Combinatorial rank of

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• 作者：M.L. Dí ; az Sosa^a ; ^b ; ^{mlds@apolo.acatlan.unam.mx" class="auth_mail" title="E-mail the corresponding author} ; V. Kharchenko^a ; ^b
• 关键词：Combinatorial representation ; Quantum group ; Skew primitive elements ; PBW basis
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：448
• 期：Complete
• 页码：48-73
• 全文大小：508 K

文摘

In general, an intersection of two Hopf ideals of a Hopf algebra is not a Hopf ideal. By this reason, one may not define a Hopf ideal generated by a set of elements, and the Hopf algebras do not admit a usual combinatorial representation by generators and relations. Nevertheless, Heyneman–Radford theorem implies that each nonzero Hopf ideal of a pointed Hopf algebra has nonzero skew primitive element. Each ideal generated by skew primitive elements is a Hopf ideal. Therefore, the Heyneman–Radford theorem allows one to define a combinatorial representation step-by step by means of skew primitive relations. By definition the combinatorial rank is the minimal number of steps in that representation. We prove that the combinatorial rank of the multiparameter version of the Frobenius–Lusztig kernel of type D_n$D_n$, e507eb59d4150c518487bf81ba556" title="Click to view the MathML source">n>3$n>3$ equals ⌊log₂⁡(2n−3)⌋+1$\lfloor {\log }_2(2n-3)\rfloor +1$, provided that q   has a finite multiplicative order greater than 2. It is already known that the combinatorial rank of the multiparameter version of the Frobenius–Lusztig kernel u_q(sl_n+1)$u_q({\mathfrak{sl}}_{n+1})$ of type 16e" title="Click to view the MathML source">A_n$A_n$ equals ⌊log₂⁡n⌋+1$\lfloor {\log }_{2}n\rfloor +1$, whereas in case u_q(so_2n+1)$u_q({\mathfrak{so}}_{2n+1})$ of type B_n$B_n$, it is equal to ⌊log₂⁡(n−1)⌋+2$\lfloor {\log }_2(n-1)\rfloor +2$.