Central invariants and enveloping algebras of braided Hom-Lie algebras

Let (H,?) be a monoidal Hom-Hopf algebra and HH HYD the Hom-Yetter-Drinfeld category over (H,?). Then in this paper, we first introduce the definition of braided Hom-Lie algebras and show that each monoidal Hom-algebra in HH HYD gives rise to a braided Hom-Lie algebra. Second, we prove that if (A,?) is a sum of two H-commutative monoidal Hom-subalgebras, then the commutator Hom-ideal [A,A] of A is nilpotent. Also, we study the central invariant of braided Hom-Lie algebras as a generalization of generalized Lie algebras. Finally, we obtain a construction of the enveloping algebras of braided Hom-Lie algebras and show that the enveloping algebras are H-cocommutative Hom-Hopf algebras.

GENERALIZED HOM-LIE STRUCTURE OF MONOIDAL HOM-ALGEBRAS IN A CATEGORY

In this paper, we study the structure of monoidal Hom-Lie algebras in the category Hℳ of H-modules for a triangular Hopf algebra (H, R) and in particular the H-Lie structure of a monoidal Hom-algebra in Hℳ by analogy with that of generalized Lie algebras.

ON F-THEORY QUIVER MODELS AND KAC–MOODY ALGEBRAS

We discuss quiver gauge models with bi-fundamental and fundamental matter obtained from F-theory compactified on ALE spaces over a four-dimensional base space. We focus on the base geometry which consists of intersecting F0 = CP1 × CP1 Hirzebruch complex surfaces arranged as Dynkin graphs classified by three kinds of Kac–Moody (KM) algebras: ordinary, i.e. finite-dimensional, affine and indefinite, in particular hyperbolic. We interpret the equations defining these three classes of generalized Lie algebras as the anomaly cancelation condition of the corresponding N = 1 F-theory quivers in four dimensions. We analyze in some detail hyperbolic geometries obtained from the affine [Formula: see text] base geometry by adding a node, and we find that it can be used to incorporate fundamental fields to a product of SU-type gauge groups and fields.

Topics on n-ary Algebraic Structures

We review the basic definitions and properties of two types of n-ary structures, the Generalized Lie Algebras (GLA) and the Filippov (≡ n-Lie) algebras (FA), as well as those of their Poisson counterparts, the Generalized Poisson (GPS) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology complexes relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends Whitehead’s lemma to all n ≥ 2, n = 2 being the original Lie algebra case. Some comments onn-Leibniz algebras are also made.

ANTI-COMMUTATIVE ALGEBRAS WITH SKEW-SYMMETRIC IDENTITIES

Generalizing Lie algebras, we consider anti-commutative algebras with skew-symmetric identities of degree > 3. Given a skew-symmetric polynomial f, we call an anti-commutative algebra f-Lie if it satisfies the identity f = 0. If sn is a standard skew-symmetric polynomial of degree n, then any s4-Lie algebra is f-Lie if deg f ≥ 4. We describe a free anti-commutative super-algebra with one odd generator. We exhibit various constructions of generalized Lie algebras, for example: given any derivations D, F of an associative commutative algebra U, the algebras (U, D ∧ F) and (U, id ∧ D2) are s4-Lie. An algebra (U, id ∧ D3 - 2D ∧ D2) is s'5-Lie, where s'5 is a non-standard skew-symmetric polynomial of degree 5.