Diamond distances in Nottingham algebras

Nottingham algebras are a class of just-infinite-dimensional, modular, [Formula: see text]-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree [Formula: see text], and the second occurs in degree [Formula: see text], a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to [Formula: see text]. A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper, we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra [Formula: see text] can be assigned a type, in such a way that the degrees and types of the diamonds completely describe [Formula: see text]. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals [Formula: see text]. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type [Formula: see text].

Modules of the Intermediate Series over Not-finitely Graded Lie Algebras W(Γ)

Let [Formula: see text] be a class of not-finitely graded Lie algebras related to generalized Virasoro algebras with basis [Formula: see text], which satisfies relations [Formula: see text] and [Formula: see text]. In this paper, [Formula: see text]-modules of the intermediate series satisfying some conditions are constructed and classified. We also obtain modules of the intermediate series over the related Lie superalgebra.

Duality Hierarchies and Differential Graded Lie Algebras

The gauge theories underlying gauged supergravity and exceptional field theory are based on tensor hierarchies: generalizations of Yang-Mills theory utilizing algebraic structures that generalize Lie algebras and, as a consequence, require higher-form gauge fields. Recently, we proposed that the algebraic structure allowing for consistent tensor hierarchies is axiomatized by ‘infinity-enhanced Leibniz algebras’ defined on graded vector spaces generalizing Leibniz algebras. It was subsequently shown that, upon appending additional vector spaces, this structure can be reinterpreted as a differential graded Lie algebra. We use this observation to streamline the construction of general tensor hierarchies, and we formulate dynamics in terms of a hierarchy of first-order duality relations, including scalar fields with a potential.