Triangular Lie Bialgebras Research Articles

Deformations and Homotopy Theory of Relative Rota\u2013Baxter Lie Algebras

We determine the L_infty -algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying mathsf {Lie}mathsf {Rep} pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota–Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a homotopy relative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related to pre-Lie_infty -algebras.

Deformations and Their Controlling Cohomologies of $${\mathcal{O}}$$ O -Operators

$${\mathcal{O}}$$ -operators are important in broad areas of mathematics and physics, such as integrable systems, the classical Yang–Baxter equation, pre-Lie algebras and splitting of operads. In this paper, a deformation theory of $${\mathcal{O}}$$ -operators is established in consistence with the general principles of deformation theories. On the one hand, $${\mathcal{O}}$$ -operators are shown to be characterized as the Maurer–Cartan elements in a suitable graded Lie algebra. A given $${\mathcal{O}}$$ -operator gives rise to a differential graded Lie algebra whose Maurer–Cartan elements characterize deformations of the given $${\mathcal{O}}$$ -operator. On the other hand, a Lie algebra with a representation is identified from an $${\mathcal{O}}$$ -operator T such that the corresponding Chevalley–Eilenberg cohomology controls deformations of T, thus can be regarded as an analogue of the Andre–Quillen cohomology for the $${\mathcal{O}}$$ -operator. Thereafter, linear and formal deformations of $${\mathcal{O}}$$ -operators are studied. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order n deformations of an $${\mathcal{O}}$$ -operator are also characterized in terms of the new cohomology theory. Applications are given to deformations of Rota–Baxter operators of weight 0 and skew-symmetric r-matrices for the classical Yang–Baxter equation. For skew-symmetric r-matrices, there is an independent Maurer–Cartan characterization of the deformations as well as an analogue of the Andre–Quillen cohomology, which turn out to have an explicit relationship with the ones obtained as $${\mathcal{O}}$$ -operators associated to the coadjoint representations. Finally, linear deformations of skew-symmetric r-matrices and their corresponding triangular Lie bialgebras are studied.

Lie bialgebra structures on derivation Lie algebra over quantum tori

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W ⊗ W) is trivial.

Lie comodules and the constructions of Lie bialgebras

In this paper, we first give a direct sum decomposition of Lie comodules, and then according to the Lie comodule theory, construct some (triangular) Lie bialgebras through Lie coalgebras.

Invariant star products on nondegenerate triangular Lie bialgebras over formal power series rings

We obtain a bijection between the set of equivalent classes of invariant star products on a non-degenerate triangular finite dimensional Lie bialgebra over the formal power series ring and the set , working in the framework developed by Etingof–Kazhdan for the quantization of Lie bialgebras. Two of the corresponding triangular Hopf algebras over the ring are isomorphic if and only if the invariant star products defining them are equivalent. Therefore, when t = ℏ, we obtain a set of triangular Hopf quantized universal enveloping algebras which can also be seen as quantizations of the deformation algebra . Additionally, two of them are isomorphic if and only if the above invariant star products are equivalent.

Lie bialgebras of generalized Witt type

In a paper by Michaelis a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are classified. It is proved that, for any Lie algebra $W$ of generalized Witt type, all Lie bialgebras on $W$ are coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group $H^1(W,W \otimes W)$ is trivial.

Existence of triangular Lie bialgebra structures II

We characterize finite-dimensional Lie algebras over an arbitrary field of characteristic zero which admit a non-trivial (quasi-) triangular Lie bialgebra structure.

On the Fedosov deformation quantization beyond the regular Poisson manifolds

A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang–Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF's is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson brackets on two-plane.

Quantization of triangular Lie bialgebras

We derive a universal twisting element for an arbitrary triangularγ-matrix using a simple analogue of the Fedosov quantization method.

Existence of solutions of the classical yang- baxter equation for a real lie Algebra

We characterize finite-dimensional Lie algebras over the real numbers for which the classical Yang-Baxter equation has a non-trivial skew-symmetric solution (resp. a non-trivial solution with invariant symmetric part). Equivalently, we obtain a characterization of those finite-dimensional real Lie algebras which admit a non-trivial (quasi-) triangular Lie bialgebra structure.

Poisson–Lie Diffeomorphism Groups

We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of ℝ n . Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra W n of formal vector fields on ℝ n . We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson–Lie diffeomorphism groups gives rise to large classes of compatible Poisson structures on ℝ n , thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff(ℝ m ) × FDiffℝ n ) gives rise to classes of compatible Poisson structures on the space J ∞(ℝ m ,ℝ n ) of infinite jets of smooth maps ℝ m → ℝ n , which makes it also a Poisson space for this action. Poisson modules of generalized densities are also constructed. Initial steps towards a classification of these structures are taken.

Existence of triangular Lie bialgebra structures

We characterize finite-dimensional Lie algebras over an algebraically closed field of arbitrary characteristic which admit a non-trivial (quasi-) triangular Lie bialgebra structure.

Classical Yang-Baxter equation and low dimensional triangular Lie bialgebras

All solutions of the constant classical Yang-Baxter equation (CYBE) in Lie algebra L with dim L ≤ 3 are obtained and the sufficient and necessary conditions for which ( L, [ ], Δ r , r) is a coboundary (or triangular) Lie bialgebra are given. The strongly symmetric elements in L ⊗ L are found and they are all solutions of the CYBE in L with dim L ≤ 3.

Lie bialgebroids and Poisson groupoids

Lie bialgebras arise as infinitesimal invariants of Poisson Lie groups. A Lie bialgebra is a Lie algebra g with a Lie algebra structure on the dual g∗ which is compatible with the Lie algebra g in a certain sense. For a Poisson group G, the multiplicative Poisson structure π induces a Lie algebra structure on the Lie algebra dual g∗ which makes (g, g∗) into a Lie bialgebra. In fact, there is a one-one correspondence between Poisson Lie groups and Lie bialgebras if the Lie groups are assumed to be simply connected [7], [16], [19]. The importance of Poisson Lie groups themselves arises in part from their role as classical limits of quantum groups [8] and in part because they provide a class of Poisson structures for which the realization problem is tractable [15]. Poisson groupoids were introduced by Weinstein [24] as a generalization of both Poisson Lie groups and the symplectic groupoids which arise in the integration of arbitrary Poisson manifolds [4], [11]. He noted that the Lie algebroid dual A∗G of a Poisson groupoid G itself has a Lie algebroid structure, but did not develop the infinitesimal structure further. In this paper we introduce and study a natural infinitesimal invariant for Poisson groupoids, the Lie bialgebroids of the title. Our ultimate purpose is to develop a Lie theory for Poisson groupoids parallel to that for Poisson groups. In this paper we are primarily concerned with the first half of this process; that is, with the construction of the Lie bialgebroid of a Poisson groupoid. After the preliminary §2, in which we describe the generalization to arbitrary Lie algebroids of the exterior calculus and Schouten calculus, in §3 we define a Lie bialgebroid to be a Lie algebroid A whose dual A∗ is also equipped with a Lie algebroid structure, such that the coboundary operator d∗ : A −→ ∧(A) associated to A∗ satisfies a cocycle equation with respect to Γ(A), the Lie algebra of sections of A. This is clearly a straightforward extension of the concept of a Lie bialgebra [16] but cannot be formalized in Lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general Lie algebroid. Most of §3 is devoted to proving that this definition is self-dual: if (A,A∗) is a Lie bialgebroid, then (A∗, A) is also. In §4, we briefly consider the special case of Lie bialgebroids satisfying a triangularity condition, which include some important examples such as the usual triangular Lie bialgebras and Lie bialgebroids associated to Poisson manifolds. The techniques used in §§2–4 are similar to those known for Lie bialgebras. It would be possible, by suitably generalizing the proof for Poisson groups, to prove