Verma Modules over Quantum Torus Lie Algebras

2010 ◽  
Vol 62 (2) ◽  
pp. 382-399 ◽  
Author(s):  
Rencai Lü ◽  
Kaiming Zhao
Keyword(s):  
Lie Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Functions ◽  
Verma Modules ◽  
Central Extensions ◽  
Quantum Torus ◽  
Infinite Dimensional ◽  
Quantum Tori ◽  
Necessary And Sufficient

AbstractRepresentations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras . The center of now is generally infinite dimensional.In this paper, Z-graded Verma modules over and their corresponding irreducible highest weight modules are defined for some linear functions . Necessary and sufficient conditions for to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules e to be irreducible are obtained.

scholarly journals n-Point Virasoro algebras and their modules of densities

2014 ◽  
Vol 16 (03) ◽  
pp. 1350047 ◽  
Author(s):  
Ben Cox ◽  
Xiangqian Guo ◽  
Rencai Lu ◽  
Kaiming Zhao
Keyword(s):  
Large Class ◽  
Lie Algebras ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Virasoro Algebra ◽  
Necessary And Sufficient Conditions ◽  
Genus Zero ◽  
Central Extensions ◽  
Necessary And Sufficient ◽  
Universal Central Extensions

In this paper we introduce and study n-point Virasoro algebras, [Formula: see text], which are natural generalizations of the classical Virasoro algebra and have as quotients multipoint genus zero Krichever–Novikov type algebras. We determine necessary and sufficient conditions for the latter two such Lie algebras to be isomorphic. Moreover we determine their automorphisms, their derivation algebras, their universal central extensions, and some other properties. The list of automorphism groups that occur is Cn, Dn, A4, S4 and A5. We also construct a large class of modules which we call modules of densities, and determine necessary and sufficient conditions for them to be irreducible.

scholarly journals Simple superelliptic Lie algebras

2017 ◽  
Vol 19 (03) ◽  
pp. 1650032 ◽  
Author(s):  
Ben Cox ◽  
Xiangqian Guo ◽  
Rencai Lu ◽  
Kaiming Zhao
Keyword(s):  
Lie Algebras ◽  
Riemann Surfaces ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Central Extensions ◽  
Birational Equivalence ◽  
Necessary And Sufficient ◽  
Universal Central Extensions

Let [Formula: see text], [Formula: see text]. Then we have the algebraic curve [Formula: see text], and its coordinate algebras (the Riemann surfaces) [Formula: see text] and [Formula: see text] The Lie algebras [Formula: see text] and [Formula: see text] are called the [Formula: see text]th superelliptic Lie algebras associated to [Formula: see text]. In this paper, we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras, which will help to understand the birational equivalence of some algebraic curves of the form [Formula: see text].

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.

scholarly journals A remark on irreducible representations of Lie algebras

1979 ◽  
Vol 27 (3) ◽  
pp. 332-336 ◽  
Author(s):  
JU. A. Bahturin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Irreducible Representations ◽  
Bounded Degree ◽  
Precise Form ◽  
Representations Of Lie Algebras ◽  
Necessary And Sufficient

AbstractIn addition to the results of the paper (Bachturin (1974)) we give the precise form of the necessary and sufficient conditions ensuring that all irreducible representations of a Lie algebra were of finite bounded degree.

scholarly journals POINTED HOPF ALGEBRAS WITH CLASSICAL WEYL GROUPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250066
Author(s):  
SHOUCHUAN ZHANG ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weyl Groups ◽  
Drinfeld Modules ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras ◽  
Necessary And Sufficient

We prove that Nichols algebras of irreducible Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊nsupported by 𝕊nare infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊nsupported by A to be finite dimensional.

scholarly journals Post-Lie algebra structures for nilpotent Lie algebras

2018 ◽  
Vol 28 (05) ◽  
pp. 915-933
Author(s):  
Dietrich Burde ◽  
Christof Ender ◽  
Wolfgang Alexander Moens
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nilpotent Lie Algebras ◽  
Dimension Formula ◽  
Necessary And Sufficient

We study post-Lie algebra structures on [Formula: see text] for nilpotent Lie algebras. First, we show that if [Formula: see text] is nilpotent such that [Formula: see text], then also [Formula: see text] must be nilpotent, of bounded class. For post-Lie algebra structures [Formula: see text] on pairs of [Formula: see text]-step nilpotent Lie algebras [Formula: see text] we give necessary and sufficient conditions such that [Formula: see text] defines a CPA-structure on [Formula: see text], or on [Formula: see text]. As a corollary, we obtain that every LR-structure on a Heisenberg Lie algebra of dimension [Formula: see text] is complete. Finally, we classify all post-Lie algebra structures on [Formula: see text] for [Formula: see text], where [Formula: see text] is the three-dimensional Heisenberg Lie algebra.

scholarly journals The Equivalence of Datko and Lyapunov Properties for (h,k)-Trichotomic Linear Discrete-Time Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Claudia-Luminiţa Mihiţ ◽  
Mihail Megan ◽  
Traian Ceauşu
Keyword(s):  
Discrete Time ◽  
General Property ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Dimensional ◽  
Discrete Time Systems ◽  
Exponential Trichotomy ◽  
Necessary And Sufficient ◽  
Time Systems

The aim of this paper is to characterize a general property of(h,k)-trichotomy through some Lyapunov functions for linear discrete-time systems in infinite dimensional spaces. Also, we apply the results to illustrate necessary and sufficient conditions for nonuniform exponential trichotomy and nonuniform polynomial trichotomy.

scholarly journals On the extensions of infinite-dimensional representations of Lie semigroups

2002 ◽  
Vol 29 (4) ◽  
pp. 195-207
Author(s):  
Adolf R. Mirotin
Keyword(s):  
Lie Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Unitary Representations ◽  
Local Representation ◽  
Infinite Dimensional ◽  
Lie Semigroup ◽  
Lie Semigroups ◽  
Necessary And Sufficient

The necessary and sufficient conditions have been obtained for extendability of a Banach representation of a generating Lie semigroupSto a local representation of the Lie groupGgenerated bySwhen the tangent wedge ofSis a Lie semialgebra. The most convenient conditions we obtain correspond to the case of unitary representations. In this case, we give a criterion of global extendability ifGis exponential and solvable.

scholarly journals A note on induced central extensions

1979 ◽  
Vol 20 (3) ◽  
pp. 411-420 ◽  
Author(s):  
L.R. Vermani
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Central Extension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Central Extensions ◽  
Baer Sum ◽  
Necessary And Sufficient

A characterization of induced central extensions which gives an explicit relationship between induced central extensions and n-stem extensions is obtained. Using the characterization, necessary and sufficient conditions for a central extension of an abelian group by a nilpotent group of class n to be a Baer sum of an induced central extension and an extension of class n are obtained.

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