The cohomology ring of the 12-dimensional Fomin–Kirillov algebra
The cohomology ring of the 12-dimensional Fomin–Kirillov algebra
复制标题
12 维 Fomin-Kirilov 代数的上同调环
DOI：
10.1016/j.aim.2016.01.001
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发表时间：
2014
影响因子：
1.7
通讯作者：
C. Vay
中科院分区：
文献类型：
--
作者：
D. Ştefan;C. Vay
定义12维Fomin-Kirillov代数FK 3为生成元a，B和c满足关系a2 = B 2= c2 = 0和a B+ B c+ ca = 0= B a+ c B+ ac的二次代数.通过A. Milinski和H.- J. Schneider，这个代数同构于对称群S3上与Yetter-Drinfeld模V相关的Nichols代数，对应于所有转置和符号表示的共轭类。利用这一证明，我们计算了Ext FK 3 <$（k，k）的上同调环，证明了它是一个系数在V的对称辫子代数中的多项式环S [X].作为应用，我们还计算了玻色化FK 3 #k S 3及其对偶的上同调环，它们是72维的普通Hopf代数.
Abstract The 12-dimensional Fomin–Kirillov algebra FK 3 is defined as the quadratic algebra with generators a, b and c which satisfy the relations a 2= b 2= c 2= 0 and a b+ b c+ c a= 0= b a+ c b+ a c. By a result of A. Milinski and H.-J. Schneider, this algebra is isomorphic to the Nichols algebra associated to the Yetter–Drinfeld module V, over the symmetric group S 3, corresponding to the conjugacy class of all transpositions and the sign representation. Exploiting this identification, we compute the cohomology ring Ext FK 3⁎(k, k), showing that it is a polynomial ring S [X] with coefficients in the symmetric braided algebra of V. As an application we also compute the cohomology rings of the bosonization FK 3# k S 3 and of its dual, which are 72-dimensional ordinary Hopf algebras.