Let G be a semi-simple simply-connected complex algebraic group and T ⊂ B a maximal torus and a Borel subgroup respectively. Let h = Lie T be the Cartan subalgebra of the Lie algebra Lie G, and W := N(T )/T the Weyl group associated to the pair (G, T ), where N(T ) is the normalizer of T in G. We can view any element w = w mod T ∈ W as the element (denoted by the corresponding German character)...

In this paper, we continue our study of the maximal bounded Z-filtrations of a complex semisimple Lie algebra L. Specifically, we discuss the functionals which give rise to such filtrations, and we show that they are related to certain semisimple subalgebras of L of full rank. In this way, we determine the “order” of these functionals and count them without the aid of computer computations. The...

The five simple exceptional complex Lie superalgebras of vector fields are described. One of them, kas, is new; the other four are explicitly described for the first time. All nonisomorphic maximal subalgebras of finite codimension of these Lie superalgebras, i.e., all other realizations of these Lie superalgebras as Lie superalgebras of vector fields, are also described; there are 14 of them a...

In a recent study of Engel Lie rings, Serena Cicalò and Willem de Graaf have given a practical set of conditions for an additively finitely generated Lie ring L to satisfy an Engel condition. We present a simpler and more direct proof of this fact. Then we generalize it to a result in the language of tensor algebra, which can be applied to other contexts.

Let g be the Lie superalgebra gl(m|n). We show how to associate a gl(m|n) weight Λ to a composite partition ν;μ with composite Young diagram F (ν;μ). Based upon the definition of critical representations, the notion of “critical composite partition” is introduced. It is shown that for critical composite partitions (subject to a technical restriction) the corresponding gl(m|n) representation VΛ ...

The problem of non-solvable contractions of Lie algebras is analyzed. By means of a stability theorem, the problem is shown to be deeply related to the embeddings among semisimple Lie algebras and the resulting branching rules for representations. With this procedure, we determine all deformations of indecomposable Lie algebras having a nontrivial Levi decomposition onto semisimple Lie algebras...

The uniformity, for the family of exceptional Lie algebras g, of the decompositions of the powers of their adjoint representations is well-known now for powers up to the fourth. The paper describes an extension of this uniformity for the totally antisymmetrised n-th powers up to n = 9, identifying (see Tables 3 and 6) families of representations with integer eigenvalues 5, . . . , 9 for the qua...

Brundan and Kleshchev recently introduced a new family of presentations of the Yangian Y (gln) associated to the general linear Lie algebra gln, and thus provided a fresh approach to its study. In this article, we would like to show how some of their ideas can be fruitfully extended to consider the Yangian Y (glm|n) associated to the Lie superalgebra glm|n. In particular, we give a new proof of...

The concept of superintegrability in quantum mechanics is extended to the case of a particle with spin s = 1/2 interacting with one of spin s = 0. Non-trivial superintegrable systems with 8and 9-dimensional Lie algebras of first-order integrals of motion are constructed in twoand three-dimensional spaces, respectively.

We show that the vertex algebra W1+∞ with central charge −1 is isomorphic to a tensor product of the simple W3 algebra with central charge −2 and a Heisenberg vertex algebra generated by a free bosonic field. We construct a family of irreducible modules of the W3 algebra with central charge −2 in terms of free fields and calculate the full character formulas of these modules with respect to the...