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].

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 Criterion for unlimited growth of critical multidimensional stochastic models

2016 ◽  
Vol 48 (4) ◽  
pp. 972-988 ◽  
Author(s):  
Etienne Adam
Keyword(s):  
Large Class ◽  
Stochastic Models ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Positive Probability ◽  
Size Dependent ◽  
Unlimited Growth ◽  
Necessary And Sufficient

AbstractWe give a criterion for unlimited growth with positive probability for a large class of multidimensional stochastic models. As a by-product, we recover the necessary and sufficient conditions for recurrence and transience for critical multitype Galton–Watson with immigration processes and also significantly improve some results on multitype size-dependent Galton–Watson processes.

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
Author(s):  
Nicholas S. Ford
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Fixed Ring ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

A Family of Non-weight Modules over the Virasoro Algebra

2020 ◽  
Vol 27 (04) ◽  
pp. 807-820
Author(s):  
Guobo Chen
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Virasoro Algebra ◽  
Necessary And Sufficient Conditions ◽  
Weight Module ◽  
Highest Weight Module ◽  
Isomorphism Classes ◽  
Highest Weight ◽  
Necessary And Sufficient ◽  
Weight Modules

In this paper, we consider the tensor product modules of a class of non-weight modules and highest weight modules over the Virasoro algebra. We determine the necessary and sufficient conditions for such modules to be simple and the isomorphism classes among all these modules. Finally, we prove that these simple non-weight modules are new if the highest weight module over the Virasoro algebra is non-trivial.

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.

On Zipf's law

1975 ◽  
Vol 12 (3) ◽  
pp. 425-434 ◽  
Author(s):  
Michael Woodroofe ◽  
Bruce Hill
Keyword(s):  
Probability Distribution ◽  
Large Class ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Zipf’S Law ◽  
Zipf's Law ◽  
Necessary And Sufficient ◽  
Positive Integers

A Zipf's law is a probability distribution on the positive integers which decays algebraically. Such laws describe (approximately) a large class of phenomena. We formulate a model for such phenomena and, in terms of our model, give necessary and sufficient conditions for a Zipf's law to hold.

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.

Hendricks libraries and Tsetlin piles

1982 ◽  
Vol 14 (01) ◽  
pp. 37-55 ◽  
Author(s):  
Jacques-Edouard Dies
Keyword(s):  
Markov Chains ◽  
Large Class ◽  
Special Class ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stationary Measure ◽  
Sufficient Condition ◽  
Positive Recurrence ◽  
Necessary And Sufficient

In order to study the transience of Hendricks libraries, we introduce and study a special class of Markov chains, the Tsetlin d-piles, generalizing Tsetlin libraries and briefly defined as follows: a 1-pile is a Tsetlin library and a d-pile is a Tsetlin library where each book is replaced by a (d − 1)-pile. We give a stationary measure of these chains and establish the necessary and sufficient conditions for positive recurrence and transience. Finally, the study of d-piles allows us to determine a sufficient condition for transience of quite a large class of Hendricks libraries.

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