Modular quantizations of Lie algebras of Cartan type K via Drinfeld twists of Jordanian type

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• 作者：Zhaojia Tong^a ; ^b ; ^{tongzhaojia@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Naihong Hu^c ; ¹ ; ^{nhhu@math.ecnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 17B37 ; 17B62 ; secondary ; 17B50
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：450
• 期：Complete
• 页码：102-151
• 全文大小：679 K

文摘

We construct explicit Drinfel'd twists of Jordanian type for the generalized Cartan type K Lie algebras in characteristic 0 and obtain the corresponding quantizations, especially their integral forms. By making modular reductions including modulo p and modulo p  -restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra $\mathbf{u}(\mathbf{K}(2n+1;\underset{\_}{1}))$ for the restricted simple Lie algebra of Cartan type K in characteristic p  . They are new families of noncommutative and noncocommutative Hopf algebras of dimension p^p2n+1+1$p^{p^{2n+1}+1}$ (if $2n+4\not\equiv 0(\text{mod}p)$) or p^p2n+1$p^{p^{2n+1}}$ (if $2n+4\equiv 0(\text{mod}p)$) over a truncated p-polynomials ring, which also contain the well-known Radford algebras (see [20]) as Hopf subalgebras. Some open questions are proposed.