Graded torsion-free 𝔰𝔩2(ℂ)-modules of rank 2

2021 ◽  
pp. 2250229
Author(s):  
Yuri Bahturin ◽  
Abdallah Shihadeh
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Rank 2 ◽  
Torsion Free

In this paper, we explore the possibility of endowing simple infinite-dimensional [Formula: see text]-modules by the structure of graded modules. The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in 7, 8.

scholarly journals Graded modules over simple Lie algebras

2019 ◽  
Vol 53 (supl) ◽  
pp. 45-86
Author(s):  
Yuri Bahturin ◽  
Mikhail Kochetov ◽  
Abdallah Shihadeh
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Arbitrary Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Simple Lie Algebras ◽  
Simple Modules ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Necessary And Sufficient

The paper is devoted to the study of graded-simple modules and gradings on simple modules over finite-dimensional simple Lie algebras. In general, a connection between these two objects is given by the so-called loop construction. We review the main features of this construction as well as necessary and sufficient conditions under which finite-dimensional simple modules can be graded. Over the Lie algebra sl2(C), we consider specific gradings on simple modules of arbitrary dimension.

On Derivations of Lie Algebras

1976 ◽  
Vol 28 (1) ◽  
pp. 174-180 ◽  
Author(s):  
Stephen Berman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Characteristic Zero ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Cartan Matrices

A well known result in the theory of Lie algebras, due to H. Zassenhaus, states that if is a finite dimensional Lie algebra over the field K such that the killing form of is non-degenerate, then the derivations of are all inner, [3, p. 74]. In particular, this applies to the finite dimensional split simple Lie algebras over fields of characteristic zero. In this paper we extend this result to a class of Lie algebras which generalize the split simple Lie algebras, and which are defined by Cartan matrices (for a definition see § 1). Because of the fact that the algebras we consider are usually infinite dimensional, the method we employ in our investigation is quite different from the standard one used in the finite dimensional case, and makes no reference to any associative bilinear form on the algebras.

INFINITE-DIMENSIONAL GENERALIZATIONS OF SIMPLE LIE ALGEBRAS

1990 ◽  
Vol 05 (24) ◽  
pp. 1967-1977 ◽  
Author(s):  
E. S. FRADKIN ◽  
V. YA. LINETSKY
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Virasoro Algebra ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Higher Spin ◽  
Higher Spin Theory

Infinite-dimensional algebras associated with simple finite-dimensional Lie algebra g are considered. Higher-spin generalizations of sl(2) are studied in detail. Those of the Virasoro algebra are viewed as their "analytic continuations". Applications in higher-spin theory and in conformal QFT are discussed.

The Torsion Free Pieri Formula

1998 ◽  
Vol 50 (2) ◽  
pp. 266-289 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Tensor Product ◽  
Arbitrary Degree ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional ◽  
Pieri Formula ◽  
Free Modules ◽  
Central Characters ◽  
Torsion Free

AbstractCentral to the study of simple infinite dimensional g𝓵(n, C)-modules having finite dimensional weight spaces are the torsion free modules. All degree 1 torsion free modules are known. Torsion free modules of arbitrary degree can be constructed by tensoring torsion free modules of degree 1 with finite dimensional simple modules. In this paper, the central characters of such a tensor product module are shown to be given by a Pieri-like formula, complete reducibility is established when these central characters are distinct and an example is presented illustrating the existence of a nonsimple indecomposable submodule when these characters are not distinct.

scholarly journals Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class

1999 ◽  
Vol 67 (2) ◽  
pp. 157-184 ◽  
Author(s):  
A. Caranti ◽  
S. Mattarei
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Loop Algebra ◽  
Maximal Class ◽  
Graded Lie Algebras ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Wide Range ◽  
Graded Lie Algebra

AbstractWe investigate a class of infinite-dimensional, modular, graded Lie algebra in which the homogeneous components have dimension at most two. A subclass of these algebras can be obtained via a twisted loop algebra construction from certain finite-dimensional, simple Lie algebras of Albert-Frank type.Another subclass of these algebras is strictly related to certain graded Lie algebras of maximal class, and exhibits a wide range of behaviours.

scholarly journals Jet modules for the centerless Virasoro-like algebra

2019 ◽  
Vol 18 (01) ◽  
pp. 1950002 ◽  
Author(s):  
Xiangqian Guo ◽  
Genqiang Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Block Type ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Infinite Dimensional Lie Algebra ◽  
Area Preserving

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a [Formula: see text]-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over [Formula: see text]. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of [Irreducible representations for toroidal Lie algebras, J. Algebras 221 (1999) 188–231; Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004) 23–42]. Consequently, indecomposable jet modules are described using modules over the algebra [Formula: see text], which is the “positive part” of a Block type algebra studied first by [Some infinite-dimensional simple Lie algebras in characteristic [Formula: see text] related to those of Block, J. Pure Appl. Algebra 127(2) (1998) 153–165] and recently by [A [Formula: see text]-graded generalization of the Witt-algebra, preprint; Classification of simple Lie algebras on a lattice, Proc. London Math. Soc. 106(3) (2013) 508–564]).

scholarly journals Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type

10.37236/933 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Gregg Musiker ◽  
James Propp
Keyword(s):  
Lie Algebras ◽  
Cluster Algebras ◽  
Laurent Polynomials ◽  
Finite Dimensional ◽  
Combinatorial Interpretations ◽  
Rank 2 ◽  
The Individual ◽  
Affine Type ◽  
Positive Integers ◽  
Two Parameter

Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the $(b,c)$ family, possesses the Laurentness property: for all $b,c$, each term of the $(b,c)$ sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers $b,c$ satisfy $bc < 4$, the recurrence is related to the root systems of finite-dimensional rank $2$ Lie algebras; when $bc>4$, the recurrence is related to Kac-Moody rank $2$ Lie algebras of general type. Here we investigate the borderline cases $bc=4$, corresponding to Kac-Moody Lie algebras of affine type. In these cases, we show that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. By providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.

scholarly journals Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1737
Author(s):  
Mariia Myronova ◽  
Jiří Patera ◽  
Marzena Szajewska
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Coxeter Groups ◽  
Reflection Groups ◽  
Geometrical Structures ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Branching Rules ◽  
Definition Of ◽  
Representation Orbit

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

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