ı-quantum groups,arising from quantum symmetric pairs,are coideal subalgebras of quantum groups.ı-quantum groups are a vast generalization of quantum groups,as quantum groups can be viewed asıquantum groups of diagona...ı-quantum groups,arising from quantum symmetric pairs,are coideal subalgebras of quantum groups.ı-quantum groups are a vast generalization of quantum groups,as quantum groups can be viewed asıquantum groups of diagonal type.Recently,the braid group symmetries and Drinfeld new presentations of quantum groups have been generalized to affineı-quantum groups.In this paper,we construct PBW type bases for splitı-quantum groups of type ADE,based on their braid group symmetries and Drinfeld new presentations.This can be viewed as anı-analogue of the PBW-basis for affine quantum groups,and it generalizes the PBW-basis ofı-quantum groups of finite type.展开更多
设S=(?)S_n一是-Z分次环,X是S的中心中一个次为1的正则齐次元.那么下列结论成立:(a)Sr的商环A=S/(1—X)S是一个滤过环,A上的滤(升)链定义为F_nA=S_n+(1—X)S/(1—X)S,n∈Z;(b)与A相关联的分次环G(A)=(?)F_nA/F_(n-1)A与 S/XS之间有一个...设S=(?)S_n一是-Z分次环,X是S的中心中一个次为1的正则齐次元.那么下列结论成立:(a)Sr的商环A=S/(1—X)S是一个滤过环,A上的滤(升)链定义为F_nA=S_n+(1—X)S/(1—X)S,n∈Z;(b)与A相关联的分次环G(A)=(?)F_nA/F_(n-1)A与 S/XS之间有一个显然的分次环同构;(c)A的Rees环(?)=(?)F_nA与S之间有一个显然的分次环同构.设R=(?)R_n是一个Z-分次环,那么R的外齐次化是R上的多项式环S=R[t]但此对S具有“混合分次”:S_n={sum from i+j=n to (α_it^j),α_i∈R_i},n∈Z.显然t是S中的一个次为1的中心正则齐次元,但此时S的商环A=S/(1-t)S作为滤过环同构于R,这里R具有(升)滤链F_nR=(?)R_i,n∈Z,G(A)(?)展开更多
A class of two-parameter quantum algebras Ur,s（s[（m｜n）） is constructed. It is shown that Ur,s（s[（m｜n）） is a Hopf superalgebra. Then the PBW basis of Ur,s（s[（m｜n）） is described. For this purpose, some co...A class of two-parameter quantum algebras Ur,s（s[（m｜n）） is constructed. It is shown that Ur,s（s[（m｜n）） is a Hopf superalgebra. Then the PBW basis of Ur,s（s[（m｜n）） is described. For this purpose, some commutative relations of root vectors of U^＋r,s（s[（m｜n）） are given展开更多
The aim of this paper is to investigate different radicals(Wedderburn radical,lower nil radical,Levitzky radical,upper nil radical,the set of all nilpotent elements,the sum of all nil left ideals)of the noncommutative...The aim of this paper is to investigate different radicals(Wedderburn radical,lower nil radical,Levitzky radical,upper nil radical,the set of all nilpotent elements,the sum of all nil left ideals)of the noncommutative rings known as skew Poincare–Birkhoff–Witt extensions.We characterize minimal prime ideals of these rings and prove that the Kothe’s conjecture holds for these extensions.Finally,we establish the transfer of several ring-theoretical properties(reduced,symmetric,reversible,2-primal)from the coefficients ring of a skew PBW extension to the extension itself.展开更多
In this paper,we discuss some open problems of non-commutative algebra and noncommutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings.More exactly,we will analyze the isomorphism ...In this paper,we discuss some open problems of non-commutative algebra and noncommutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings.More exactly,we will analyze the isomorphism arising in the investigation of the Gelfand–Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain.The Serre’s conjecture will be discussed for a particular class of skew PBW extensions.The questions about the Noetherianity and the Zariski cancellation property of Artin–Schelter regular algebras will be reformulated for semi-graded rings.Advances for the solution of some of the problems are included.展开更多
Poincaré-Birkhoff-Witt(PBW)deformations of Artin-Schelter regular algebras are skew CalabiYau.We prove that the Nakayama automorphisms of such PBW deformations can be obtained from their homogenizations.Some Cala...Poincaré-Birkhoff-Witt(PBW)deformations of Artin-Schelter regular algebras are skew CalabiYau.We prove that the Nakayama automorphisms of such PBW deformations can be obtained from their homogenizations.Some Calabi-Yau properties are generalized without Koszul assumption.We also show that the Nakayama automorphisms of such PBW deformations control Hopf actions on them.展开更多
文摘ı-quantum groups,arising from quantum symmetric pairs,are coideal subalgebras of quantum groups.ı-quantum groups are a vast generalization of quantum groups,as quantum groups can be viewed asıquantum groups of diagonal type.Recently,the braid group symmetries and Drinfeld new presentations of quantum groups have been generalized to affineı-quantum groups.In this paper,we construct PBW type bases for splitı-quantum groups of type ADE,based on their braid group symmetries and Drinfeld new presentations.This can be viewed as anı-analogue of the PBW-basis for affine quantum groups,and it generalizes the PBW-basis ofı-quantum groups of finite type.
文摘设S=(?)S_n一是-Z分次环,X是S的中心中一个次为1的正则齐次元.那么下列结论成立:(a)Sr的商环A=S/(1—X)S是一个滤过环,A上的滤(升)链定义为F_nA=S_n+(1—X)S/(1—X)S,n∈Z;(b)与A相关联的分次环G(A)=(?)F_nA/F_(n-1)A与 S/XS之间有一个显然的分次环同构;(c)A的Rees环(?)=(?)F_nA与S之间有一个显然的分次环同构.设R=(?)R_n是一个Z-分次环,那么R的外齐次化是R上的多项式环S=R[t]但此对S具有“混合分次”:S_n={sum from i+j=n to (α_it^j),α_i∈R_i},n∈Z.显然t是S中的一个次为1的中心正则齐次元,但此时S的商环A=S/(1-t)S作为滤过环同构于R,这里R具有(升)滤链F_nR=(?)R_i,n∈Z,G(A)(?)
基金The second author was supported by Beijing Natural Science Foundation （Grant No. 1122006）, the Specialized Research Fund for the Doctoral Program of Higher Education of China （Grant No. 20111103110011） and partially supported by the National Natural Science Foundation of China （Grant No. 11271043, 11471186）.
文摘A class of two-parameter quantum algebras Ur,s（s[（m｜n）） is constructed. It is shown that Ur,s（s[（m｜n）） is a Hopf superalgebra. Then the PBW basis of Ur,s（s[（m｜n）） is described. For this purpose, some commutative relations of root vectors of U^＋r,s（s[（m｜n）） are given
基金The first author was supported by the research fund of Facultad de Ciencias,Code HERMES 41535,Universidad Nacional de Colombia,Bogota,Colombia。
文摘The aim of this paper is to investigate different radicals(Wedderburn radical,lower nil radical,Levitzky radical,upper nil radical,the set of all nilpotent elements,the sum of all nil left ideals)of the noncommutative rings known as skew Poincare–Birkhoff–Witt extensions.We characterize minimal prime ideals of these rings and prove that the Kothe’s conjecture holds for these extensions.Finally,we establish the transfer of several ring-theoretical properties(reduced,symmetric,reversible,2-primal)from the coefficients ring of a skew PBW extension to the extension itself.
文摘In this paper,we discuss some open problems of non-commutative algebra and noncommutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings.More exactly,we will analyze the isomorphism arising in the investigation of the Gelfand–Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain.The Serre’s conjecture will be discussed for a particular class of skew PBW extensions.The questions about the Noetherianity and the Zariski cancellation property of Artin–Schelter regular algebras will be reformulated for semi-graded rings.Advances for the solution of some of the problems are included.
基金supported by National Natural Science Foundation of China(Grant No.11271319)
文摘Poincaré-Birkhoff-Witt(PBW)deformations of Artin-Schelter regular algebras are skew CalabiYau.We prove that the Nakayama automorphisms of such PBW deformations can be obtained from their homogenizations.Some Calabi-Yau properties are generalized without Koszul assumption.We also show that the Nakayama automorphisms of such PBW deformations control Hopf actions on them.
