypergeometric function ${}_{4}F_{3}.$ A universal formula for the degree of $X$ is then deduced.

Test elements of direct sums and free products of free Lie algebras

We give a characterization of test elements of a direct sum of free Lie algebras in terms of test elements of the factors. In addition, we construct certain types of test elements and we prove that in a free product of free Lie algebras, product of the homogeneous test elements of the factors is also a test element.

Matrix Representation for Seven-Dimensional Nilpotent Lie Algebras

This paper is concerned with finding linear representations for seven-dimensional real, indecomposable nilpotent Lie algebras. We consider the first 39 algebras presented in Gong’s classification which was based on the upper central series dimensions. For each algebra, we give a corresponding matrix Lie group, a representation of the Lie algebra in terms of left-invariant vector field and left-invariant one forms.

On certain generalizations of the Schrödinger-Virasoro algebra

We study certain generalizations of the Schrödinger-Virasoro algebra, introduced by Roger and Unterberger, in the context of vertex algebras. As the main result, we associate these Lie algebras with vertex algebras and their twisted modules with respect to semi-simple (not necessarily finite order) automorphisms, and we give a connection among these Lie algebras in terms of twisted current algebras of conformal Lie algebras.

Lie algebraic approach to the time-dependent quantum general harmonic oscillator and the bi-dimensional charged particle in time-dependent electromagnetic fields

We discuss the one-dimensional, time-dependent general quadratic Hamiltonian and the bi-dimensional charged particle in time-dependent electromagnetic fields through the Lie algebraic approach. Such method consists in finding a set of generators that form a closed Lie algebra in terms of which it is possible to express a quantum Hamiltonian and therefore the evolution operator. The evolution operator is then the starting point to obtain the propagator as well as the explicit form of the Heisenberg picture position and momentum operators. First, the set of generators forming a closed Lie algebra is identified for the general quadratic Hamiltonian. This algebra is later extended to study the Hamiltonian of a charged particle in electromagnetic fields exploiting the similarities between the terms of these two Hamiltonians. These results are applied to the solution of five different examples: the linear potential which is used to introduce the Lie algebraic method, a radio frequency ion trap, a Kanai–Caldirola-like forced harmonic oscillator, a charged particle in a time dependent magnetic field, and a charged particle in constant magnetic field and oscillating electric field. In particular we present exact analytical expressions that are fitting for the study of a rotating quadrupole field ion trap and magneto-transport in two-dimensional semiconductor heterostructures illuminated by microwave radiation. In these examples we show that this powerful method is suitable to treat quadratic Hamiltonians with time dependent coefficients quite efficiently yielding closed analytical expressions for the propagator and the Heisenberg picture position and momentum operators.

Reduced-order dynamic observer error linearisation: a natural extension of observer error linearisation

This paper introduces the concept of reduced-order dynamic observer error linearisation (RDOEL) for multi-output systems, which is the problem of transforming a nonlinear system into a nonlinear observer canonical form with the aid of auxiliary dynamics. The proposed RDOEL framework is not only a modified version of dynamic observer error linearisation (DOEL) but also a natural extension of observer error linearisation (OEL). We provide three necessary conditions for the RDOEL problem, and then derive a necessary and sufficient condition described in terms of Lie algebras of vector fields. Furthermore, from the result, we also give a geometric necessary and sufficient condition for the OEL problem, which has not yet been completely established in the case where a diffeomorphism on the output of general form is considered.

Self-similar associative algebras

Famous self-similar groups were constructed by Grigorchuk, Gupta and Sidki, these examples lead to interesting examples of associative algebras. The authors suggested examples of self-similar Lie algebras in terms of differential operators. Recently Sidki introduced an example of an associative algebra of self-similar matrices.We construct families of self-similar associative algebras CΩ, DΩ, generalizing the example of Sidki. We prove that our algebras are Z⊕Z-graded and have polynomial growth. Our approach is the weight strategy developed by the authors for self-similar Lie algebras and their envelopes. In particular, we obtain similar triangular decompositions into direct sums of three subalgebras C=C+⊕C0⊕C−, D=D+⊕D0⊕D−. We prove that some of our algebras are direct sums of two locally nilpotent subalgebras C=C+⊕C−, D=D+⊕D−. We show that in some cases the zero components C0, D0 are nontrivial and not nil algebras.We show that our construction includes the example of Sidki and the examples of self-similar Lie algebras and their associative hulls constructed by the authors before.

COTRIPLE HOMOLOGY OF CROSSED 2-CUBES OF LIE ALGEBRAS

The cotriple homology of crossed 2-cubes of Lie algebras is constructed and investigated. Namely, we calculate the cotriple homology of an inclusion crossed 2-cube of Lie algebras in terms of the bi-relative Chevalley–Eilenberg homologies. We also define in a natural way the Chevalley–Eilenberg homology of crossed 2-cubes of Lie algebras and study the relationship between cotriple and Chevalley–Eilenberg homologies for any crossed 2-cube of Lie algebras. We show that low-dimensional cyclic homologies of associative algebras are calculated in terms of the cotriple homology of crossed 2-cubes of Lie algebras.

YOUNG TABLEAUX, CANONICAL BASES, AND THE GINDIKIN-KARPELEVICH FORMULA

A combinatorial description of the crystal B(infinity) for finite-dimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig's parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young tableaux.

Representations of some quantum tori Lie subalgebras

In this paper, we define the q-analog Virasoro-like Lie subalgebras in x∞ = a∞(b∞, c∞, d∞). The embedding formulas into x∞ are introduced. Irreducible highest weight representations of \documentclass[12pt]{minimal}\begin{document}$\tilde{\mathcal {A}}_{q}$\end{document}Ãq, \documentclass[12pt]{minimal}\begin{document}$\tilde{\mathcal {B}}_{q}$\end{document}B̃q, and \documentclass[12pt]{minimal}\begin{document}$\tilde{\mathcal {C}}_{q}$\end{document}C̃q-series of the q-analog Virasoro-like Lie algebras in terms of vertex operators are constructed. We also construct the polynomial representations of the \documentclass[12pt]{minimal}\begin{document}$\tilde{\mathcal {A}}_{q}$\end{document}Ãq, \documentclass[12pt]{minimal}\begin{document}$\tilde{\mathcal {B}}_{q}$\end{document}B̃q, \documentclass[12pt]{minimal}\begin{document}$\tilde{\mathcal {C}}_{q}$\end{document}C̃q, and \documentclass[12pt]{minimal}\begin{document}$\tilde{\mathcal {D}}_{q}$\end{document}D̃q-series of the q-analog Virasoro-like Lie algebras.

Split abelian chief factors and first degree cohomology for Lie algebras

In this paper we investigate the relation between the multiplicities of split abelian chief factors of finite-dimensional Lie algebras and first degree cohomology. In particular, we obtain a characterization of modular solvable Lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors. The analogues of these results are well known in the modular representation theory of finite groups. An important tool in the proof of these results is a refinement of a non-vanishing theorem of Seligman for the first degree cohomology of non-solvable finite-dimensional Lie algebras in prime characteristic. As an application we derive several results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module.

Constructing (n + 1)-Lie algebras from n-Lie algebras

n-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. In this paper, we construct (n + 1)-Lie algebras by two-dimensional extensions of metric n-Lie algebras, and construct (n + 1)-Lie algebras in terms of n-Lie algebras and certain linear functions. Along this approach, we obtain (n + 1)-Lie algebras by n-Lie algebras with the generating index n, and 4-Lie algebras by non-nilpotent 3-Lie algebras with the generating index 4.

Indirect Controllability of Quantum Systems; A Study of Two Interacting Quantum Bits

A quantum mechanical system S is indirectly controlled when the control affects an ancillary system A and the evolution of S is modified through the interaction with A only. A study of indirect controllability gives a description of the set of states that can be obtained for S with this scheme. In this paper, we study the indirect controllability of quantum systems in the finite dimensional case. After discussing the relevant definitions, we give a general necessary condition for controllability in Lie algebraic terms. We present a detailed treatment of the case where both systems, S and A, are two-dimensional (qubits). In particular, we characterize the dynamical Lie algebra associated with S + A, extending previous results, and prove that complete controllability of S + A and an appropriate notion of indirect controllability are equivalent properties for this system. We also prove several further indirect controllability properties for the system of two qubits, and illustrate the role of the Lie algebraic analysis in the study of reachable states.