Representations of Bihom-Lie Algebras

A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.

Revisiting the conformally soft sector with celestial diamonds

Celestial diamonds encode the structure of global conformal multiplets in 2D celestial CFT and offer a natural language for describing the conformally soft sector. The operators appearing at their left and right corners give rise to conformally soft factorization theorems, the bottom corners correspond to conserved charges, and the top corners to conformal dressings. We show that conformally soft charges can be expressed in terms of light ray integrals that select modes of the appropriate conformal weights. They reside at the bottom corners of memory diamonds, and ascend to generalized currents. We then identify the top corners of the associated Goldstone diamonds with conformal Faddeev-Kulish dressings and compute the sub-leading conformally soft dressings in gauge theory and gravity which are important for finding nontrivial central extensions. Finally, we combine these ingredients to speculate on 2D effective descriptions for the conformally soft sector of celestial CFT.

Cohomologies of n-Lie Algebras with Derivations

The goal of this paper is to study cohomological theory of n-Lie algebras with derivations. We define the representation of an n-LieDer pair and consider its cohomology. Likewise, we verify that a cohomology of an n-LieDer pair could be derived from the cohomology of associated LeibDer pair. Furthermore, we discuss the (n−1)-order deformations and the Nijenhuis operator of n-LieDer pairs. The central extensions of n-LieDer pairs are also investigated in terms of the first cohomology groups with coefficients in the trivial representation.

On deformations and extensions of Diff(S2)

We investigate the algebra of vector fields on the sphere. First, we find that linear deformations of this algebra are obstructed under reasonable conditions. In particular, we show that hs[λ], the one-parameter deformation of the algebra of area-preserving vector fields, does not extend to the entire algebra. Next, we study some non-central extensions through the embedding of $$ \mathfrak{vect} $$ vect (S2) into $$ \mathfrak{vect} $$ vect (ℂ*). For the latter, we discuss a three parameter family of non-central extensions which contains the symmetry algebra of asymptotically flat and asymptotically Friedmann spacetimes at future null infinity, admitting a simple free field realization.

3d gravity in Bondi-Weyl gauge: charges, corners, and integrability

We introduce a new gauge and solution space for three-dimensional gravity. As its name Bondi-Weyl suggests, it leads to non-trivial Weyl charges, and uses Bondi-like coordinates to allow for an arbitrary cosmological constant and therefore spacetimes which are asymptotically locally (A)dS or flat. We explain how integrability requires a choice of integrable slicing and also the introduction of a corner term. After discussing the holographic renormalization of the action and of the symplectic potential, we show that the charges are finite, symplectic and integrable, yet not conserved. We find four towers of charges forming an algebroid given by $$ \mathfrak{vir}\oplus \mathfrak{vir}\oplus $$ vir ⊕ vir ⊕ Heisenberg with three central extensions, where the base space is parametrized by the retarded time. These four charges generate diffeomorphisms of the boundary cylinder, Weyl rescalings of the boundary metric, and radial translations. We perform this study both in metric and triad variables, and use the triad to explain the covariant origin of the corner terms needed for renormalization and integrability.

A generalized Contou-Carrère symbol and its reciprocity laws in higher dimensions

We generalize Contou-Carrère symbols to higher dimensions. To an ( n + 1 ) (n+1) -tuple f 0 , … , f n ∈ A ( ( t 1 ) ) ⋯ ( ( t n ) ) × f_0,\dots ,f_n \in A((t_1))\cdots ((t_n))^{\times } , where A A denotes an algebra over a field k k , we associate an element ( f 0 , … , f n ) ∈ A × (f_0,\dots ,f_n) \in A^{\times } , extending the higher tame symbol for k = A k = A , and earlier constructions for n = 1 n = 1 by Contou-Carrère, and n = 2 n = 2 by Osipov–Zhu. It is based on the concept of higher commutators for central extensions by spectra. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic K K -theory, and prove a version of Parshin–Kato reciprocity for higher Contou-Carrère symbols.

Central Extension of Left Alternative Algebras and the Second Cohomology Group

The purpose of this paper is to prove that the second cohomology group H 2 A , F of a left alternative algebra A over an algebraically closed field F of characteristic 0 can be interpreted as the set of equivalent classes of one-dimensional central extensions of A .