(Quasi-)Poisson enveloping algebras

We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.     

Cohomological construction of quantized universal enveloping algebras

Given an associative algebra and the category of its finite dimensional modules, additional structures on the algebra induce corresponding ones on the category . Thus, the structure of a rigid quasi-tensor (braided monoidal) category on is induced by an algebra homomorphism (comultiplication), coassociative up to conjugation by (associativity constraint) and cocommutative up to conjugation by (commutativity constraint), together with an antiautomorphism (antipode) of satisfying the compatibility conditions. A morphism of quasi-tensor structures is given by an element with suitable induced actions on , and . Drinfeld defined such a structure on for any semisimple Lie algebra with the usual comultiplication and antipode but nontrivial and , and proved that the corresponding quasi-tensor category is isomomorphic to the category of representations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra (QUE), .

In the paper we give a direct cohomological construction of the which reduces to the trivial associativity constraint, without any assumption on the prior existence of a strictly coassociative QUE. Thus we get a new approach to the DJ quantization. We prove that can be chosen to satisfy some additional invariance conditions under (anti)automorphisms of , in particular, gives an isomorphism of rigid quasi-tensor categories. Moreover, we prove that for pure imaginary values of the deformation parameter, the elements , and can be chosen to be formal unitary operators on the second and third tensor powers of the regular representation of the Lie group associated to with depending only on even powers of the deformation parameter. In addition, we consider some extra properties of these elements and give their interpretation in terms of additional structures on the relevant categories.

     

On the bosonization ofL-operators for the quantum affine algebraU q (sl 2)

Some relations between different objects associated with quantum affine algebras are reviewed. It is shown that the Frenkel-Jing bosonization of a new realization of the quantum affine algebra as well as bosonization of L-operators for this algebra can be obtained from Zamolodchikov-Faddeev algebras defined by the quantum R-matrix satisfying unitarity and crossing-symmetry conditions.On leave of absence from the ITP, Kiev 252143, Ukraine. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 64–77, July, 1995.     

NON-COMMUTATIVE POISSON ALGEBRA STRUCTURES ON LIE ALGEBRA sln(fCq) WITH NULLITY M

Non-commutative Poisson algebras are the algebras having both an associa-tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson alg...     

Ideals in tensor powers of the enveloping algebra u(sl 2)

Let U = U(_sl₂)^?n be the tensor power of n copies of the enveloping algebra U(sl ₂) over an arbitrary field K of characteristic zero. In this paper we list the prime ideals of U by generators and classify them by height. If Z is the center of U and J is a prime ideal of Z, there are exactly 2₅ prime ideals I of U with I ∩ Z = J, where 0 ≤ s = s(J) ≤ n is an integer. Indeed, with respect to inclusion, they form a lattice isornorphic to the lattice of subsets of a set. When J is a maximal ideal of Z, there are only finitely many two-sided ideals of U containing J, They are presented by generators and their lattice is described, In particular, for each such J there exists a unique maximal ideal of U containing J and a unique ideal of U minimal with respect to the property that it properly contains JU. Similar results are given in the case when U is the tensor product of infinitely many copies of U(sl ₂).     

Variational construction of unbounded orbits in Lagrangian systems

We show the existence of unbounded orbits in perturbations of generic geodesic flow in T2 by a generic periodic potential. Different from previous work such as in Mather (1997), the initial values of the orbits obtained here are not required sufficiently large.     

Lagrangian Grassmann manifold Λ(2)

Based on the relationship between symplectic group Sp(2) and Λ(2), we provide an intuitive explanation (model) of the 3-dimensional Lagrangian Grassmann manifold Λ(2), the singular cycles of Λ(2), and the special Lagrangian Grassmann manifold SΛ(2). Under this model, we give a formula of the rotation paths dened by Arnold.     

On the centralizer ofK in the universal enveloping algebra of SO(n,1) and SU(n,1)

LetG _o be a non compact real semisimple Lie group with finite center, and letU U(g) ^K denote the centralizer inU U(g) of a maximal compact subgroupK _o ofG _o. To study the algebraU U(g) ^K , B. Kostant suggested to consider the projection mapP:U U(g)→U(k)⊗U(a), associated to an Iwasawa decompositionG _o=K _o A _o N _o ofG _o, adapted toK _o. WhenP is restricted toU U(g) ^K J. Lepowsky showed thatP becomes an injective anti-homomorphism ofU U(g) ^K intoU(k) ^M ⊗U(a). HereU(k) ^M denotes the centralizer ofM _o inU(k),M _o being the centralizer ofA _o inK _o. To pursue this idea further it is necessary to have a good characterization of the image ofU U(g) ^K inU(k)^M×U(a). In this paper we describe such image whenG _o=SO(n,1)_e or SU(n,1). This is acomplished by establishing a (minimal) set of equations satisfied by the elements in the image ofU U(g) ^K , and then proving that they are enough to characterize such image. These equations are derived on one hand from the intertwining relations among the principal series representations ofG _o given by the Kunze-Stein interwining operators, and on the other hand from certain imbeddings among Verma modules. This approach should prove to be useful to attack the general case. Supported in part by Fundación Antorchas     

K(a)hler流形上的Lagrange力学

讨论了K(a)hler流形上的Lagrange力学,并给出Lagrange算子、Lagrange方程、作用泛函、Hamilton原理和Hamilton方程等复的数学形式.     

Orbits of Lagrangian Subalgebras of the Double sl(2, R)

We describe the variety of Lagrangian subalgebras of the Drinfeld double for an arbitrary bialgebra structure on sl(2, R). We determine the irreducible components and the orbit structure under the natural action of the group SL(2, R)     

Lagrangian decomposability of some two-generator subgroups of PU(2,1)

We describe isometry groups of the complex hyperbolic plane generated by two loxodromic motions. We give then a condition for such a group to be decomposable as a group generated by 3 antiholomorphic involutions, and use this decomposition to describe a 3-dimensional ball in the $\mathrm{PU}(2,1)$ Teichmüller space of the once punctured torus. To cite this article: P. Will, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Space-time locality in theSp(2)-symmetric Lagrangian formalism

The existence of a local solution to the Sp(2) master equation for the gauge field theory is proved in the perturbation theory under standard regularity assumptions for the action. The arbitrariness of solutions to the Sp(2) master equation is described, provided they are proper. The effective action can be chosen to be Sp(2) invariant and (under the additional assumption that the gauge transformation generators are Lorentz tensors) Lorentz invariant. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115. No. 3, pp. 373–388, June, 1998.     

Equivalence of Lagrangian and HamiltonianSp(2)-symmetric BRST quantizations

The existence of a local solution to the quantum Sp(2) master equation and the equivalence of the Lagrangian and Hamiltonian Sp(2) quantizations are proved in the framework of perturbation theory. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 137–148, January, 1997.     

Centers of reduced enveloping algebras

We compute the inverse image of a functional in the Zassenhaus variety. We apply this computation to describe the category of representations for a regular functional. Received November 11, 1997; in final form February 9, 1998