LINEARIZATION OF POISSON–LIE STRUCTURES ON THE 2D EUCLIDEAN AND (1 + 1) POINCARÉ GROUPS

The paper deals with linearization problem of Poisson-Lie structures on the \((1+1)\) Poincaré and \(2D\) Euclidean groups. We construct the explicit form of linearizing coordinates of all these Poisson-Lie structures. For this, we calculate all Poisson-Lie structures on these two groups mentioned above, through the correspondence with Lie Bialgebra structures on their Lie algebras which we first determine.

Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation

This paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang–Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang–Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by some concrete examples.

Jacobi-Lie T-plurality

We propose a Leibniz algebra, to be called DD^++, which is a generalization of the Drinfel’d double. We find that there is a one-to-one correspondence between a DD^++ and a Jacobi–Lie bialgebra, extending the known correspondence between a Lie bialgebra and a Drinfel’d double. We then construct generalized frame fields E_A{}^M\in\text{O}(D,D)\times\mathbb{R}^+EAM∈O(D,D)×ℝ+ satisfying the algebra \hat{\pounds}_{E_A}E_B = - X_{AB}{}^C\,E_C£̂EAEB=−XABCEC, where X_{AB}{}^CXABC are the structure constants of the DD^++ and \hat{\pounds}£̂ is the generalized Lie derivative in double field theory. Using the generalized frame fields, we propose the Jacobi–Lie TT-plurality and show that it is a symmetry of double field theory. We present several examples of the Jacobi–Lie TT-plurality with or without Ramond–Ramond fields and the spectator fields.

Lie super-bialgebra structures on the N = 2 super-BMS3 algebra

In this paper, Lie super-bialgebra structures on the [Formula: see text] super-BMS3 algebra are investigated. By computing derivations from the super-BMS3 to the tensor product of its adjoint module, we obtain that all Lie bialgebra structures on the [Formula: see text] super-BMS3 Algebra are triangular coboundary.

Double Constructions of Compatible Associative Algebras

A compatible associative algebra is a pair of associative algebras satisfying that any linear combination of the two associative products is still an associative product. We construct a compatible associative algebra with a decomposition into the direct sum of the underlying vector space of another compatible associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant. This compatible associative algebra is equivalent to a certain bialgebra structure of compatible associative algebras, which is an analogue of a Lie bialgebra. Many properties of the bialgebra are presented. In particular, the coboundary bialgebra theory leads to the system of associative Yang–Baxter equations in compatible associative algebras, which is an analogue of the classical Yang–Baxter equation in a Lie algebra. Furthermore, the bialgebra can also be regarded as a “compatible version” of antisymmetric infinitesimal bialgebras, that is, a pair of antisymmetric infinitesimal bialgebras satisfying any linear combination of them is still an antisymmetric infinitesimal bialgebra.

Some remarks on invariant Poisson quasi-Nijenhuis structures on Lie groups

We study right-invariant (respectively, left-invariant) Poisson quasi-Nijenhuis structures on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-qn structures on the corresponding Lie algebra [Formula: see text]. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all [Formula: see text]-[Formula: see text] structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between [Formula: see text]-[Formula: see text] structures and the generalized complex structures on the Lie algebras [Formula: see text] and also the solutions of modified Yang–Baxter equation (MYBE) on the double of Lie bialgebra [Formula: see text]. The results are applied to some relevant examples.