Finite Simple Lie Research Articles

Conformal Triple Derivations and Triple Homomorphisms of Lie Conformal Algebras

Let [Formula: see text] be a finite Lie conformal algebra. We investigate the conformal derivation algebra [Formula: see text], conformal triple derivation algebra [Formula: see text] and generalized conformal triple derivation algebra [Formula: see text], focusing mainly on the connections among these derivation algebras. We also give a complete classification of (generalized) conformal triple derivation algebras on all finite simple Lie conformal algebras. In particular, [Formula: see text], where [Formula: see text] is a finite simple Lie conformal algebra. But for [Formula: see text], we obtain a conclusion that is closely related to [Formula: see text]. Finally, we introduce the definition of a triple homomorphism of Lie conformal algebras. Triple homomorphisms of all finite simple Lie conformal algebras are also characterized.

Irreducible modules over finite simple Lie pseudoalgebras IV. Non-primitive pseudoalgebras

Let d \subset d' be finite-dimensional Lie algebras, H = U(d), H'=U(d') the corresponding universal enveloping algebras endowed with the cocommutative Hopf algebra structure. We show that if L is a primitive Lie pseudoalgebra over H then all finite irreducible L' = Cur_H^{H'} L-modules are of the form Cur_H^{H'} V, where V is an irreducible L-module, with a single class of exceptions. Indeed, when L = H(d, \chi, \omega), we introduce non current L'-modules that are obtained by modifying the current pseudoaction with an extra term depending on an element t \in \d' \setminus d, which must satisfy some technical conditions. This, along with results from [BDK1, BDK2, BDK3], completes the classification of finite irreducible modules of finite simple Lie pseudoalgebras over the universal enveloping algebra of a finite-dimensional Lie algebra.

Tensor categories of affine Lie algebras beyond admissible levels

We show that if V is a vertex operator algebra such that all the irreducible ordinary V-modules are $$C_1$$ -cofinite and all the grading-restricted generalized Verma modules for V are of finite length, then the category of finite length generalized V-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) $$\mathfrak {g}$$ at level k and the category $$KL_k(\mathfrak {g})$$ of its finite length generalized modules, we discover several families of $$KL_k(\mathfrak {g})$$ at non-admissible levels k, having braided tensor category structures. In particular, $$KL_k(\mathfrak {g})$$ has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories $$KL_k(\mathfrak {g})$$ , including the category $$KL_{-1}(\mathfrak {sl}_n)$$ . Using these results, we construct a rigid tensor category structure on a full subcategory of $$KL_1(\mathfrak {sl}(n|m))$$ consisting of objects with semisimple Cartan subalgebra actions.

Generalized conformal derivations of Lie conformal algebras

Let [Formula: see text] be a finite Lie conformal algebra. The purpose of this paper is to investigate the conformal derivation algebra [Formula: see text], the conformal quasiderivation algebra [Formula: see text] and the generalized conformal derivation algebra [Formula: see text]. The generalized conformal derivation algebra is a natural generalization of the conformal derivation algebra. Obviously, we have the following tower [Formula: see text], where [Formula: see text] is the general Lie conformal algebra. Furthermore, we mainly research the connection of these generalized conformal derivations. Finally, the conformal [Formula: see text]-derivations of Lie conformal algebras are studied. Moreover, we obtain some connections between several specific generalized conformal derivations and the conformal [Formula: see text]-derivations. In addition, all conformal [Formula: see text]-derivations of finite simple Lie conformal algebras are characterized.

Finite irreducible representations of map Lie conformal algebras

In this paper, we study the finite representation theory of the map Lie conformal algebra [Formula: see text], where G is a finite simple Lie conformal algebra and A is a commutative associative algebra with unity over [Formula: see text]. In particular, we give a complete classification of nontrivial finite irreducible conformal modules of [Formula: see text] provided A is finite-dimensional.

Leibniz H-pseudoalgebras

We define a Leibniz H-pseudoalgebra over a cocommutative Hopf algebra H, and a tensor product of Leibniz H-pseudoalgebras, and prove that every Leibniz H-pseudoalgebra of free rank one over H is a Lie H-pseudoalgebra. To characterize simple Leibniz H-pseudoalgebras, we define the Leibniz cohomology of Leibniz H-pseudoalgebras and determine some simple Leibniz H-pseudoalgebras by calculating Leibniz cohomological groups of finite simple Lie H-pseudoalgebras defined in [2].

Young tableaux and crystal [formula omitted] for the exceptional Lie algebra types

We study the crystal base B ( ∞ ) associated with the negative part of the quantum group for finite simple Lie algebras of types E 6 , E 7 , E 8 , and F 4 . After describing the crystal B ( ∞ ) as a union of highest weight crystals, we give an explicit description of B ( ∞ ) in terms of Young tableaux.

Irreducible modules over finite simple Lie pseudoalgebras I. Primitive pseudoalgebras of type [formula omitted] and [formula omitted

One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which C [ ∂ ] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra. The finite (i.e., finitely generated over H ) simple Lie pseudoalgebras were classified in our previous work [B.Bakalov, A.D’Andrea, V.G. Kac, Theory of finite pseudoalgebras, Adv. Math. 162 (2001) 1–140]. In a series of papers, starting with the present one, we classify all irreducible finite modules over finite simple Lie pseudoalgebras.

Classification of finite simple Lie conformal superalgebras

The notion of a Lie conformal superalgebra encodes an axiomatic descrption of singular parts of the operator product expansions of chiral fields in conformal field theory. In the paper we give a detailed proof of the classification of all finite simple Lie conformal superalgebras. We also classify all their central extensions.

Some examples of locally finite simple Lie algebras

Some examples of locally finite simple Lie algebras

Fourier transforms of nilpotently supported invariant functions on a finite simple Lie algebra

Fourier transforms of nilpotently supported invariant functions on a finite simple Lie algebra