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We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group $U_{q} (2)$ , which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators $h_{j} : = I_{j} \otimes α \otimes I_{n - 2 - j}$ on ${(ℂ ³)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra $ℋ_{q, n} (2)$ associated with the quantum group $U_{q} (2)$ . The purpose of this note is to present the construction.

On a diffeological group realization of certain generalized symmetrizable Kac-Moody Lie algebras.

Leslie, Joshua (2003)

Journal of Lie Theory

On a family of operators and their Lie algebras.

Sanders, Jan A., Wang, Jing Ping (2002)

Journal of Lie Theory

On a general difference Galois theory I

Shuji Morikawa (2009)

Annales de l’institut Fourier

We know well difference Picard-Vessiot theory, Galois theory of linear difference equations. We propose a general Galois theory of difference equations that generalizes Picard-Vessiot theory. For every difference field extension of characteristic $0$ , we attach its Galois group, which is a group of coordinate transformation.

On a lie algebra of polynomials

S. Andreadakis (1966)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

On a new normalization for tractor covariant derivatives

Matthias Hammerl, Petr Somberg, Vladimír Souček, Josef Šilhan (2012)

Journal of the European Mathematical Society

A regular normal parabolic geometry of type $G / P$ on a manifold $M$ gives rise to sequences $D_{i}$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative $\nabla^{ω}$ on the corresponding tractor bundle $V$ , where $ω$ is the normal Cartan connection. The first operator $D_{0}$ in the sequence is overdetermined and it is well known that $\nabla^{ω}$ yields the prolongation of this operator in the homogeneous case $M = G / P$ . Our first main result...

On a nilpotent Lie superalgebra which generalizes Qn.

José María Ancochea Bermúdez, Otto Rutwig Campoamor (2002)

Revista Matemática Complutense

In Gilg (2000, 2001) the author introduces the notion of filiform Lie superalgebras, generalizing the filiform Lie algebras studied by Vergne in the sixties. In these appers, the superalgebras whose even part is isomorphic to the model filiform Lie algebra Ln are studied and classified in low dimensions. Here we consider a class of superalgebras whose even part is the filiform, naturally graded Lie algebra Qn, which only exists in even dimension as a consequence of the centralizer property. Certain...

On a quantum differential structure.

Guerrero, Berenice (1998)

Revista Colombiana de Matemáticas

On a theorem of E. Cartan

B. Морозов (1943)

Matematiceskij sbornik

On a theorem of Kontsevich.

Conant, James, Vogtmann, Karen (2003)

Algebraic & Geometric Topology

On affine Kac-Moody Lie algebras

Thomas N. Vougiouklis (1985)

Commentationes Mathematicae Universitatis Carolinae

On automorphisms of Weyl algebra

L. Makar-Limanov (1984)

Bulletin de la Société Mathématique de France

On basic and axiomatic ranks of nilpotent varieties of Lie algebras

Juri Bahturin (1982)

Banach Center Publications

On bounded generalized Harish-Chandra modules

Ivan Penkov, Vera Serganova (2012)

Annales de l’institut Fourier

Let $𝔤$ be a complex reductive Lie algebra and $𝔨 \subset 𝔤$ be any reductive in $𝔤$ subalgebra. We call a $(𝔤, 𝔨)$ -module $M$ bounded if the $𝔨$ -multiplicities of $M$ are uniformly bounded. In this paper we initiate a general study of simple bounded $(𝔤, 𝔨)$ -modules. We prove a strong necessary condition for a subalgebra $𝔨$ to be bounded (Corollary 4.6), i.e. to admit an infinite-dimensional simple bounded $(𝔤, 𝔨)$ -module, and then establish a sufficient condition for a subalgebra $𝔨$ to be bounded (Theorem 5.1). As a result we are able to...

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