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Gradedness of the set of rook placements in A n - 1

Mikhail V. Ignatev (2021)

Communications in Mathematics

A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in A n - 1 with respect to a slightly different order and prove that this poset is...

Graphs associated with nilpotent Lie algebras of maximal rank.

Eduardo Díaz, Rafael Fernández-Mateos, Desamparados Fernández-Ternero, Juan Núñez (2003)

Revista Matemática Iberoamericana

In this paper, we use the graphs as a tool to study nilpotent Lie algebras. It implies to set up a link betwcen graph theory and Lie theory. To do this, it is already known that every nilpotent Lie algebra of maximal rank is associated with a generalized Cartan matrix A and it ils isomorphic to a quotient of the positive part n+ of the KacMoody algebra g(A). Then, if A is affine, we can associate n+ with a directed graph (from now on, we use the term digraph) and we can also associate a subgraph...

Groupe de Galois différentiel local et représentation adjointe

Elie Compoint, Anne Duval (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

Dans cet article on s’intéresse à la représentation adjointe du tore exponentiel sur l’algèbre de Lie du groupe de Galois différentiel local. Nous proposons un algorithme pour réduire les sous-espaces poids de dimension supérieure à 1 à des sous-espaces de racines. Ce faisant, on construit un tore (en général) maximal qui contient le tore exponentiel. Au cours de ce travail on est amené à étudier la régularité du tore exponentiel dans le groupe de Galois local.

Growth of some varieties of Leibniz-Poisson algebras

Ratseev, S. M. (2011)

Serdica Mathematical Journal

2010 Mathematics Subject Classification: 17A32, 17B63.Let V be a variety of Leibniz-Poisson algebras over an arbitrary field whose ideal of identities contains the identities {{x1,y1},{x2,y2},ј,{xm,ym}} = 0, {x1,y1}·{x2,y2}· ј ·{xm,ym} = 0 for some m. It is shown that the exponent of V exists and is an integer.

Hall algebra of morphism category

QingHua Chen, Liwang Zhang (2024)

Czechoslovak Mathematical Journal

This paper investigates a universal PBW-basis and a minimal set of generators for the Hall algebra ( C 2 ( 𝒫 ) ) , where C 2 ( 𝒫 ) is the category of morphisms between projective objects in a finitary hereditary exact category 𝒜 . When 𝒜 is the representation category of a Dynkin quiver, we develop multiplication formulas for the degenerate Hall Lie algebra , which is spanned by isoclasses of indecomposable objects in C 2 ( 𝒫 ) . As applications, we demonstrate that contains a Lie subalgebra isomorphic to the central extension...

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

For any positive integer n , let A n be a linearly oriented quiver of type A with n vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories n + 1 and n , where n + 1 and n are the two extriangulated categories corresponding to the representation category of A n + 1 and the morphism category of projective representations of A n , respectively. As a by-product,...

Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products

Pavel Etingof, Wee Liang Gan, Victor Ginzburg, Alexei Oblomkov (2007)

Publications Mathématiques de l'IHÉS

The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras...

Heisenberg algebra and a graphical calculus

Mikhail Khovanov (2014)

Fundamenta Mathematicae

A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose Grothendieck ring contains an integral form of the Heisenberg algebra in infinitely many variables. We construct bases of the vector spaces of morphisms between products of generating objects in this category.

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