共查询到20条相似文献,搜索用时 78 毫秒
1.
Hong Wei Yang Bao Shu Yin Yong Fang 《International Journal of Theoretical Physics》2011,50(3):671-681
A class of new Lie algebra B
3 is constructed, which is far different from the known Lie algebra A
n−1. Based on the corresponding loop algebra [(B3)\tilde]\tilde{B_{3}}, the generalized mKdV hierarchy is established. In order to look for the Hamiltonian structure of such integrable system,
a generalized trace functional of matrices is introduced, whose special case is just the well-known trace identity. Finally,
its expanding integrable model is worked out by use of an enlarged Lie algebra. 相似文献
2.
The generalized Bessel functions (GBF) are framed within the context of the representation Q(ω,m
0) of the three-dimensional Lie algebra
. The analysis has been carried out by generalizing the formalism relevant to Bessel functions. New generating relations and identities involving various forms of GBF are obtained. Certain known results are also mentioned as special cases.Mathematics Subject Classifications (2000) 33C10, 33C80, 33E20. 相似文献
3.
We classify simple linearly compact n-Lie superalgebras with n > 2 over a field ${\mathbb{F}}We classify simple linearly compact n-Lie superalgebras with n > 2 over a field
\mathbbF{\mathbb{F}} of characteristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive
\mathbbZ{\mathbb{Z}}-graded Lie superalgebras of the form L=?j=-1n-1 Lj{L=\oplus_{j=-1}^{n-1} L_j}, where dim L
n−1 = 1, L
−1 and L
n−1 generate L, and [L
j
, L
n−j−1] = 0 for all j, thereby reducing it to the known classification of simple linearly compact Lie superalgebras and their
\mathbbZ{\mathbb{Z}}-gradings. The list consists of four examples, one of them being the n + 1-dimensional vector product n-Lie algebra, and the remaining three infinite-dimensional n-Lie algebras. 相似文献
4.
Anthony?M.?Bloch Vasile?Br?nz?nescu Arieh?Iserles Jerrold?E.?Marsden Tudor?S.?Ratiu 《Communications in Mathematical Physics》2009,290(2):399-435
For a given skew symmetric real n × n matrix N, the bracket [X, Y]
N
= XNY − YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry,
integrability, and linearizability of the Hamiltonian system , or equivalently in Lax form, the equation on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure
of Sym(n, N). This is established by finding another compatible Poisson structure.
If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [·, ·]) is Lie algebra isomorphic with the symplectic Lie algebra associated to the symplectic form on given by N
−1. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying
symplectic group Sp(n, N
−1). Second, the trace of the product of matrices defines a non-invariant non-degenerate inner product on Sym(n, N) which identifies it with its dual. Therefore Sym(n, N) carries a natural Lie-Poisson structure as well as a compatible “frozen bracket” structure. The Poisson diffeomorphism from
Sym(n, N) to maps our system to a Mischenko-Fomenko system, thereby providing another proof of its integrability if N is invertible with distinct eigenvalues. Third, there is a second ad-invariant inner product on Sym(n, N); using it to identify Sym(n, N) with itself and composing it with the dual of the Lie algebra isomorphism with , our system becomes a Mischenko- Fomenko system directly on Sym(n, N).
If N is invertible and has distinct eigenvalues, it is shown that this geodesic flow on Sym(n, N) is linearized on the Prym subvariety of the Jacobian of the spectral curve associated to a Lax pair formulation with parameter
of the system. If, on the other hand, N has nullity one and distinct eigenvalues, in spite of the fact that the system is completely integrable, it is shown that
the flow does not linearize on the Jacobian of the spectral curve.
Research partially supported by NSF grants CMS-0408542 and DMS-0604307.
Research partially supported by the Swiss SCOPES grant IB7320-110721/1, 2005-2008, and MEdC Contract 2-CEx 06-11-22/25.07.2006.
Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932.
Research partially supported by the Swiss NSF and the Swiss SCOPES grant IB7320-110721/1. 相似文献
5.
André Lichnerowicz 《Letters in Mathematical Physics》1979,3(6):495-502
The twisted products play an important role in Quantum Mechanics [1, 2]. We introduce here a distinction between Vey *ν-products and strong Vey *ν-products and prove that each *ν-product is equivalent to a Vey *ν-product. If b
3(W)=0, the symplectic manifold (W, F) admits strong Vey *ν-products. If b
2(W)=0, all *ν-products are equivalent as well as the Vey Lie algebras. In the general case, we characterize the formal Lie algebras which
are generated by a *ν-product and we prove that the existence of a *ν-product is equivalent to the existence of a formal Lie algebra infinitesimally equivalent to a Vey Lie algebra at the first
order. 相似文献
6.
Christopher L. Rogers 《Letters in Mathematical Physics》2012,100(1):29-50
A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n + 1. In previous work with Baez and Hoffnung, we described how the ‘higher analogs’ of the algebraic and geometric structures
found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, just as a symplectic manifold gives
a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L
∞-algebras: graded vector spaces equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi
identity. Here, we generalize our previous result. Given an n-plectic manifold, we explicitly construct a corresponding Lie n-algebra on a complex consisting of differential forms whose multi-brackets are specified by the n-plectic structure. We also show that any n-plectic manifold gives rise to another kind of algebraic structure known as a differential graded Leibniz algebra. We conclude
by describing the similarities between these two structures within the context of an open problem in the theory of strongly
homotopy algebras. We also mention a possible connection with the work of Barnich, Fulp, Lada, and Stasheff on the Gelfand–Dickey–Dorfman
formalism. 相似文献
7.
X. M. Song Z. G. Huang H. Q. Lu Q. Q. Sun 《International Journal of Theoretical Physics》2008,47(4):865-872
In dilatonic cosmology model, we study the behavior of attractor solution in ω–ω′ plane, which is defined by the equation of state parameter for the dark energy and its derivative with respect to N (the logarithm of the scale factor a). This is a good method which is useful to the study of classifying the dynamical dark energy models including “freezing”
and “thawing” model. We find that our model belongs to “freezing” type model classified in ω–ω′ plane. We show mathematically the property of attractor solutions which correspond to ω
σ
=−1, Ω
σ
=1. The present values of energy density parameter
,
and
are 0.715001, 0.284972 and 0.00002706 respectively, which meet the current observations well. Finally, we can obtain that
the coupling between dilaton and matter affects the evolutive process of the Universe, but not the fate of the Universe. 相似文献
8.
J. Swoboda 《Optics and Spectroscopy》2007,103(3):343-348
Given a compact, connected Lie group G with Lie algebra
. We discuss time-optimal control of bilinear systems of the form
where H
d
, H
j
∈
, U ∈ G, and the v
j
act as control variables. The case G = SU(2
n
) has found interesting applications to questions of time-optimal control of spin systems. In this context Eq. (I) describes
the dynamics of an n-particle system with fixed drift Hamiltonian H
d
, which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters v
j
. The question of interest here is to transfer the system from a given initial state U
0 to a prescribed final state U
1 in least possible time. Denote by
the Lie algebra spanned by H
1, ..., H
m
, and by K the corresponding Lie subgroup of G. After reformulating the optimal control problem for system (I) in terms of an equivalent problem on the homogeneous space
G/K we discuss in detail time-optimal control strategies for system (I) in the case where G/K carries the structure of a Riemannian symmetric space.
The text submitted by the author in English. 相似文献
((I)) |
9.
10.
Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang–Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Koszul of global dimension 3 and Gorenstein but except for s=1 (i.e. in the two-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin–Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincaré series of this algebra
and for the dimension in degree n of the graded Lie algebra of which
is the universal enveloping algebra. In the four-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one. 相似文献
11.
The problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants. This provides
a powerful tool of analysis and it opens the road to new applications of Automorphic Lie Algebras, beyond the context of integrable
systems. In this paper it is shown that
\mathfraksl2(\mathbbC){\mathfrak{sl}_{2}(\mathbb{C})}–based Automorphic Lie Algebras associated to the icosahedral group
\mathbb I{{\mathbb I}}, the octahedral group
\mathbb O{{\mathbb O}}, the tetrahedral group
\mathbb T{{\mathbb T}}, and the dihedral group
\mathbb Dn{{\mathbb D}_n} are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition
and transvectants, Molien functions and the trace-form. This result provides a complete classification of
\mathfraksl2(\mathbbC){\mathfrak{sl}_{2}(\mathbb{C})}–based Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it
is a crucial step towards the complete classification of Automorphic Lie Algebras. 相似文献
12.
Hatem Najar 《Journal of statistical physics》2008,130(4):713-725
In this paper we continue with the investigation of the behavior of the integrated density of states of random operators of
the form H
ω
=−∇
ρ
ω
∇. In the present work we are interested in its behavior at 0, the bottom of the spectrum of H
ω
. We prove that it converges exponentially fast to the integrated density of states of some periodic operator
. Being periodic,
cannot exhibit a Lifshitz behaviour. This result relates to the result of S.M. Kozlov (Russ. Math. Surv. 34(4):168–169, 1979) and improves it.
Research partially supported by the Research Unity 01/UR/ 15-01 projects. 相似文献
13.
T. Kojima 《Physics of Atomic Nuclei》2010,73(2):367-374
We study a free field realization of the elliptic quantum algebra Uq,p($
\widehat{sl_3 }
$
\widehat{sl_3 }
) for arbitrary level k. We give the free field realization of elliptic analog of Drinfeld current associated with Uq,p($
\widehat{sl_3 }
$
\widehat{sl_3 }
) for arbitrary level k. In the limit p → 0, q → 1 our realization reproduces Wakimoto realization for the affine Lie algebra
$
\widehat{sl_3 }
$
\widehat{sl_3 }
. 相似文献
14.
A new method is proposed for setting a lower or upper limit a α
n
* on the neutron electric polarizability α
an
. It is based on the fact that the real part of the s-wave scattering amplitude changes sign near the s-wave neutron resonance at E=E*. The methods consist of the observation of the energy behavior of the forward-backward scattering asymmetry ω
1 which experiences a jump at E=E*. If the jump is such that dω
1/dE>0, then α
n
>α
n
*, while if dω
1/dE<0, then α
n
<α
n
*, and if dω
1/dE∼0, then α
n
∼α
n
*. Seven even-even nuclei are found with α
n
* from 0.5 to 3.1 in 10−3 fm3. Some details of a possible experiment with 182W are described.
Pis’ma Zh. éksp. Teor. Fiz. 70, No. 3, 171–174 (10 August 1999)
Published in English in the original Russian journal. Edited by Steve Torstveit. 相似文献
15.
Oleg I. Bogoyavlenskij 《Communications in Mathematical Physics》2008,277(2):369-384
Form-invariant solutions for the Poisson brackets of hydrodynamic type on a manifold M
n
with (2,0)-tensor g
ij
(u) of rank m ≤ n are derived. Tensor invariants of the Poisson brackets are introduced that include a vector field V (or dynamical system V) on M
n
, the Lie derivative L
V
g
ij
and symmetric (k, 0)-tensors . Several scalar invariants of the Poisson brackets are defined. A nilpotent Lie algebra structure is disclosed in the space
of 1-forms that annihilate the (2,0)-tensor g
ij
(u). Applications to the one-dimensional gas dynamics are presented. 相似文献
16.
We study the Leibniz homology of the Poisson algebra of polynomial functions over (2n
,) where is the standard symplectic structure. We identify it with certain highest-weight vectors of some
2n
(
)-modules and obtain some explicit result in low degree. 相似文献
17.
The length of instability intervals is investigated for the Hill equation y′′+ω(ω− 2∈p(x)y = 0, where p(x) has an infinite Fourier series of coefficients c
n. For any small ∈ it is shown that these lengths are completely characterized by the c
n's. 相似文献
18.
We classify all the quasifinite highest-weight modules over the central extension of the Lie algebra of matrix quantum pseudo-differential operators, and obtain them in terms of representation theory of the Lie algebra
(, R
m
) of infinite matrices with only finitely many nonzero diagonals over the algebra R
m
=
[t]/(t
m+1). We also classify the unitary ones. 相似文献
19.
T. Palev 《Czechoslovak Journal of Physics》1979,29(1):91-98
The Lie superalgebraical properties of the ordinary quantum statistics are discussed. It is indicated that the algebra generated byn pairs of Fermi operator is isomorphic to the classical simple Lie algebraB
n
, whereasn pairs of Bose operators generate the simple Lie superalgebraB(0,n). An idea of how one can introduce new classes of creation and annihilation operators that satisfy the second quantization postulates and generate other simple Lie superalgebras is given. The statistics corresponding to the Lie algebraA
n
is considered in more details.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978. 相似文献
20.
Let G be a reductive Lie group, g its Lie algebra, and M a G-manifold. Suppose 𝔸
h
(M) is a 𝕌
h
(g)-equivariant quantization of the function algebra 𝔸(M) on M. We develop a method of building 𝕌
h
(g)-equivariant quantization on G-orbits in M as quotients of 𝔸
h
(M). We are concerned with those quantizations that may be simultaneously represented as subalgebras in 𝕌*
h
(g) and quotients of 𝔸
h
(M). It turns out that they are in one-to-one correspondence with characters of the algebra 𝔸
h
(M). We specialize our approach to the situation g=gl(n,ℂ), M=End(ℂ
n
), and 𝔸
h
(M) the so-called reflection equation algebra associated with the representation of 𝕌
h
(g) on ℂ
n
. For this particular case, we present in an explicit form all possible quantizations of this type; they cover symmetric and
bisymmetric orbits. We build a two-parameter deformation family and obtain, as a limit case, the 𝕌(g)-equivariant quantization of the Kirillov-Kostant-Souriau bracket on symmetric orbits.
Received: 28 April 2002 / Accepted: 3 October 2002 Published online: 24 January 2003
RID="*"
ID="*" This research is partially supported by the Israel Academy of Sciences grant no. 8007/99-01.
Communicated by L. Takhtajan 相似文献