Let F be a field and char F = p > 3. In this paper the derivation algebras of Lie superalgebras W and S of Cartan-type over F are determined by the calculating method.
In this paper, we explicitly determine the maximal torus of the derivation algebra of a Qn filiform Lie algebra. Using the root space decomposition of DerQn, we prove that the derivation algebra of a Qn filiform Lie a...In this paper, we explicitly determine the maximal torus of the derivation algebra of a Qn filiform Lie algebra. Using the root space decomposition of DerQn, we prove that the derivation algebra of a Qn filiform Lie algebra is complete.展开更多
In this paper we explicitly determine automorphism group of filiform Lie algebra Rn to find the indecomposable solvable Lie algebras with filiform Lie algebra Rn nilradicals.We also prove that the indecomposable solva...In this paper we explicitly determine automorphism group of filiform Lie algebra Rn to find the indecomposable solvable Lie algebras with filiform Lie algebra Rn nilradicals.We also prove that the indecomposable solvable Lie algebras with filiform Rn nilradicals is complete.展开更多
In this paper we explicitly determine the derivation algebra of a quasi Rn-filiform Lie algebra and prove that a quasi Rn-filiform Lie algebra is a completable nilpotent Lie algebra.
In this paper the derivation algebra of Lie algebra Σof characteristic two is determined. Using this result we obtain the necessary and sufficient condition under that Σ of characteristic two is the restreted Lie al...In this paper the derivation algebra of Lie algebra Σof characteristic two is determined. Using this result we obtain the necessary and sufficient condition under that Σ of characteristic two is the restreted Lie algebra. Finally we prove that Σ of characteristic two isn't isomorphic the Lie algebras of characteristic two which are known by authors of this paper. But it still is a filtered deformation of the Lie algebra of H-type i.e. Its associated grated algebra GrΣ is isomorphic to H(2n + 2.r).展开更多
In this paper, we study a class of subalgebras of the Lie algebra of vector fields on n-dimensional torus, which are called the Triangular derivation Lie algebra. We give the structure and the central extension of Tri...In this paper, we study a class of subalgebras of the Lie algebra of vector fields on n-dimensional torus, which are called the Triangular derivation Lie algebra. We give the structure and the central extension of Triangular derivation Lie algebra.展开更多
In this paper, the derivation algebra of Lie superalgebra H of Caftan-type over F are determined by the calculating method in the situations of CharF = p ≥ 3 or m ≥ 2 or n ≥ 1. The main result is following： DerFH ...In this paper, the derivation algebra of Lie superalgebra H of Caftan-type over F are determined by the calculating method in the situations of CharF = p ≥ 3 or m ≥ 2 or n ≥ 1. The main result is following： DerFH = adH（H＂ ＋ Fh） （{（adDi）^pt ｜ i = 1,2,…,m, t=1,2,…,ti-1}）.展开更多
Let F be a field, n ≥ 3, N（n,F） the strictly upper triangular matrix Lie algebra consisting of the n × n strictly upper triangular matrices and with the bracket operation {x, y} = xy-yx. A linear map φ on N（...Let F be a field, n ≥ 3, N（n,F） the strictly upper triangular matrix Lie algebra consisting of the n × n strictly upper triangular matrices and with the bracket operation {x, y} = xy-yx. A linear map φ on N（n,F） is said to be a product zero derivation if {φ（x）,y] ＋ [x, φ（y）] = 0 whenever {x, y} = 0,x,y ∈ N（n,F）. In this paper, we prove that a linear map on N（n, F） is a product zero derivation if and only if φ is a sum of an inner derivation, a diagonal derivation, an extremal product zero derivation, a central product zero derivation and a scalar multiplication map on N（n, F）.展开更多
Let t be a positive integer and A be a hereditary abelian category satisfying some finiteness conditions.We define the semi-derived Ringel-Hall algebra of A from the category C_(Z/t)(A)of Z/t-graded complexes and obta...Let t be a positive integer and A be a hereditary abelian category satisfying some finiteness conditions.We define the semi-derived Ringel-Hall algebra of A from the category C_(Z/t)(A)of Z/t-graded complexes and obtain a natural basis of the semi-derived Ringel-Hall algebra.Moreover,we describe the semiderived Ringel-Hall algebra by the generators and defining relations.In particular,if t is an odd integer,we show an embedding of the derived Hall algebra of the odd-periodic relative derived category in the extended semi-derived Ringel-Hall algebra.展开更多
We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis,and the corresponding irreducible el...We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis,and the corresponding irreducible elements forms a PBW type basis of the Ringel-Hall algebra.We aim to generalize this result to the derived Hall algebra DH(An)of type An.First,we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category D^b(An)using the Auslander-Reiten quiver of D^b(An),and then we prove that all possible compositions between these skew commutator relations are trivial.As an application,we give a PBW type basis of DH(An).展开更多
Let R be an arbitrary commutative ring with identity. Denote by t the Lie algebra over R consisting of all upper triangular n by n matrices and let b be the Lie subalgebra of t consisting of all matrices of trace 0. T...Let R be an arbitrary commutative ring with identity. Denote by t the Lie algebra over R consisting of all upper triangular n by n matrices and let b be the Lie subalgebra of t consisting of all matrices of trace 0. The aim of this paper is to give an explicit description of the derivation algebras of the Lie algebras t and b, respectively.展开更多
We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt ...We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.展开更多
In this paper, we determine the derivation algebra and the automorphism group of the original deformative Schrodinger-Virasoro algebra, which is the semi-direct product Lie algebra of the Witt algebra and its tensor d...In this paper, we determine the derivation algebra and the automorphism group of the original deformative Schrodinger-Virasoro algebra, which is the semi-direct product Lie algebra of the Witt algebra and its tensor density module Ig（a, b）.展开更多
Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321： 3601-3619] constructed a new family of finite-dimensional modular Lie superalgebra Ω. Let Ω denote the even part of the Lie superalgebra Ω. We first give the gen...Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321： 3601-3619] constructed a new family of finite-dimensional modular Lie superalgebra Ω. Let Ω denote the even part of the Lie superalgebra Ω. We first give the generator sets of the Lie algebra Ω. Then, we reduce the derivation of Ω to a certain form. With the reduced derivation and a torus of Ω, we determine the derivation algebra of Ω.展开更多
Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized： （L） ≌ （R＋/（[R, R] A R＋））＊ and Der（L） = Inn（L） ＋ Der（L）0 = Inn（L） ＋ SDer（R）, which ...Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized： （L） ≌ （R＋/（[R, R] A R＋））＊ and Der（L） = Inn（L） ＋ Der（L）0 = Inn（L） ＋ SDer（R）, which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field F.展开更多
Given an odd-periodic algebraic triangulated category, we compare Bridgeland's Hall algebra in the sense of Bridgeland(2013) and Gorsky(2014), and the derived Hall algebra in the sense of Ten(2006), Xiao and Xu(20...Given an odd-periodic algebraic triangulated category, we compare Bridgeland's Hall algebra in the sense of Bridgeland(2013) and Gorsky(2014), and the derived Hall algebra in the sense of Ten(2006), Xiao and Xu(2008) and Xu and Chen(2013), and show that the former one is the twisted form of the tensor product of the latter one and a suitable group algebra.展开更多
Let L be the symplectic algebra or the orthogonal algebra over a commutative ring R, h the maximal torus of L consisting of all diagonal matrices in L, and b the standard Borel subalgebra of L containing h. In this pa...Let L be the symplectic algebra or the orthogonal algebra over a commutative ring R, h the maximal torus of L consisting of all diagonal matrices in L, and b the standard Borel subalgebra of L containing h. In this paper, we first determine the intermediate algebras between h and b, then for such an intermediate algebra, we give an explicit description on its derivations, provided that R is a commutative ring with identity and 2 is invertible in R.展开更多
We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough p...We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K^b(P), and finding an example such that D_(hf)~b(A)≠K^b(P). We realize the bounded derived category D^b(A) as a Verdier quotient of the relative derived category D_C^b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT <∞ such that ~⊥T is finite, then D^b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.展开更多
文摘Let F be a field and char F = p > 3. In this paper the derivation algebras of Lie superalgebras W and S of Cartan-type over F are determined by the calculating method.
文摘In this paper, we explicitly determine the maximal torus of the derivation algebra of a Qn filiform Lie algebra. Using the root space decomposition of DerQn, we prove that the derivation algebra of a Qn filiform Lie algebra is complete.
文摘In this paper we explicitly determine automorphism group of filiform Lie algebra Rn to find the indecomposable solvable Lie algebras with filiform Lie algebra Rn nilradicals.We also prove that the indecomposable solvable Lie algebras with filiform Rn nilradicals is complete.
文摘In this paper we explicitly determine the derivation algebra of a quasi Rn-filiform Lie algebra and prove that a quasi Rn-filiform Lie algebra is a completable nilpotent Lie algebra.
文摘In this paper the derivation algebra of Lie algebra Σof characteristic two is determined. Using this result we obtain the necessary and sufficient condition under that Σ of characteristic two is the restreted Lie algebra. Finally we prove that Σ of characteristic two isn't isomorphic the Lie algebras of characteristic two which are known by authors of this paper. But it still is a filtered deformation of the Lie algebra of H-type i.e. Its associated grated algebra GrΣ is isomorphic to H(2n + 2.r).
基金Supported by the National Natural Science Foundation of China (Grant No. 11171294)the Natural Science Foundation of Heilongjiang Province (Grant No. A201013)the Fund of Heilongjiang Education Committee(Grant No. 11541268)
文摘In this paper, we study a class of subalgebras of the Lie algebra of vector fields on n-dimensional torus, which are called the Triangular derivation Lie algebra. We give the structure and the central extension of Triangular derivation Lie algebra.
基金Supported by the Natural Science Foundation of the Henan Institute of Science and Technology（06057）
文摘In this paper, the derivation algebra of Lie superalgebra H of Caftan-type over F are determined by the calculating method in the situations of CharF = p ≥ 3 or m ≥ 2 or n ≥ 1. The main result is following： DerFH = adH（H＂ ＋ Fh） （{（adDi）^pt ｜ i = 1,2,…,m, t=1,2,…,ti-1}）.
基金Supported by the National Natural Science Foundation of China(Grant No.11101084)the Natural Science Foundation of Fujian Province(Grant No.2013J01005)
文摘Let F be a field, n ≥ 3, N（n,F） the strictly upper triangular matrix Lie algebra consisting of the n × n strictly upper triangular matrices and with the bracket operation {x, y} = xy-yx. A linear map φ on N（n,F） is said to be a product zero derivation if {φ（x）,y] ＋ [x, φ（y）] = 0 whenever {x, y} = 0,x,y ∈ N（n,F）. In this paper, we prove that a linear map on N（n, F） is a product zero derivation if and only if φ is a sum of an inner derivation, a diagonal derivation, an extremal product zero derivation, a central product zero derivation and a scalar multiplication map on N（n, F）.
基金supported by National Natural Science Foundation of China(Grant Nos.12001107 and 11821001)University Natural Science Project of Anhui Province(Grant No.KJ2021A0661)+1 种基金University Outstanding Youth Research Project in Anhui Province(Grant No.2022AH020082)Scientific Research and Innovation Team Project of Fuyang Normal University(Grant No.TDJC2021009)。
文摘Let t be a positive integer and A be a hereditary abelian category satisfying some finiteness conditions.We define the semi-derived Ringel-Hall algebra of A from the category C_(Z/t)(A)of Z/t-graded complexes and obtain a natural basis of the semi-derived Ringel-Hall algebra.Moreover,we describe the semiderived Ringel-Hall algebra by the generators and defining relations.In particular,if t is an odd integer,we show an embedding of the derived Hall algebra of the odd-periodic relative derived category in the extended semi-derived Ringel-Hall algebra.
基金Supported by the Natural Science Foundation of China(Grant No.11861061)。
文摘We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis,and the corresponding irreducible elements forms a PBW type basis of the Ringel-Hall algebra.We aim to generalize this result to the derived Hall algebra DH(An)of type An.First,we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category D^b(An)using the Auslander-Reiten quiver of D^b(An),and then we prove that all possible compositions between these skew commutator relations are trivial.As an application,we give a PBW type basis of DH(An).
基金the National Natural Scieace Foundation of China(10071078).
文摘Let R be an arbitrary commutative ring with identity. Denote by t the Lie algebra over R consisting of all upper triangular n by n matrices and let b be the Lie subalgebra of t consisting of all matrices of trace 0. The aim of this paper is to give an explicit description of the derivation algebras of the Lie algebras t and b, respectively.
基金This work was supported in part by the National Natural Science Foundation of China （Grant No. 11371245） and the Natural Science Foundation of Hebei Province, China （Grant No. A2014201006）.
文摘We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 10825101, 10861004, 11101266）, SMSTC grant no. 12XD1405000, Fundamental Research Funds for the Central Universities, and Science ＆ Technology Program of Shanghai Maritime University.
文摘We determine the derivation algebra and the automorphism group of the generalized topological N = 2 superconformal algebra.
文摘In this paper, we determine the derivation algebra and the automorphism group of the original deformative Schrodinger-Virasoro algebra, which is the semi-direct product Lie algebra of the Witt algebra and its tensor density module Ig（a, b）.
文摘Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321： 3601-3619] constructed a new family of finite-dimensional modular Lie superalgebra Ω. Let Ω denote the even part of the Lie superalgebra Ω. We first give the generator sets of the Lie algebra Ω. Then, we reduce the derivation of Ω to a certain form. With the reduced derivation and a torus of Ω, we determine the derivation algebra of Ω.
文摘Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized： （L） ≌ （R＋/（[R, R] A R＋））＊ and Der（L） = Inn（L） ＋ Der（L）0 = Inn（L） ＋ SDer（R）, which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field F.
基金supported by National Natural Science Foundation of China(Grant Nos.11301533 and 11471177)
文摘Given an odd-periodic algebraic triangulated category, we compare Bridgeland's Hall algebra in the sense of Bridgeland(2013) and Gorsky(2014), and the derived Hall algebra in the sense of Ten(2006), Xiao and Xu(2008) and Xu and Chen(2013), and show that the former one is the twisted form of the tensor product of the latter one and a suitable group algebra.
文摘Let L be the symplectic algebra or the orthogonal algebra over a commutative ring R, h the maximal torus of L consisting of all diagonal matrices in L, and b the standard Borel subalgebra of L containing h. In this paper, we first determine the intermediate algebras between h and b, then for such an intermediate algebra, we give an explicit description on its derivations, provided that R is a commutative ring with identity and 2 is invertible in R.
基金supported by National Natural Science Foundation of China(Grant Nos.11271251 and 11431010)
文摘We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K^b(P), and finding an example such that D_(hf)~b(A)≠K^b(P). We realize the bounded derived category D^b(A) as a Verdier quotient of the relative derived category D_C^b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT <∞ such that ~⊥T is finite, then D^b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.
