q-Deformations of 3-Lie Algebras

• 来源期刊: 代数集刊（英文版）
• 关键词: q-3-Lie algebra q-deformation q-Jacobi identity
• 年，卷（期）: 2017,（3）
• 页码范围: 519-540
• 页数: 22页
• 语种: 英文
• DOI: 10.1142/S1005386717000347

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,Jq),where q ∈ F and q ≠ 0,is a vector space A over a field F with 3-ary linear multiplications [,,]q and [,,]'q from A(x)3 to A,and a map Jq:A(x)5 → F satisfying the q-Jacobi identity Jq(x1,x2,x3,x4,xs)[x1,x2,[x3,x4,x5]q]'q Jq(x4,x5,x1,x2,x3)[x4,xs,[x1,x2,x3]q]'q for all xi ∈ 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 (Derq C[x,x-1],[,,]q,[,,]'q,Jq),and (Derδq C[x,x-1],[,,]q,[,,]'q,Jq) are obtained.