We extend the asymptotic formula given by Eskin,Mozes and Shah in 1996 for counting integral matrices with a given irreducible characteristic polynomial to the case of counting elements in a maximal order of a certain...We extend the asymptotic formula given by Eskin,Mozes and Shah in 1996 for counting integral matrices with a given irreducible characteristic polynomial to the case of counting elements in a maximal order of a certain central simple algebra with a given irreducible characteristic polynomial.展开更多
Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A?F[D] from any pair of a commutative associative algebra,A with an identity element...Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A?F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras.展开更多
In this paper, through a meticulous description of finite root system,a concrete comultiplication with an explicit action on the basis elements of finitedimensional simple Lie algebras of type A; D; E is constructed. ...In this paper, through a meticulous description of finite root system,a concrete comultiplication with an explicit action on the basis elements of finitedimensional simple Lie algebras of type A; D; E is constructed. Then any finitedimensional simple Lie algebra of type A; D; E is endowed with a new generalizedLie coalgebra splitting. This construction verifies the known existence of a co-splitLie structure on any finite dimensional complex simple Lie algebra.展开更多
Let g be a complex simple Lie algebra of rank ι, b the standard Borel subalgebra. An invertible map on Ь is said to preserve abelian ideals if it maps each abelian ideal to some such ideal of the same dimension. In ...Let g be a complex simple Lie algebra of rank ι, b the standard Borel subalgebra. An invertible map on Ь is said to preserve abelian ideals if it maps each abelian ideal to some such ideal of the same dimension. In this article, by using some results of Chevalley groups, the theory of root systems and root space decomposition, the author gives an explicit description on such maps of Ь.展开更多
A map φ on a Lie algebra g is called to be commuting if [φ(x), x] = 0 for all x ∈ g. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgeb...A map φ on a Lie algebra g is called to be commuting if [φ(x), x] = 0 for all x ∈ g. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, we prove that a linear mapφon P is commuting if and only if φ is a scalar multiplication map on P.展开更多
The third order explicit autonomous differential equations named as jerk equations represent an interesting subclass of dynamical systems that can exhibit many major features of the regular and chaotic motion. In this...The third order explicit autonomous differential equations named as jerk equations represent an interesting subclass of dynamical systems that can exhibit many major features of the regular and chaotic motion. In this paper, we show that an algebraically simple system, the Genesio system can be recast into a jerky dynamics and its jerk equation can be derived from one-dimensional Newtonian equation. We also investigate the global dynamical properties of the corresponding jerk system.展开更多
An algeber L is said to be simple, if its congruence lattice Con L reduces to the 2-elementchain {ω,v}. This paper describes the structure of finite simple Ockham algebras.
Abstract Let n ≥ 3. The complex Lie algebra, which is attached to a unit form xixj and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A~, and r...Abstract Let n ≥ 3. The complex Lie algebra, which is attached to a unit form xixj and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A~, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.展开更多
For any finitely generated unital commutative associative algebra R over C and any complex finite-dimensional simple Lie algebra g with a fixed Cartan subalgebra h,we classify all g⊗R-modules on U(h)such that h as a s...For any finitely generated unital commutative associative algebra R over C and any complex finite-dimensional simple Lie algebra g with a fixed Cartan subalgebra h,we classify all g⊗R-modules on U(h)such that h as a subalgebra of g⊗R,acts on U(h)by the multiplication.We construct these modules explicitly and study their module structures.展开更多
The author constructs a class of indecomposable non-degenerate solvable Lie Algebras corresponding to a Cartan matrix A over the field of complex numbers and we determine all their derivations.
Let F be an algebracially closed field of characteristic p】2, and L be the p<sup>n</sup>-dimensional Zassenhaus algebra with the maximal invariant subalgebra L<sub>0</sub> and the standard fil...Let F be an algebracially closed field of characteristic p】2, and L be the p<sup>n</sup>-dimensional Zassenhaus algebra with the maximal invariant subalgebra L<sub>0</sub> and the standard filtration {L<sub>i</sub>}|<sub>i=-1</sub><sup>p<sup>n</sup>-2</sup>. Then the number of isomorphism classes of simple L-modules is equal to that of simple L<sub>0</sub>-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than p<sup>n</sup>.展开更多
We find a new representation of the simple Lie algebra of type E6 on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this r...We find a new representation of the simple Lie algebra of type E6 on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen＇s idea of mixed product, we construct a new functor from Ds-Mod to E6-Mod. A condition for the functor to map a finite-dimensional irreducible Ds-module to an infinite-dimensional irreducible E6-module is obtained. Our results yield explicit constructions of certain infinite-dimensional irreducible weight E6-modules with finite-dimensional weight subspaces. In our approach, the idea of Kostant＇s characteristic identities plays a key role.展开更多
By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra E7 into a sum of irreducible s...By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra E7 into a sum of irreducible submodules. This essentially gives a partial differential equation proof of a combinatorial identity on the dimensions of certain irreducible modules of E7. We also determine two three-parameter families of irreducible submodules in the solution space of Cartan＇s well-known fourth-order Ez-invariant partial differential equation.展开更多
The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytic...The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors.We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors.We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue.We verify our results via random numerical examples.展开更多
A monomial basis and a filtration of subalgebras for the universal enveloping algebra U（gl） of a complex simple Lie algebra gl of type Bl and Cl are given, and the decomposition of the Weyl module V （λ） as a U（g...A monomial basis and a filtration of subalgebras for the universal enveloping algebra U（gl） of a complex simple Lie algebra gl of type Bl and Cl are given, and the decomposition of the Weyl module V （λ） as a U（gl）-module into a direct sum of Weyl modules V （μ）’s as U（gl-1）modules is described. In particular, a new multiplicity formula for the Weyl module V （λ） is obtained in this note.展开更多
文摘We extend the asymptotic formula given by Eskin,Mozes and Shah in 1996 for counting integral matrices with a given irreducible characteristic polynomial to the case of counting elements in a maximal order of a certain central simple algebra with a given irreducible characteristic polynomial.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19801037) a Fund from National Education Ministry of China.
文摘Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A?F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras.
基金The Anhui Province College Excellent Young Talents Fund(2013SQRL071ZD)
文摘In this paper, through a meticulous description of finite root system,a concrete comultiplication with an explicit action on the basis elements of finitedimensional simple Lie algebras of type A; D; E is constructed. Then any finitedimensional simple Lie algebra of type A; D; E is endowed with a new generalizedLie coalgebra splitting. This construction verifies the known existence of a co-splitLie structure on any finite dimensional complex simple Lie algebra.
基金Supported by the Doctor Foundation of Henan Polytechnic University(B2010-93)Supported by the National Natural Science Foundation of China(11126121)+2 种基金Supported by the Natural Science Foundation of Henan Province(112300410120)Supported by the Natural Science Research Program of Education Department of Henan Province(201lB110016)Supported by the Applied Mathematics Provincial-level Key Discipline of Henan Province of Henau Polytechuic University
文摘Let g be a complex simple Lie algebra of rank ι, b the standard Borel subalgebra. An invertible map on Ь is said to preserve abelian ideals if it maps each abelian ideal to some such ideal of the same dimension. In this article, by using some results of Chevalley groups, the theory of root systems and root space decomposition, the author gives an explicit description on such maps of Ь.
基金Supported by the National Natural Science Foundation of China（Ill01084） Supported by the Fujian Province Natural Science Foundation of China
文摘A map φ on a Lie algebra g is called to be commuting if [φ(x), x] = 0 for all x ∈ g. Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, we prove that a linear mapφon P is commuting if and only if φ is a scalar multiplication map on P.
文摘The third order explicit autonomous differential equations named as jerk equations represent an interesting subclass of dynamical systems that can exhibit many major features of the regular and chaotic motion. In this paper, we show that an algebraically simple system, the Genesio system can be recast into a jerky dynamics and its jerk equation can be derived from one-dimensional Newtonian equation. We also investigate the global dynamical properties of the corresponding jerk system.
文摘An algeber L is said to be simple, if its congruence lattice Con L reduces to the 2-elementchain {ω,v}. This paper describes the structure of finite simple Ockham algebras.
文摘Abstract Let n ≥ 3. The complex Lie algebra, which is attached to a unit form xixj and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A~, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.
基金supported by the NSF of China(11931009,12161141001,12171132 and 11771410)Innovation Program for Quantum Science and Technology(2021ZD0302902).
文摘For any finitely generated unital commutative associative algebra R over C and any complex finite-dimensional simple Lie algebra g with a fixed Cartan subalgebra h,we classify all g⊗R-modules on U(h)such that h as a subalgebra of g⊗R,acts on U(h)by the multiplication.We construct these modules explicitly and study their module structures.
文摘The author constructs a class of indecomposable non-degenerate solvable Lie Algebras corresponding to a Cartan matrix A over the field of complex numbers and we determine all their derivations.
基金Supported in part by the National Natural Science Foundation of China Grant 19801022the Scientifictechnological Major Project of Educational Ministry of China, Grant 99036.
文摘Let F be an algebracially closed field of characteristic p】2, and L be the p<sup>n</sup>-dimensional Zassenhaus algebra with the maximal invariant subalgebra L<sub>0</sub> and the standard filtration {L<sub>i</sub>}|<sub>i=-1</sub><sup>p<sup>n</sup>-2</sup>. Then the number of isomorphism classes of simple L-modules is equal to that of simple L<sub>0</sub>-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than p<sup>n</sup>.
基金Supported by NSFC(Grant Nos.11171324 and 11321101)
文摘We find a new representation of the simple Lie algebra of type E6 on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen＇s idea of mixed product, we construct a new functor from Ds-Mod to E6-Mod. A condition for the functor to map a finite-dimensional irreducible Ds-module to an infinite-dimensional irreducible E6-module is obtained. Our results yield explicit constructions of certain infinite-dimensional irreducible weight E6-modules with finite-dimensional weight subspaces. In our approach, the idea of Kostant＇s characteristic identities plays a key role.
基金Supported by NSFC(Grant Nos.11171324 and 11321101)
文摘By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra E7 into a sum of irreducible submodules. This essentially gives a partial differential equation proof of a combinatorial identity on the dimensions of certain irreducible modules of E7. We also determine two three-parameter families of irreducible submodules in the solution space of Cartan＇s well-known fourth-order Ez-invariant partial differential equation.
基金the National Natural Science Foundation of China(No.11271084)International Cooperation Project of Shanghai Municipal Science and Technology Commission(No.16510711200).
文摘The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors.We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors.We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue.We verify our results via random numerical examples.
基金Supported by the National Natural Science Foundation of China (Grant No.10671142)
文摘A monomial basis and a filtration of subalgebras for the universal enveloping algebra U（gl） of a complex simple Lie algebra gl of type Bl and Cl are given, and the decomposition of the Weyl module V （λ） as a U（gl）-module into a direct sum of Weyl modules V （μ）’s as U（gl-1）modules is described. In particular, a new multiplicity formula for the Weyl module V （λ） is obtained in this note.
