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1.
Flag partial differential equations naturally appear in the problem of decomposing the polynomial algebra (symmetric tensor) over an irreducible module of a Lie algebra into the direct sum of its irreducible submodules. Many important linear partial differential equations in physics and geometry are also of flag type. In this paper, we use the grading technique in algebra to develop the methods of solving such equations. In particular, we find new special functions by which we are able to explicitly give the solutions of the initial value problems of a large family of constant-coefficient linear partial differential equations in terms of their coefficients. As applications to representations of Lie algebras, we find certain explicit irreducible polynomial representations of the Lie algebras $sl(n,\mathbb {F}),\;so(n,\mathbb {F})$ and the simple Lie algebra of type G 2. 相似文献
2.
We completely solve the equivalence problem for Euler-Bernoulli equation using Lie symmetry analysis. We show that the quotient of the symmetry Lie algebra of the Bernoulli equation by the infinite-dimensional Lie algebra spanned by solution symmetries is a representation of one of the following Lie algebras: 2 A1, A1⊕ A2, 3 A1, or A3,3⊕ A1. Each quotient symmetry Lie algebra determines an equivalence class of Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity, we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters (lineal mass density and flexural rigidity). For each class, we provide a simple representative and we explicitly construct transformations that maps a class member to its representative. The maximally symmetric class described by the four-dimensional quotient symmetry Lie algebra A3,3⊕ A1 corresponds to Euler-Bernoulli equations homeomorphic to the uniform one (constant lineal mass density and flexural rigidity). We rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations reducible to the uniform unit beam. Our models extend and emphasize the symmetry flavor of Gottlieb's iso-spectral beams [H.P.W. Gottlieb, Isospectral Euler-Bernoulli beam with continuous density and rigidity functions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 413 (1987) 235-250]. 相似文献
3.
In this paper, we define a P-twisted affine Lie algebra, and construct its realizations by twisted vertex operators. 相似文献
4.
In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by “direct unions.” Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A1, B, C, and BC. 相似文献
5.
The invariance of non-linear partial differential equations under a certain infinite-dimensional Lie algebra AN( z) in N spatial dimensions is studied. The special case A1(2) was introduced in [J. Stat. Phys. 75 (1994) 1023] and contains the Schrödinger Lie algebra sch1 as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of AN( z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of AN( z) such that the well-known Monge-Ampère equation is the required additional condition. New exact solutions are found for some representatives of this class. 相似文献
6.
Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module
that are nullified by positive root vectors. In this article, we use partial differential equations to explicitly find all
the singular vectors of the polynomial representation of the simple Lie algebra of type F
4 over its 26-dimensional basic irreducible module, which also supplements a proof of the completeness of Brion’s abstractly
described generators. Moreover, we show that the number of irreducible submodules contained in the space of homogeneous harmonic
polynomials with degree k ⩾ 2 is greater than or equal to 〚 k/3〛 + 〚( k − 2)/3〛 + 2. 相似文献
7.
In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E 0(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric. 相似文献
8.
This article is devoted to an extensive study of an infinite-dimensional Lie algebra
, introduced in [14] in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra
of invariance of the free Schr?dinger equation and the central charge-free Virasoro algebra Vect( S1). We call
the Schr?dinger-Virasoro Lie algebra. We study its representation theory: realizations as Lie symmetries of field equations, coadjoint representation, coinduced
representations in connection with Cartan’s prolongation method (yielding analogues of the tensor density modules for Vect( S1)). We also present a detailed cohomological study, providing in particular a classification of deformations and central extensions;
there appears a non-local cocycle.
Communicated by Petr Kulish
Daniel Arnaudon, in memoriam
Submitted: January 17, 2006; Accepted: March 21, 2006 相似文献
9.
A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g0, where g0 is some element of the grading group G such that g0 = 0 or 4 g0≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras. 相似文献
11.
We construct a Lie algebra G by using a semi-direct sum of Lie algebra G1 with Lie algebra G2. A direct application to the TD hierarchy leads to a novel hierarchy of integrable couplings of the TD hierarchy. Furthermore, the generalized variational identity is applied to Lie algebra G to obtain quasi-Hamiltonian structures of the associated integrable couplings. 相似文献
12.
We obtain some necessary and sufficient conditions for the non- x-autonomy of the basic Lie algebra of a system of first-order linear differential equations with constant coefficients over
the field of complex numbers. Using these results, we establish the x-autonomy of the basic Lie algebra of some systems of equations of mathematical physics. 相似文献
13.
We study the Lie structure of graded associative algebras. Essentially, we analyze the relation between Lie and associative graded ideals, and between Lie and associative graded derivations. Gathering together results on both directions, we compute maximal graded algebras of quotients of graded Lie algebras that arise from associative algebras. We also show that the Lie algebra Der gr ( A) of graded derivations of a graded semiprime associative algebra is strongly non-degenerate (modulo a certain ideal containing the center of Der gr ( A)). 相似文献
15.
The present paper is devoted to the classification of infinite-dimensional naturally graded Lie algebras that are narrow in the sense of Zelmanov and Shalev [9]. Such Lie algebras are Lie algebras of slow linear growth. In the theory of nonlinear hyperbolic partial differential equations the notion of the characteristic Lie algebra of equation is introduced [3]. Two graded Lie algebras n1 and n2 from our list, that are positive parts of the affine Kac–Moody algebras A1(1) and A2(2), respectively, are isomophic to the characteristic Lie algebras of the sinh-Gordon and Tzitzeika equations [6]. We also note that questions relating to narrow and slowly growing Lie algebras have been extensively studied in the case of a field of positive characteristic [2]. 相似文献
16.
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 ZhuH 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 ZhuH 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.
“I am an old man, and I know that a definition cannot be so complicated.” I.M. Gelfand (after a talk on vertex algebras in
his Rutgers seminar) 相似文献
18.
We prove that the Lie algebra of the McCool group M3 is torsion free. As a result, we are able to give a presentation for the Lie algebra of M3. Furthermore, M3 is a Magnus group. 相似文献
19.
For a restricted Lie algebra L over a field of characteristic p > 0 we study the Lie nilpotency index t L ( u( L)) of its restricted universal enveloping algebra u( L). In particular, we determine an upper and a lower bound for t L ( u( L)). Finally, under the assumption that L is p-nilpotent and finite-dimensional, we establish when the Lie nilpotency index of u( L) is maximal. Communicated by I. Shestakov. 相似文献
20.
We show that the structure of the Lie symmetry algebra of a system of n linear second-order ordinary differential equations with constant coefficients depends on at most n-1 parameters. The tools used are Jordan canonical forms and appropriate scaling transformations. We put our approach to test by presenting a simple proof of the fact that the dimension of the symmetry Lie algebra of a system of two linear second-order ordinary differential with constant coefficients is either 7, 8 or 15. Also, we establish for the first time that the dimension of the symmetry Lie algebra of a system of three linear second-order ordinary differential equations with constant coefficients is 10, 12, 13 or 24. 相似文献
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