Leandro Cagliero

FAITHFUL REPRESENTATIONS OF MINIMAL DIMENSION OF CURRENT HEISENBERG LIE ALGEBRAS

Given a Lie algebra 𝔤 over a field of characteristic zero k, let μ(𝔤) = min{dim π : π is a faithful representation of 𝔤}. Let 𝔥m be the Heisenberg Lie algebra of dimension 2m + 1 over k and let k[t] be the polynomial algebra in one variable. Given m ∈ ℕ and p ∈ k[t], let 𝔥m, p = 𝔥m ⊗ k[t]/(p) be the current Lie algebra associated to 𝔥m and k[t]/(p), where (p) is the principal ideal in k[t] generated by p. In this paper we prove that . We also prove a result that gives information about the structure of a commuting family of operators on a finite dimensional vector space. From it is derived the well-known theorem of Schur on maximal abelian subalgebras of 𝔤𝔩(n, k).

Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials

For a two-parameter family of lower triangular matrices with entries involving Jacobi polynomials an explicit inverse is given, with entries involving a sum of two Jacobi polynomials. The formula simplifies in the Gegenbauer case and then one choice of the parameter solves an open problem in a recent paper by Koelink, van Pruijssen & Roman. The two-parameter family is closely related to two two-parameter groups of lower triangular matrices, of which we also give the explicit generators. Another family of pairs of mutually inverse lower triangular matrices with entries involving Jacobi polynomials, unrelated to the family just mentioned, was given by J.?Koekoek & R.?Koekoek (1999). We show that this last family is a limit case of a pair of connection relations between Askey-Wilson polynomials having one of their four parameters in common.

The cohomology of the cotangent bundle of Heisenberg groups

Abstract Given a parabolic subalgebra g1×n of a semisimple Lie algebra, Kostant (Ann. Math. 1963) and Griffiths (Acta Math. 1963) independently computed the g1 invariants in the cohomology group of n with exterior adjoint coefficients. By a theorem of Bott (Ann. Math. 1957), this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms. In this paper we determine explicitly the full module structure, over the symplectic group, of the cohomology group of the Heisenberg Lie algebra with exterior adjoint coefficients. This is the cohomology of the cotangent bundle of the Heisenberg group.

Indecomposable Modules of 2-step Solvable Lie Algebras in Arbitrary Characteristic

Let F be an algebraically closed field and consider the Lie algebra 𝔤 = ⟨ x ⟩ ⋉ 𝔞, where ad x acts diagonalizably on the abelian Lie algebra 𝔞. Refer to a 𝔤-module as admissible if [𝔤, 𝔤] acts via nilpotent operators on it, which is automatic if chr(F) = 0. In this article, we classify all indecomposable 𝔤-modules U which are admissible as well as uniserial, in the sense that U has a unique composition series.

The Nash–Moser theorem of Hamilton and rigidity of finite dimensional nilpotent Lie algebras

Abstract We apply the Nash–Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space H k - nil 2 ( g , g ) , and certain subspaces of it, that provide fine information about the deformations of g in the variety of k -step nilpotent Lie algebras. Then we focus on degenerations and rigidity in the variety of k -step nilpotent Lie algebras of dimension n with n ≤ 7 and, in particular, we obtain rigid Lie algebras and rigid curves in the variety of 3-step nilpotent Lie algebras of dimension 7. We also recover some known results and point out a possible error in a published article related to this subject.

Classification of finite dimensional uniserial representations of conformal Galilei algebras

With the aid of the

Nilradicals of parabolic subalgebras admitting symplectic structures

6j

Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

-symbol, we classify all uniserial modules of

Free 2-step nilpotent Lie algebras and indecomposable representations

\mathfrak{sl}(2)\ltimes \mathfrak{h}_{n}

A lower bound for faithful representations of nilpotent Lie algebras

, where