3-Lie algebras have close relationships with many important fields in mathemat- ics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The ...3-Lie algebras have close relationships with many important fields in mathemat- ics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of char- acteristic zero are provided.展开更多
In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and ...In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.展开更多
In this paper, based on the existing research results, we obtain the unary extension 3-Lie algebras by one-dimensional extension of the known Lie algebra L. For two known 3-Lie algebras <em>H</em>, <em&...In this paper, based on the existing research results, we obtain the unary extension 3-Lie algebras by one-dimensional extension of the known Lie algebra L. For two known 3-Lie algebras <em>H</em>, <em>M</em>, the (<i>μ</i>, <i>ρ</i>, <i>β</i>)-extension of <em>H</em> through <em>M</em> is given, and the necessary and sufficient conditions for the (<i>μ</i>, <i>ρ</i>, <i>β</i>)-extension algebra of <em>H</em> through <em>M</em> being 3-Lie algebra are obtained, and the structural characteristics and properties of these two kinds of extended 3-Lie algebras are given.展开更多
q-Deformations of 3-Lie algebras and representations of q-3-Lie algebras are discussed. A q-3-Lie algebra （A, [, ,]q, [, ,]＇q, Ja）, where q ∈ F and q ≠ 0, is a vector space A over a field F with 3-cry linear mult...q-Deformations of 3-Lie algebras and representations of q-3-Lie algebras are discussed. A q-3-Lie algebra （A, [, ,]q, [, ,]＇q, Ja）, where q ∈ F and q ≠ 0, is a vector space A over a field F with 3-cry linear multiplications [,, ]q and [,, ]＇q from A 3 to A, and a map Jq ： A 5→ F satisfying the q-Jacobi identity Jq（x1, x2, x3, x4, x5）[x1, x2, [x3, x4, x5]q]＇q = Jq（x4, x5, x1, x2, x3）[x4, x5, [x1, x2, x3]q]＇q for all x1 ∈ A. If the multiplications satisfy that [,,]q = [,,]＇q and [,,]q is skew-symmetry, then （A, [,,]q, Jq） is called a type （I）-q-3- Lie algebra. From q-Lie algebras, group algebras and commutative associative algebras, q-3-Lie algebras and type （I）-q-3-Lie algebras are constructed. Also, the non-trivial one- dimensional central extension of q-3-Lie algebras is investigated, and new q-3-Lie algebras （DerqC[x,x-1], [, ,]q, [, ,]＇q, Jq）, and （Derδq C[x,x-1], [, ,]q, [,,]＇q, Jq） are obtained.展开更多
Associative multiplications of cubic matrices are provided.The N3-dimensional3-Lie algebras are realized by cubic matrices,and structures of the 3-Lie algebras are studied.
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
基金partially supported by NSF(11371245)of ChinaNSF(A2010000194)of Hebei Province
文摘3-Lie algebras have close relationships with many important fields in mathemat- ics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of char- acteristic zero are provided.
文摘In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.
文摘In this paper, based on the existing research results, we obtain the unary extension 3-Lie algebras by one-dimensional extension of the known Lie algebra L. For two known 3-Lie algebras <em>H</em>, <em>M</em>, the (<i>μ</i>, <i>ρ</i>, <i>β</i>)-extension of <em>H</em> through <em>M</em> is given, and the necessary and sufficient conditions for the (<i>μ</i>, <i>ρ</i>, <i>β</i>)-extension algebra of <em>H</em> through <em>M</em> being 3-Lie algebra are obtained, and the structural characteristics and properties of these two kinds of extended 3-Lie algebras are given.
文摘q-Deformations of 3-Lie algebras and representations of q-3-Lie algebras are discussed. A q-3-Lie algebra （A, [, ,]q, [, ,]＇q, Ja）, where q ∈ F and q ≠ 0, is a vector space A over a field F with 3-cry linear multiplications [,, ]q and [,, ]＇q from A 3 to A, and a map Jq ： A 5→ F satisfying the q-Jacobi identity Jq（x1, x2, x3, x4, x5）[x1, x2, [x3, x4, x5]q]＇q = Jq（x4, x5, x1, x2, x3）[x4, x5, [x1, x2, x3]q]＇q for all x1 ∈ A. If the multiplications satisfy that [,,]q = [,,]＇q and [,,]q is skew-symmetry, then （A, [,,]q, Jq） is called a type （I）-q-3- Lie algebra. From q-Lie algebras, group algebras and commutative associative algebras, q-3-Lie algebras and type （I）-q-3-Lie algebras are constructed. Also, the non-trivial one- dimensional central extension of q-3-Lie algebras is investigated, and new q-3-Lie algebras （DerqC[x,x-1], [, ,]q, [, ,]＇q, Jq）, and （Derδq C[x,x-1], [, ,]q, [,,]＇q, Jq） are obtained.
基金supported by the National Natural Science Foundation of China(No.11371245)the Natural Science Foundation of Hebei Province(No.A2010000194)
文摘Associative multiplications of cubic matrices are provided.The N3-dimensional3-Lie algebras are realized by cubic matrices,and structures of the 3-Lie algebras are studied.
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.
