Paolo Papi

A characterization of a special ordering in a root system

We give necessary and sufficient conditions for an ordering of a set of positive roots in a root system R to be associated to a reduced expression of an element of the Weyl group of R . Finally we characterize the sets of positive roots which can be given such an ordering.

-nilpotent -ideals in () having a fixed class of nilpotence: combinatorics and enumeration

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n + 1, C). We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q,t)-analogue of the Catalan number C n . These (q,t)-Catalan numbers count, on the one hand, ad-nilpotent ideals with respect to dimension and class of nilpotence and, on the other hand, admit interpretations in terms of natural statistics on Dyck paths.

Enumeration of ad-nilpotent ideals of a Borel subalgebra in type A by class of nilpotence

Abstract We provide a formula affording the number of ad-nilpotent ideals of a Borel subalgebra of sl (n+1, C ) having a fixed class of nilpotence.

A Combinatorial Approach to the Fusion Process for the Symmetric Group

We give a detailed account of Cheredniks fusion process for the symmetric group using as a key tool the combinatorics ofcompatible orderson the set of inversions of permutations.

Conformal embeddings of affine vertex algebras in minimal

Abstract We find all values of k ∈ C , for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra W k ( g , θ ) is conformal, where g is a basic simple Lie superalgebra and −θ its minimal root. In particular, it turns out that if W k ( g , θ ) does not collapse to its affine part, then the possible values of these k are either − 2 3 h ∨ or − h ∨ − 1 2 , where h ∨ is the dual Coxeter number of g for the normalization ( θ , θ ) = 2 . As an application of our results, we present a realization of simple affine vertex algebra V − n + 1 2 ( s l ( n + 1 ) ) inside the tensor product of the vertex algebra W n − 1 2 ( s l ( 2 | n ) , θ ) (also called the Bershadsky–Knizhnik algebra) with a lattice vertex algebra.

W

We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups, and, as an application of these formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description of the real forms of basic classical type Lie superalgebras.

-algebras I: structural results

Abstract We give explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra and of its twisted analogues, corresponding to symplectic and orthogonal Lie algebras. In particular, we develop the fusion procedure for twisted Yangians. For the non-twisted Yangian, this procedure goes back to the works of Cherednik.

Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs

Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras

Finite vs. Infinite Decompositions in Conformal Embeddings

{V_{\mathbf{k}}({\mathfrak{g}}^0)\subset V_{k}({\mathfrak{g}})}