Nicolai Reshetikhin
Phase space of a characteristic Hamiltonian system is a symplectic leaf of a factorizable Poisson Lie group. Its Hamiltonian is a restriction to the symplectic leaf of a function on the group which is invariant with respect to conjugations. It is shown in this paper that such system is always integrable.
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We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the Gelfand-Tsetlin variables. We introduce the abelian algebra of collective hamiltonians defined by a chain of nested subalgebras and prove complete integrability. By construction, these models are integrable with respect to both Poisson structures. The eigenvalues of the Nijenhuis…
The complete list of Poisson-Lie structures on 4-d Galilei group is presented.
Poisson-Lie (PL) dynamical r-matrices are generalizations of dynamical r-matrices, where the base is a Poisson-Lie group. We prove analogues of basic results for these r-matrices, namely constructions of (quasi)Poisson groupoids and of Poisson homogeneous spaces. We introduce a class of PL dynamical r-matrices, associated to nondegenerate Lie bialgebras with a splitting; this is a generalization of trigonometric r-matrices with an abelian base. We prove a composition theorem for PL dynamical r-matrices, and construct quantizations of the polarized PL dynamical r-matrices. This way, we obtain q…
In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications.
Let $\g$ be a finite dimensional complex Lie algebra and $ł\subset \g$ a Lie subalgebra equipped with the structure of a factorizable quasitriangular Lie bialgebra. Consider the Lie group $\Exp ł$ with the Semenov-Tjan-Shansky Poisson bracket as a Poisson Lie manifold for the double Lie bialgebra $\Dł$. Let $\Nc_ł(0)\subset ł$ be an open domain parameterizing a neighborhood of the identity in $\Exp ł$ by the exponential map. We present dynamical $r$-matrices with values in $\g\wedge \g$ over the Poisson Lie base manifold $\Nc_ł(0)$.
We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows very singular polarizations. In particular we consider the case when the modular function is "multiplicatively integrable", i.e. when the space of leaves of the polarization inherits a groupoid structure. If suitable regularity conditions are satisfied, then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying t…
In this paper we study integrability and algebraic integrability properties of certain matrix Schrödinger operators. More specifically, we associate such an operator (with rational, trigonometric, or elliptic coefficients) to every simple Lie algebra g and every representation U of this algebra with a nonzero but finite dimensional zero weight subspace. The Calogero-Sutherland operator is a special case of this construction. Such an operator is always integrable. Our main result is that it is also algebraically integrable in the rational and trigonometric case if the representation U is highes…
We construct a Poisson isomorphism between the formal Poisson manifolds g^* and G^*, where g is a finite dimensional quasitriangular Lie bialgebra. Here g^* is equipped with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G^* with its Poisson-Lie structure. We also quantize Poisson-Lie dynamical r-matrices of Balog-Feher-Palla.
The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions(C_2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie subalgebra {\frak g} of the weight one subspace V_1 is isomorphic to the irreducible highest weight \hat{\frak g}-module L(k, 0) for a positive integer k, and V is an integrable \hat{\frak g}-module. The case in which {\frak g} is replaced by an abelian Lie subalgebra is also considered, and several consequences of integrability are discussed.
All bialgebra structures on twodimensional Galilei algebra are classified. The corresponding Lie-Poisson structures on Galilei group are found.
We provide a quasi-Poisson version of the Drinfeld's correspondence between Poisson homogeneous spaces and Lagrangian subalgebras.
The notions of transposed Hom-Poisson and Hom-pre-Lie Poisson algebras are introduced. Their bimodules and matched pairs are defined and the relevant properties and theorems are given. The notion of Manin triple of transposed Hom-Poisson algebras is introduced, and its equivalence to the transposed Hom-Poisson bialgebras is investigated. The notion of $\mathcal{O}$-operator is exploited to illustrate the relations existing between transposed Hom-Poisson and Hom-pre-Lie Poisson algebras.
All bialgebra structures for centrally extended Galilei algebra are classified. The corresponding Lie-Poisson structures on centrally extended Galilei group are found.
We show that Riemann surfaces, and separated variables immediately provide classical Poisson commuting Hamiltonians. We show that Baxter's equations for separated variables immediately provide quantum commuting Hamiltonians. The construction is simple, general, and does not rely on the Yang--Baxter equation.
We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to g-quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these g-quasi-Poisson Lie groups are nilpotent radicals of parabolic subgroups. We also provide examples of moment maps in this new context coming from moduli spaces of flat connections on surfaces.
The general expression for the bicovariant bracket for odd generators of the external algebra on a Poisson-Lie group is given. It is shown that the graded Poisson-Lie structures derived before for $GL(N)$ and $SL(N)$ are the special cases of this bracket. The formula is the universal one and can be applied to the case of any matrix Lie group.
Let $G$ be a Poisson Lie group and $\g$ its Lie bialgebra. Suppose that $\g$ is a group Lie bialgebra. This means that there is an action of a discrete group $Γ$ on $G$ deforming the Poisson structure into coboundary equivalent ones. Starting from this we construct a non-trivial stack of Hopf-Poisson algebras and prove the existence of associated deformation quantizations. This non-trivial stack is a stack of functions on the formal Poisson group, dual of the starting $Γ$ Poisson-Lie group. To quantize this non-trivial stack we use quantization of a $Γ$ Lie bialgebra which is the infinitesimal…
The notions of vertex Lie algebra and vertex Poisson algebra are presented and connections among vertex Lie algebras, vertex Poisson algebras and vertex algebras are discussed.
A Poisson-Lie group acting by the coadjoint action on the dual of its Lie algebra induces on it a non-trivial class of quadratic Poisson structures extending the linear Poisson bracket on the coadjoint orbits.
Classical Lie Bialgebras for AdS/CFT Integrability by Contraction and Reduction
本研究基于d(2,1;epsilon)与sl(2)李超代数的已有结果，通过收缩和约化步骤推导AdS/CFT可积结构的经典李双代数，得到带相应经典r矩阵的目标代数，并以在壳场表示验证结果。
Classical limits of quantum groups give rise to multiplicative Poisson structures such as Poisson-Lie and quasi-Poisson structures. We relate them to the notion of a shifted Poisson structure which gives a conceptual framework for understanding classical (dynamical) $r$-matrices, quasi-Poisson groupoids and so on. We also propose a notion of a symplectic realization of shifted Poisson structures and show that Manin pairs and Manin triples give examples of such.
We introduce a dual notion of the Poisson algebra by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. We show that the transposed Poisson algebra thus defined not only shares common properties of the Poisson algebra, including the closure under taking tensor products and the Koszul self-duality as an operad, but also admits a rich class of identities. More significantly, a transposed Poisson algebra naturally arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. Consequently, the classic constructio…
Affine Classical Lie Bialgebras for AdS/CFT Integrability
本文将 gl(2|2) 李双代数的收缩与约化构造推广至仿射代数 sl(2)^(1)×d(2,1;α)^(1)，用以刻画 AdS/CFT 与 Hubbard 模型可积性中 r-矩阵偏离差分形式及 Lorentz boost 对称性。
This work is motivated by a result of Drinfeld on Poisson homogeneous spaces. For each Poisson manifold $P$ with a Poisson action by a Poisson Lie group $G$, we describe a Lie algebroid structure on the direct sum vector bundle $P \times {\frak g} \oplus T^*P$, where ${\frak g}$ is the Lie algebra of $G$. It is built out of the transformation Lie algebroid $P \times {\frak g}$ and the cotangent bundle Lie algebroid $T^*P$ together with a pair of representations of them on each other. When the action of $G$ on $P$ is transitive, the kernel of the anchor map of this Lie algebroid gives a Lie alg…
Generalized AKS scheme of integrability via vertex algebra
本文通过定义顶点李代数的经典R-矩阵，提出构造新顶点李代数的系统方法，并利用Feigin-Frenkel中心建立无穷维可积系统的新方案。
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