We prove a formula for the multiplicity of the irreducible representation V (n) of sl(2, C) as a direct summand of its own exterior cube ΛV (n). From this we determine that V (n) occurs exactly once as a summand of ΛV (n) if and only if n = 3, 5, 6, 7, 8, 10. These representations admit a unique sl(2)-invariant alternating ternary structure obtained from the projection ΛV (n) → V (n). We calcul...

We prove that for every automata algebra of exponential growth, the associated Lie algebra contains a free subalgebra. For n ≥ 1, let Ln+2 be a Lie algebra with generator set x1, . . . , xn+2 and the following relations: for k ≤ n, any commutator of length k which consists of fewer than k different symbols from {x1, . . . , xn+2} is zero. As an application of this result about automata algebras...

Let G0 denote a compact semisimple Lie algebra and U a finite dimensional real G0 module. The vector space N0 = U ⊕ G0 admits a canonical 2-step nilpotent Lie algebra structure with [N0,N0] = G0 and an inner product 〈, 〉, unique up to scaling, for which the elements of G0 are skew symmetric derivations of N0. Let N0 denote the corresponding simply connected 2-step nilpotent Lie group with Lie a...

A primeness criterion due to Bell is shown to apply to the universal enveloping algebra of the Cartan type Lie superal-gebras S(V) and e S(V ; t) when dim V is even. This together with other recent papers yields Theorem. Let L be a nite-dimensional simple Lie superalgebra over an algebraically closed eld of characteristic zero. Then L satisses Bell's criterion (so that U(L) is prime), unless L ...

We study the Leibniz $n$-algebra $\textbf{U}_n(\mathfrak{L})$, whose multiplication is defined via bracket of a algebra $\mathfrak{L}$ as $[x_1,\dots,x_n]=[x_1,[\dots, [x_{n-2},[x_{n-1},x_n]]\dots]]$. show that $\textbf{U}_n(\mathfrak{L})$ simple if and only Lie algebra. An analogue Levi's theorem for algebras in $\textbf{U}_n(\textbf{Lb})$ established it proven $n$-kernel any semisimple $\text...

In this paper, we study the shape and reduced algebra for Lie superalgebra $${\mathfrak {sl}}(m,n)$$ . We define quasistandard Young tableaux, they give a combinatorial basis algebra, called diamond cone. There is natural bijective map from set of semistandard tableaux with $$\lambda $$ to smaller shape: cone compatible stratification algebra.

I show that the projective Schur functions may be interpreted as bispherical functions of either the triple (q(n), q(n)⊕q(n), q(n)), where q(n) is the “odd” (queer) analog of the general linear Lie algebra, or the triple (pe(n), gl(n|n), pe(n)), where pe(n) is the periplectic Lie superalgebra which preserves the nondegenerate odd bilinear form (either symmetric or skew-symmetric). Making use of...

In this paper we classify indecomposable modules for the Lie algebra of vector fields on a torus that admit a compatible action of the algebra of functions. An important family of such modules is given by spaces of jets of tensor fields. 0. Introduction. In recent years there was a substantial progress in representation theory of infinitedimensional Lie algebras of rank n > 1, toroidal Lie alge...

We obtain inductive and enumerative formulas for the multiplicities of weights spin module Clifford algebra a Levi subalgebra in complex semisimple Lie algebra. Our involve only matrices tableaux, our techniques combine linear algebra, theory, combinatorics. Moreover, this suggests relationship with nilpotent orbits. The case special $\mathfrak{sl}(n,{\mathbb C})$ is emphasized.