arXiv subjects

Paolo Papi

Publications and source records attributed to Paolo Papi.

45 records · Page 3Linked to original sources

arXiv2009

ad-nilpotent ideals containing a fixed number of simple root spaces

We give formulas for the number of ad-nilpotent ideals of a Borel subalgebra of a Lie algebra of type B or D containing a fixed number of root spaces attached to simple roots. This result solves positively a conjecture of Panyushev (cf. D. Panyushev, ad-nilpotent ideals: generators and duality, J. of Alg., to appear) and affords a complete knowledge of the above statistics for any simple Lie algebra. We also study the restriction of the above statistics to the abelian ideals of a Borel subalgebra, obtaining uniform results for any simple Lie algebra.

math.RT↗

arXiv2007

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting

We extend classical results of Kostant and al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.

math.RT↗

arXiv2007

Casimir operators, abelian subspaces and u-cohomology

This survey paper is an exposition of old and recent results of Kostant and al. on the relationships between the exterior algebra of a simple Lie algebra and the action of the Casimir operator on it. Our exposition relies on u-cohomology and it is basically self-contained.

math.RT↗

arXiv2006

The $\hat W$-orbit of $ρ$, Kostant's formula for powers of the Euler product and affine Weyl groups as permutations of Z

Let an affine Weyl group $\hat W$ act as a group of affine transformations on a real vector space V. We analyze the $\hat W$-orbit of a regular element in V and deduce applications to Kostant's formula for powers of the Euler product and to the representations of $\hat W$ as permutations of the integers.

math.RT↗

arXiv2006

Decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of Z_2-graded Lie algebras

We give uniform formulas for the branching rules of level 1 modules over orthogonal affine Lie algebras for all conformal pairs associated to symmetric spaces. We also provide a combinatorial intepretation of these formulas in terms of certain abelian subalgebras of simple Lie algebras.

math-ph↗

arXiv2004

Abelian subalgebras in Z_2-graded Lie algebras and affine Weyl groups

Let g=g_0+ g_1 be a simple Z_2-graded Lie algebra and let b_0 be a fixed Borel subalgebra of g_0. We describe and enumerate the abelian b_0-stable subalgebras of g_1.

math.RT↗

arXiv2001

ad-Nilpotent ideals of a Borel subalgebra II

We provide an explicit bijection between the ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra g and the orbits of \check{Q}/(h+1)\check{Q} under the Weyl group (\check{Q} being the coroot lattice and h the Coxeter number of g). From this result we deduce in a uniform way a counting formula for the ad-nilpotent ideals.

math.RT↗

arXiv2000

Enumeration of ad-nilpotent $\frak b$-ideals for simple Lie algebras

We provide explicit formulas for the number of ad-nilpotent ideals of a Borel subalgebra of a complex simple Lie algebra having fixed class of nilpotence.

math.RA↗

arXiv2000

ad-nilpotent $\frak b$-ideals in sl(n) having a fixed class of nilpotence: combinatorics and enumeration

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of $sl(n+1,\Bbb C)$. 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.

math.RA↗