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
Let F be an algebraically closed field of characteristic p 〉 3, and g be the Witt algebra over F. Let N be the nilpotent cone of g. An explicit description of N is given, so that the conjugacy classes of Borel subalg...Let F be an algebraically closed field of characteristic p 〉 3, and g be the Witt algebra over F. Let N be the nilpotent cone of g. An explicit description of N is given, so that the conjugacy classes of Borel subalgebras of g under the automorphism group of g are determined. In contrast with only one conjugacy class of Borel subalgebras in a classical simple Lie algebra, there are two conjugacy classes of Borel subalgebras in g. The representatives of conjugacy classes of Borel subalgebras, i.e.,the so-called standard Borel subalgebras, are precisely given.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
Let g = W1 be the Witt algebra over an algebraically closed field k of characteristic p 〉 3, and let ∮(g) = {(x,y) ∈ g×g [x,y] = 0} be the commuting variety of g. In contrast with the case of classical Lie...Let g = W1 be the Witt algebra over an algebraically closed field k of characteristic p 〉 3, and let ∮(g) = {(x,y) ∈ g×g [x,y] = 0} be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473-484], we show that the variety ∮(g) is reducible, and not equidimensional. Irreducible components of ∮(g) and their dimensions are precisely given. As a consequence, the variety ∮(g) is not normal.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
Let F be a field of characteristic zero. W_n = F[t_1^(+-1), t_2^(+-1), ...,t_n^(+-1)] (partial deriv)/((partial deriv)t_1) + ... + F[t_1^(+-1), t_2^(+-1), ..., t_n^(+-1)](partial deriv)/((partial deriv)t_n) is the Wit...Let F be a field of characteristic zero. W_n = F[t_1^(+-1), t_2^(+-1), ...,t_n^(+-1)] (partial deriv)/((partial deriv)t_1) + ... + F[t_1^(+-1), t_2^(+-1), ..., t_n^(+-1)](partial deriv)/((partial deriv)t_n) is the Witt algebra over F, W_n^+ = F[t_1, t_2 ..., t_n](partial deriv)/((partial deriv)t_1) + ... + F[t_1, t_2 ..., t_n] (partial deriv)/((partialderiv)t_n) is Lie subalgebra of W_n. It is well known both W_n and W_n^+ are simple infinitedimensional Lie algebra. In Zhao's paper, it was conjectured that End(W_n^+) - {0} = Aut(W_n^+) andit was proved that the validity of this conjecture implies the validity of the well-known Jacobianconjecture. In this short note, we check the conjecture above for n = 1. We show End(W_1^+) - {0} =Aut(W_1^+).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.
基金Supported by National Natural Science Foundation of China(Grant Nos.11201293 and 11271130)the Innovation Program of Shanghai Municipal Education Commission(Grant Nos.13YZ077 and 12ZZ038)
文摘Let F be an algebraically closed field of characteristic p 〉 3, and g be the Witt algebra over F. Let N be the nilpotent cone of g. An explicit description of N is given, so that the conjugacy classes of Borel subalgebras of g under the automorphism group of g are determined. In contrast with only one conjugacy class of Borel subalgebras in a classical simple Lie algebra, there are two conjugacy classes of Borel subalgebras in g. The representatives of conjugacy classes of Borel subalgebras, i.e.,the so-called standard Borel subalgebras, are precisely given.
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.
文摘Let g = W1 be the Witt algebra over an algebraically closed field k of characteristic p 〉 3, and let ∮(g) = {(x,y) ∈ g×g [x,y] = 0} be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473-484], we show that the variety ∮(g) is reducible, and not equidimensional. Irreducible components of ∮(g) and their dimensions are precisely given. As a consequence, the variety ∮(g) is not normal.
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Let F be a field of characteristic zero. W_n = F[t_1^(+-1), t_2^(+-1), ...,t_n^(+-1)] (partial deriv)/((partial deriv)t_1) + ... + F[t_1^(+-1), t_2^(+-1), ..., t_n^(+-1)](partial deriv)/((partial deriv)t_n) is the Witt algebra over F, W_n^+ = F[t_1, t_2 ..., t_n](partial deriv)/((partial deriv)t_1) + ... + F[t_1, t_2 ..., t_n] (partial deriv)/((partialderiv)t_n) is Lie subalgebra of W_n. It is well known both W_n and W_n^+ are simple infinitedimensional Lie algebra. In Zhao's paper, it was conjectured that End(W_n^+) - {0} = Aut(W_n^+) andit was proved that the validity of this conjecture implies the validity of the well-known Jacobianconjecture. In this short note, we check the conjecture above for n = 1. We show End(W_1^+) - {0} =Aut(W_1^+).