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 derivation algebra of a quasi Rn-filiform Lie algebra and prove that a quasi Rn-filiform Lie algebra is a completable nilpotent Lie algebra.
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
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 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
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
Let L be the derivation Lie algebra of C[t1^±1 , t2^±1 ]. Given a triangle decomposition L = L+ η + L-, we define a nonsingular Lie algebra homomorphism φ : L+ → C and the universal Whittaker L-module...Let L be the derivation Lie algebra of C[t1^±1 , t2^±1 ]. Given a triangle decomposition L = L+ η + L-, we define a nonsingular Lie algebra homomorphism φ : L+ → C and the universal Whittaker L-module We of type φ. We obtain all Whittaker vectors and submodules of We. Moreover, all simple Whittaker L-modules of type φ are determined.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
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 A be a finitary hereditary abelian category with enough projectives.We study the Hall algebra of complexes of fixed size over projectives.Explicitly,we first give a relation between Hall algebras of complexes of f...Let A be a finitary hereditary abelian category with enough projectives.We study the Hall algebra of complexes of fixed size over projectives.Explicitly,we first give a relation between Hall algebras of complexes of fixed size and cyclic complexes.Second,we characterize the Hall algebra of complexes of fixed size by generators and relations,and relate it to the derived Hall algebra of A.Finally,we give the integration map on the Hall algebra of 2-term complexes over projectives.展开更多
This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspon...This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.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
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 derivation algebra of a quasi Rn-filiform Lie algebra and prove that a quasi Rn-filiform Lie algebra is a completable nilpotent Lie algebra.
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.
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).
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.11571145 and 11271165)the Youth Foundation of National Natural Science Foundation of China(Grant Nos.11101350 and 11302052)the Natural Science Foundation of Fujian Province(Grant No.2010J05001)
文摘Let L be the derivation Lie algebra of C[t1^±1 , t2^±1 ]. Given a triangle decomposition L = L+ η + L-, we define a nonsingular Lie algebra homomorphism φ : L+ → C and the universal Whittaker L-module We of type φ. We obtain all Whittaker vectors and submodules of We. Moreover, all simple Whittaker L-modules of type φ are determined.
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Supported by the National Natural Science Foundation of China(Grant No.11801273)Natural Science Foundation of Jiangsu Province of China(Grant No.BK20180722)。
文摘Let A be a finitary hereditary abelian category with enough projectives.We study the Hall algebra of complexes of fixed size over projectives.Explicitly,we first give a relation between Hall algebras of complexes of fixed size and cyclic complexes.Second,we characterize the Hall algebra of complexes of fixed size by generators and relations,and relate it to the derived Hall algebra of A.Finally,we give the integration map on the Hall algebra of 2-term complexes over projectives.
基金Supported by China Scholarship Council(Grant No.201206125047)China Postdoctoral Science Foundation Funded Project(Grant No.2012M520715)the Fundamental Research Funds for the Central Universities(Grant No.HIT.NSRIF.201462)
文摘This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.
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.