In the Ringel-Hall algebra of Dynkin type,the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Grobner-Shirshov basis and the set of the corresponding irreducibl...In the Ringel-Hall algebra of Dynkin type,the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Grobner-Shirshov basis and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel-Hall algebra.We aim to generalize this result to the reduced Drinfeld double Hall algebra of type A_(n).First,we compute a minimal Grobner-Shirshov basis for the reduced Drinfeld double Hall algebra of type An by proving that all possible compositions between the commutator relations are trivial.Then,by taking the corresponding irreducible monomials,we construct a PBW-type basis for the reduced Drinfeld double Hall algebra of type A_(n).展开更多
We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Grobner-Shirshov bases method for Lie algebras to solve the i...We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Grobner-Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras （Lie di-algebras）. As another application, we prove an analogue of the Poincare-Birkhoff-Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra （CTD!-algebra）.展开更多
In this paper, by using PBW bases for the twisted generic composition algebras of affine type, we prove that the set of the skew-commutator relations of the iso-classes of indecomposable representations forms a minima...In this paper, by using PBW bases for the twisted generic composition algebras of affine type, we prove that the set of the skew-commutator relations of the iso-classes of indecomposable representations forms a minimal GrSbner-Shirshov basis for the twisted generic composition algebras of affine type.展开更多
Let R be a ring with identity 1. Jacobson＇s lemma states that for any a, b ∈ R, if 1 - ab is invertible then so is 1 - ba. Jacobson＇s lemma has suitable analogues for several types of generalized inverses, e.g., Dr...Let R be a ring with identity 1. Jacobson＇s lemma states that for any a, b ∈ R, if 1 - ab is invertible then so is 1 - ba. Jacobson＇s lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Grobner-Shirshov basis theory to obtain the inverse of 1 - ab in terms of （1 - ba）-1, assuming the latter exists.展开更多
In this paper,by using the Anick resolution and Grobner-Shirshov basis for quantized enveloping algebra of type B2,we compute the minimal projective resolution of the trivial module of Uq^+(B2),and as an application w...In this paper,by using the Anick resolution and Grobner-Shirshov basis for quantized enveloping algebra of type B2,we compute the minimal projective resolution of the trivial module of Uq^+(B2),and as an application we compute the global dimension of Uq^+(B2).展开更多
For exceptional Lie superalgebras D（2, 1; α）, G（3） and F（4）, GrSbner-Shirshov bases and the corresponding super-Lyndon-Shirshov monomial bases are constructed.
In this paper,we give a Grobner-Shirshov basis of quantum group of type C3 by using the Ringel-Hall algebra approach.For this,first we compute all skew-commutator relations between the isoclasses of indecomposable rep...In this paper,we give a Grobner-Shirshov basis of quantum group of type C3 by using the Ringel-Hall algebra approach.For this,first we compute all skew-commutator relations between the isoclasses of indecomposable reprersentations of Ringel-Hall algebras of type C3 by using an“inductive”method.Precisely,we do not use the traditional way of computing the skew-commutative relations,that is first compute all Hall polynomials then compute the corresponding skew-commutator relations;contrarily,we compute the“easier”skew-commutator relations which corresponding to those exact sequences with middile term indecomposable or the split exact sequences first,then“inductive”others from these“easier”ones and this in turn gives Hall polynomials as a byproduct.Then we prove that the set of these relations is closed under composition.So they constitutes a minimal Grobner-Shirshov basis of the positive part of quantum group of type C3.Dually,we get a Grobner-Shirshov basis of the negative part of quantum group of type C3.And finally we give a Grobner-Shirshov basis for the whole quantum group of type C3.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11861061).
文摘In the Ringel-Hall algebra of Dynkin type,the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Grobner-Shirshov basis and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel-Hall algebra.We aim to generalize this result to the reduced Drinfeld double Hall algebra of type A_(n).First,we compute a minimal Grobner-Shirshov basis for the reduced Drinfeld double Hall algebra of type An by proving that all possible compositions between the commutator relations are trivial.Then,by taking the corresponding irreducible monomials,we construct a PBW-type basis for the reduced Drinfeld double Hall algebra of type A_(n).
基金Supported by NSFC (Nos.11771190,1861051,12071191)the Natural Science Foundation of Gansu Province (No.20JR5RA249)the Natural Science Foundation of Shandong Province (No.ZR2020MA002)。
文摘We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Grobner-Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras （Lie di-algebras）. As another application, we prove an analogue of the Poincare-Birkhoff-Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra （CTD!-algebra）.
基金Supported by National Science Foundation of China （Grant No. 11061033, 11361056）.
文摘In this paper, by using PBW bases for the twisted generic composition algebras of affine type, we prove that the set of the skew-commutator relations of the iso-classes of indecomposable representations forms a minimal GrSbner-Shirshov basis for the twisted generic composition algebras of affine type.
文摘Let R be a ring with identity 1. Jacobson＇s lemma states that for any a, b ∈ R, if 1 - ab is invertible then so is 1 - ba. Jacobson＇s lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Grobner-Shirshov basis theory to obtain the inverse of 1 - ab in terms of （1 - ba）-1, assuming the latter exists.
文摘In this paper,by using the Anick resolution and Grobner-Shirshov basis for quantized enveloping algebra of type B2,we compute the minimal projective resolution of the trivial module of Uq^+(B2),and as an application we compute the global dimension of Uq^+(B2).
文摘For exceptional Lie superalgebras D（2, 1; α）, G（3） and F（4）, GrSbner-Shirshov bases and the corresponding super-Lyndon-Shirshov monomial bases are constructed.
基金This paper is supported by the National Natural Science Foundation of China(No.11061033).
文摘In this paper,we give a Grobner-Shirshov basis of quantum group of type C3 by using the Ringel-Hall algebra approach.For this,first we compute all skew-commutator relations between the isoclasses of indecomposable reprersentations of Ringel-Hall algebras of type C3 by using an“inductive”method.Precisely,we do not use the traditional way of computing the skew-commutative relations,that is first compute all Hall polynomials then compute the corresponding skew-commutator relations;contrarily,we compute the“easier”skew-commutator relations which corresponding to those exact sequences with middile term indecomposable or the split exact sequences first,then“inductive”others from these“easier”ones and this in turn gives Hall polynomials as a byproduct.Then we prove that the set of these relations is closed under composition.So they constitutes a minimal Grobner-Shirshov basis of the positive part of quantum group of type C3.Dually,we get a Grobner-Shirshov basis of the negative part of quantum group of type C3.And finally we give a Grobner-Shirshov basis for the whole quantum group of type C3.
