It is known that there exists an isogeny sort of Chevalley groups G (Σ, F) associated to any indecomposable root system Σ and any field F . In this paper the author determines all nontrivial homomorphi...It is known that there exists an isogeny sort of Chevalley groups G (Σ, F) associated to any indecomposable root system Σ and any field F . In this paper the author determines all nontrivial homomorphisms from G(Σ, k) to G(Σ, K) when the root system Σ is of type C n or G 2 , and the fields k and K are finite fields of characteristic p .展开更多
The main objective of this paper is to provide the tool rather than the classical adjoint representation of Lie algebra;which is essential in the conception of the Chevalley Eilenberg Cohomology. We introduce the noti...The main objective of this paper is to provide the tool rather than the classical adjoint representation of Lie algebra;which is essential in the conception of the Chevalley Eilenberg Cohomology. We introduce the notion of representation induced by a 2 - 3 matrix. We construct the corresponding Chevalley Eilenberg differential and we compute all its cohomological groups.展开更多
In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf sub- algebra. For a non-cosemisimple Hopf algebra ...In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf sub- algebra. For a non-cosemisimple Hopf algebra A with the Chevalley property, if its diagram is a Nichols algebra, then the diagram of its maximal pointed Hopf subalgebra is also a Nichols algebra. When A is of finite dimension, we provide a necessary and sufficient condition for A＇s diagram equaling the diagram of its maximal pointed Hopf subalgebra.展开更多
The author studies the linkage between the standardness and the standard automorphisms of ChevMley groups over rings. It is proved that if H is any standard subgroup of G（R）, then each of its automorphisms can be ex...The author studies the linkage between the standardness and the standard automorphisms of ChevMley groups over rings. It is proved that if H is any standard subgroup of G（R）, then each of its automorphisms can be extended to an automorphism of G（R, I）, restricted to an automorphism of E（R, I）, and an automorphism of E（R, I） can be extended to one of G（R, I）. The case of Chevalley groups of rank at least two is treated in this paper. Further results about the case of Chevalley groups of rank one, the case of nomcommutative ground ring and some others exceptions will appear elsewhere.展开更多
This is a pedagogical introduction to the theory of buildings of Jacques Tits and to some applications of this theory.This paper has 4 parts.In the first part we discuss incidence geometry,Coxeter systems and give two...This is a pedagogical introduction to the theory of buildings of Jacques Tits and to some applications of this theory.This paper has 4 parts.In the first part we discuss incidence geometry,Coxeter systems and give two definitions of buildings.We study in the second part the spherical and affine buildings of Chevalley groups.In the third part we deal with Bruhat-Tits theory of reductive groups over local fields.Finally we discuss the construction of the p-adic flag manifolds.展开更多
An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical.Such a semi-reductive algebraic group naturally arises and also plays a key role in the stud...An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical.Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic,and some other cases.Let G be a connected semi-reductive algebraic group over an algebraically closed field F and g=Lie(G).It turns out that G has many same properties as reductive groups,such as the Bruhat decomposition.In this note,we obtain an analogue of classical Chevalley restriction theorem for g,which says that the G-invariant ring F[g]~G is a polynomial ring if g satisfies a certain“positivity”condition suited for lots of cases we are interested in.As applications,we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.展开更多
This is the second part of a pedagogical introduction to the theory of buildings of Jacques Tits.We de ne a(B,N)pair and construct a building out of it.Then we give a description of Chevalley groups,their(B,N)pair and...This is the second part of a pedagogical introduction to the theory of buildings of Jacques Tits.We de ne a(B,N)pair and construct a building out of it.Then we give a description of Chevalley groups,their(B,N)pair and the associated buildings.We illustrates this theory with many examples from classical groups.展开更多
Let L be a simple Lie algebra with irreducible root system. having roots of different length, F be a field of charaCteristic different from 2, G = L(F) be a Chevalley group of type L over F. Denote by Φl the set of a...Let L be a simple Lie algebra with irreducible root system. having roots of different length, F be a field of charaCteristic different from 2, G = L(F) be a Chevalley group of type L over F. Denote by Φl the set of all long roots in Φl Set Gl = <xr (t); f∈EΦl,t∈F>. It is a subgroup of G generated by all the long root subgroups. This paper determines the pronormality of Gl in G when L is not of type G2.展开更多
The Harish-Chandra homomorphism for the higher congruence spherical functions algebra of Chevalley groups over p-adic fiealds is given in the case of the Levi-component of a (rational)parabolic subgroup. It is a gener...The Harish-Chandra homomorphism for the higher congruence spherical functions algebra of Chevalley groups over p-adic fiealds is given in the case of the Levi-component of a (rational)parabolic subgroup. It is a generalization for the Harish-Chandra homomorphism for the higher congurence spherical functions algebra of the groups CLn over p-adic field in the same case.展开更多
文摘It is known that there exists an isogeny sort of Chevalley groups G (Σ, F) associated to any indecomposable root system Σ and any field F . In this paper the author determines all nontrivial homomorphisms from G(Σ, k) to G(Σ, K) when the root system Σ is of type C n or G 2 , and the fields k and K are finite fields of characteristic p .
文摘The main objective of this paper is to provide the tool rather than the classical adjoint representation of Lie algebra;which is essential in the conception of the Chevalley Eilenberg Cohomology. We introduce the notion of representation induced by a 2 - 3 matrix. We construct the corresponding Chevalley Eilenberg differential and we compute all its cohomological groups.
基金Supported by the National Natural Science Foundation of China(11271319,11301126)
文摘In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf sub- algebra. For a non-cosemisimple Hopf algebra A with the Chevalley property, if its diagram is a Nichols algebra, then the diagram of its maximal pointed Hopf subalgebra is also a Nichols algebra. When A is of finite dimension, we provide a necessary and sufficient condition for A＇s diagram equaling the diagram of its maximal pointed Hopf subalgebra.
文摘The author studies the linkage between the standardness and the standard automorphisms of ChevMley groups over rings. It is proved that if H is any standard subgroup of G（R）, then each of its automorphisms can be extended to an automorphism of G（R, I）, restricted to an automorphism of E（R, I）, and an automorphism of E（R, I） can be extended to one of G（R, I）. The case of Chevalley groups of rank at least two is treated in this paper. Further results about the case of Chevalley groups of rank one, the case of nomcommutative ground ring and some others exceptions will appear elsewhere.
文摘This is a pedagogical introduction to the theory of buildings of Jacques Tits and to some applications of this theory.This paper has 4 parts.In the first part we discuss incidence geometry,Coxeter systems and give two definitions of buildings.We study in the second part the spherical and affine buildings of Chevalley groups.In the third part we deal with Bruhat-Tits theory of reductive groups over local fields.Finally we discuss the construction of the p-adic flag manifolds.
基金Supported by NSFC(Grant Nos.12071136,11671138,11771279,12101544)Shanghai Key Laboratory of PMMP(Grant No.13dz2260400)the Fundamental Research Funds of Yunnan Province(Grant No.2020J0375)。
文摘An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical.Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic,and some other cases.Let G be a connected semi-reductive algebraic group over an algebraically closed field F and g=Lie(G).It turns out that G has many same properties as reductive groups,such as the Bruhat decomposition.In this note,we obtain an analogue of classical Chevalley restriction theorem for g,which says that the G-invariant ring F[g]~G is a polynomial ring if g satisfies a certain“positivity”condition suited for lots of cases we are interested in.As applications,we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.
文摘This is the second part of a pedagogical introduction to the theory of buildings of Jacques Tits.We de ne a(B,N)pair and construct a building out of it.Then we give a description of Chevalley groups,their(B,N)pair and the associated buildings.We illustrates this theory with many examples from classical groups.
基金Project supported by the National Natural Science Foundation of China (No.19671079).
文摘Let L be a simple Lie algebra with irreducible root system. having roots of different length, F be a field of charaCteristic different from 2, G = L(F) be a Chevalley group of type L over F. Denote by Φl the set of all long roots in Φl Set Gl = <xr (t); f∈EΦl,t∈F>. It is a subgroup of G generated by all the long root subgroups. This paper determines the pronormality of Gl in G when L is not of type G2.
文摘The Harish-Chandra homomorphism for the higher congruence spherical functions algebra of Chevalley groups over p-adic fiealds is given in the case of the Levi-component of a (rational)parabolic subgroup. It is a generalization for the Harish-Chandra homomorphism for the higher congurence spherical functions algebra of the groups CLn over p-adic field in the same case.
