Factor systems are a tool for working on the extension problem of algebraic structures such as groups, Lie algebras, and associative algebras. Their applications numerous well-known in these common settings. We construct P algebra analogues to series results from W. R. Scott’s Group Theory, which gives an explicit theory factor group case. Here ranges over Leibniz, Zinbiel, diassociative, dendr...

Abstract We study pre-Lie pairs, by which we mean a pair of homotopy Lie algebra and with compatible action. Such pairs provide wealth algebraic structure, in particular can be used to analyze the part pair. Our main application motivation for this development are dg algebras hairy graphs computing rational groups mapping spaces little disks operads. show that twisting certain Maurer–Cartan ele...

The goal of this paper is to construct infinite dimensional Lie algebras using infinite product identities, and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and Lie algebras. The Lie algebras that we construct conjecturally appear in an orbifold conformal field theory with symmetries given by the monster ...

We construct double biproduct, bicrossproduct, double crossproduct, double bicrossproduct Lie bialgebras from braided Lie bialgebras. The relations between them are found. The main result generalizes Majid’s matched pair of Lie algebras, Drinfeld’s quantum double of Lie bialgebras, and Masuoka’s cross product Lie bialgebras. Some properties of double biproduct Lie bialgebras are given. In the a...

We present several embedding results for 3-graded Lie algebras and KKT algebras that are generated by two homogeneous elements of degrees 1 and −1. We also propose the canonical kernel function for a “universal Bergman kernel” which extends the usual Bergman kernel on a bounded symmetric domain to a group-valued function or, in terms of formal series, to an element in the formal completion of t...

Some examples are given of finite dimensional Lie bialgebras whose brackets and cobrackets are determined by pairs of r-matrices. The aim of this Letter is to give some low-dimensional examples of classical coboundary Lie bialgebras [1, 2] with coboundary duals. Since such structures can be specified (up to automorphisms) by pairs of r-matrices, so it is natural to call them bi-r-matrix bialgeb...

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit highly reducible unitary positive energy representations in a Fock space. The multiplicity of their irreducible componen...

The aim of this paper is to define a pair symplectic Dirac operators (D+, D–) in an algebraic setting motivated by the analogy with orthogonal representation theory. We work settings ℤ/2-graded quadratic Lie algebras

The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite grou...