The linear operations of the equivalent classes of crossed modules of Lie color algebras are studied. The set of the equivalent classes of crossed modules is proved to be a vector space, which is isomorphic with the h...The linear operations of the equivalent classes of crossed modules of Lie color algebras are studied. The set of the equivalent classes of crossed modules is proved to be a vector space, which is isomorphic with the homogeneous components of degree zero of the third cohomology group of Lie color algebras. As an application of this theory, the crossed modules of Witt type Lie color algebras is described, and the result is proved that there is only one equivalent class of the crossed modules of Witt type Lie color algebras when the abelian group Г is equal to Г＋. Finally, for a Witt type Lie color algebra, the classification of its crossed modules is obtained by the isomorphism between the third cohomology group and the crossed modules.展开更多
For any complex parameters a, b, let W（a, b） be the Lie algebra with basis {Li, Hi｜i ∈ Z} and relations [Li,Lj] = （j - i）Li＋j, [Li,Hj] = （a ＋ j ＋ bi）Hi＋j and [Hi, Hi] = 0. In this paper, we construct the W...For any complex parameters a, b, let W（a, b） be the Lie algebra with basis {Li, Hi｜i ∈ Z} and relations [Li,Lj] = （j - i）Li＋j, [Li,Hj] = （a ＋ j ＋ bi）Hi＋j and [Hi, Hi] = 0. In this paper, we construct the W（a, b） conformal algebra for some a, b and its conformal module of rank one.展开更多
Let L be a Lie algebra of Block type over C with basis {Lα,i ｜ a,i ∈ Z} and brackets [Lα,i, Lβ,j] = （β（i ＋ 1） - α（j ＋ 1））Lα＋β,i＋j. In this paper, we first construct a formal distribution Lie algebra...Let L be a Lie algebra of Block type over C with basis {Lα,i ｜ a,i ∈ Z} and brackets [Lα,i, Lβ,j] = （β（i ＋ 1） - α（j ＋ 1））Lα＋β,i＋j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with C[δ]- basis { Lα（w） ｜ α ∈ Z} and λ-brackets [Lα（w）λLβ（w）] = （αδ ＋ （α ＋β）A）Lα＋β（w）. Finally, we give a classification of free intermediate series B-modules.展开更多
In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of a certain vaccum module for the algebra W(2, 2) via the Weyl vertex alg...In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of a certain vaccum module for the algebra W(2, 2) via the Weyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra W(2, 2).展开更多
In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-alg...In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A（m）-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.展开更多
Let F be an algebracially closed field of characteristic p】2, and L be the p<sup>n</sup>-dimensional Zassenhaus algebra with the maximal invariant subalgebra L<sub>0</sub> and the standard fil...Let F be an algebracially closed field of characteristic p】2, and L be the p<sup>n</sup>-dimensional Zassenhaus algebra with the maximal invariant subalgebra L<sub>0</sub> and the standard filtration {L<sub>i</sub>}|<sub>i=-1</sub><sup>p<sup>n</sup>-2</sup>. Then the number of isomorphism classes of simple L-modules is equal to that of simple L<sub>0</sub>-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than p<sup>n</sup>.展开更多
First,we prove a decomposition formula for any multiplicative differential form on a Lie groupoid G.Next,we prove that if G is a Poisson Lie groupoid,then the spaceΩ_(mult)·(G)of multiplicative forms on G has a ...First,we prove a decomposition formula for any multiplicative differential form on a Lie groupoid G.Next,we prove that if G is a Poisson Lie groupoid,then the spaceΩ_(mult)·(G)of multiplicative forms on G has a differential graded Lie algebra(DGLA)structure.Furthermore,when combined withΩ·(M),which is the space of forms on the base manifold M of G,Ω_(mult)·(G)forms a canonical DGLA crossed module.This supplements a previously known fact that multiplicative multi-vector fields on G form a DGLA crossed module with the Schouten algebraΓ(∧·A)stemming from the Lie algebroid A of G.展开更多
基金The Natural Science Foundation of Jiangsu Province(No.BK2012736)the Natural Science Foundation of Chuzhou University(No.2010kj006Z)
文摘The linear operations of the equivalent classes of crossed modules of Lie color algebras are studied. The set of the equivalent classes of crossed modules is proved to be a vector space, which is isomorphic with the homogeneous components of degree zero of the third cohomology group of Lie color algebras. As an application of this theory, the crossed modules of Witt type Lie color algebras is described, and the result is proved that there is only one equivalent class of the crossed modules of Witt type Lie color algebras when the abelian group Г is equal to Г＋. Finally, for a Witt type Lie color algebra, the classification of its crossed modules is obtained by the isomorphism between the third cohomology group and the crossed modules.
文摘For any complex parameters a, b, let W（a, b） be the Lie algebra with basis {Li, Hi｜i ∈ Z} and relations [Li,Lj] = （j - i）Li＋j, [Li,Hj] = （a ＋ j ＋ bi）Hi＋j and [Hi, Hi] = 0. In this paper, we construct the W（a, b） conformal algebra for some a, b and its conformal module of rank one.
文摘Let L be a Lie algebra of Block type over C with basis {Lα,i ｜ a,i ∈ Z} and brackets [Lα,i, Lβ,j] = （β（i ＋ 1） - α（j ＋ 1））Lα＋β,i＋j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with C[δ]- basis { Lα（w） ｜ α ∈ Z} and λ-brackets [Lα（w）λLβ（w）] = （αδ ＋ （α ＋β）A）Lα＋β（w）. Finally, we give a classification of free intermediate series B-modules.
基金supported by National Natural Science Foundation of China (Grant Nos. 11271056, 11671056 and 11101030)National Science Foundation of Jiangsu (Grant No. BK20160403)+3 种基金National Science Foundation of Zhejiang (Grant Nos. LQ12A01005 and LZ14A010001)National Science Foundation of Shanghai (Grant No. 16ZR1425000)Beijing Higher Education Young Elite Teacher ProjectMorningside Center of Mathematics
文摘In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of a certain vaccum module for the algebra W(2, 2) via the Weyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra W(2, 2).
基金Supported by ZJNSF(Grant Nos.LY17A010015 and LZ14A010001)NNSF(Grant No.11171296)
文摘In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A（m）-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.
基金Supported in part by the National Natural Science Foundation of China Grant 19801022the Scientifictechnological Major Project of Educational Ministry of China, Grant 99036.
文摘Let F be an algebracially closed field of characteristic p】2, and L be the p<sup>n</sup>-dimensional Zassenhaus algebra with the maximal invariant subalgebra L<sub>0</sub> and the standard filtration {L<sub>i</sub>}|<sub>i=-1</sub><sup>p<sup>n</sup>-2</sup>. Then the number of isomorphism classes of simple L-modules is equal to that of simple L<sub>0</sub>-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than p<sup>n</sup>.
基金supported by National Natural Science Foundation of China(Grant No.12071241)the National Key Research and Development Program of China(Grant No.2021YFA1002000).
文摘First,we prove a decomposition formula for any multiplicative differential form on a Lie groupoid G.Next,we prove that if G is a Poisson Lie groupoid,then the spaceΩ_(mult)·(G)of multiplicative forms on G has a differential graded Lie algebra(DGLA)structure.Furthermore,when combined withΩ·(M),which is the space of forms on the base manifold M of G,Ω_(mult)·(G)forms a canonical DGLA crossed module.This supplements a previously known fact that multiplicative multi-vector fields on G form a DGLA crossed module with the Schouten algebraΓ(∧·A)stemming from the Lie algebroid A of G.
