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.

Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

The Cayley–Klein (CK) formalism is applied to the real algebra so(5) by making use of four graded contraction parameters describing, in a unified setting, 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel’d–Jimbo real Lie bialgebra for so(5) together with its Drinfel’d double structure, we obtain the corresponding CK bialgebra and the CK r-matrix coming from a Drinfel’d double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring dealing with real structures, we found that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we found 14 classical r-matrices coming from Drinfel’d doubles, obtaining new results for the de Sitter so(4,1) and anti-de Sitter so(3,2) as well as for some of their contractions. These geometric results were exhaustively applied onto the (3 + 1)D kinematical algebras, considering not only the usual (3 + 1)D spacetime but also the 6D space of lines. We established different assignations between the geometrical CK generators and the kinematical ones, which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature Λ and the speed of light c. We, finally, obtained four classes of kinematical r-matrices together with their noncommutative spacetimes and spaces of lines, comprising all κ-deformations as particular cases.

E6(6) exceptional Drinfel’d algebras

The exceptional Drinfel’d algebra (EDA) is a Leibniz algebra introduced to provide an algebraic underpinning with which to explore generalised notions of U-duality in M-theory. In essence, it provides an M-theoretic analogue of the way a Drinfel’d double encodes generalised T-dualities of strings. In this note we detail the construction of the EDA in the case where the regular U-duality group is E6(6). We show how the EDA can be realised geometrically as a generalised Leibniz parallelisation of the exceptional generalised tangent bundle for a six-dimensional group manifold G, endowed with a Nambu-Lie structure. When the EDA is of coboundary type, we show how a natural generalisation of the classical Yang-Baxter equation arises. The construction is illustrated with a selection of examples including some which embed Drinfel’d doubles and others that are not of this type.

Non-Abelian U -duality for membranes

The $T$-duality of string theory can be extended to the Poisson–Lie $T$-duality when the target space has a generalized isometry group given by a Drinfel’d double. In M-theory, $T$-duality is understood as a subgroup of $U$-duality, but the non-Abelian extension of $U$-duality is still a mystery. In this paper we study membrane theory on a curved background with a generalized isometry group given by the $\mathcal E_n$ algebra. This provides a natural setup to study non-Abelian $U$-duality because the $\mathcal E_n$ algebra has been proposed as a $U$-duality extension of the Drinfel’d double. We show that the standard treatment of Abelian $U$-duality can be extended to the non-Abelian setup. However, a famous issue in Abelian $U$-duality still exists in the non-Abelian extension.

Classification of 6D Leibniz algebras

Leibniz algebras ${\mathcal E}_n$ were introduced as an algebraic structure underlying U-duality. Algebras ${\mathcal E}_3$ derived from Bianchi 3D Lie algebras are classified here. Two types of algebras are obtained: 6D Lie algebras that can be considered an extension of the semi-Abelian 4D Drinfel’d double and unique extensions of non-Abelian Bianchi algebras. For all of the algebras explicit forms of generalized frame fields are given.

U -duality extension of Drinfel’d double

A family of algebras $\mathcal{E}_n$ that extends the Lie algebra of the Drinfel’d double is proposed. This allows us to systematically construct the generalized frame fields $E_A{}^I$ which realize the proposed algebra by means of the generalized Lie derivative, i.e., $\hat{\pounds}_{E_A}E_B{}^I =-\mathcal{F}_{AB}{}^C\,E_C{}^I$. By construction, the generalized frame fields include a twist by a Nambu–Poisson tensor. A possible application to the non-Abelian extension of $U$-duality and a generalization of the Yang–Baxter deformation are also discussed.