q-Deformations of 3-Lie Algebras  

作　　者：Ruipu Bai Lixin Lin Yan Zhang Chuangchuang Kang Ruipu Bai Lixin Lin Yan Zhang Chuangchuang Kang(College of Mathematics and Information Science, Hebei University Baoding, Hebei 071002, China)

机构地区：[1]College of Mathematics and Information Science, Hebei University Baoding, Hebei 071002, China

出　　处：《Algebra Colloquium》2017年第3期519-540,共22页代数集刊（英文版）

摘　　要：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.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.

关 键 词：q-3-Lie algebra Q-DEFORMATION q-Jacobi identity 

分 类 号：O153.3[理学—数学] O152.5[理学—基础数学]