论文标题
$ u_q（\ Mathfrak {SL} _2）$的立方狄拉克操作员
Cubic Dirac operator for $U_q(\mathfrak{sl}_2)$
论文作者
论文摘要
我们构建了$ \ mathfrak {sl} _2 $的$ q $ - 成型的Clifford代数并研究其属性。这使我们能够定义$ u_q（\ mathfrak {slfrak {sl} _2）$和相应的dirac codial dirac Cobial Dirac运营商$ d_q $的$ q $ deformed nocumutative nocuntative weil代数$ \ MATHCAL {W} _Q在经典的情况下，这是由Alekseev和Meinrenken完成的。我们表明，相对于$ u_q（\ Mathfrak {sl} _2）$ - ACTION和 *-tructures在$ \ Mathcal {W} _Q（\ Mathfrak {SL} _2 _2 _2）$上，$ d_q $是$ d_q $ central in Central in Central in of $ u_q（\ mathfrak {sl} _2）$ - action $ d_q $是不变的。 $ \ MATHCAL {W} _Q（\ Mathfrak {Sl} _2）$。我们计算〜$ u_q（\ Mathfrak {sl} _2）$的有限维和VERMA模块的立方元素的频谱和相应的DIRAC共同体。
We construct the $q$-deformed Clifford algebra of $\mathfrak{sl}_2$ and study its properties. This allows us to define the $q$-deformed noncommutative Weil algebra $\mathcal{W}_q(\mathfrak{sl}_2)$ for $U_q(\mathfrak{sl}_2)$ and the corresponding cubic Dirac operator $D_q$. In the classical case it was done by Alekseev and Meinrenken. We show that the cubic Dirac operator $D_q$ is invariant with respect to the $U_q(\mathfrak{sl}_2)$-action and *-structures on $\mathcal{W}_q(\mathfrak{sl}_2)$, moreover, the square of $D_q$ is central in $\mathcal{W}_q(\mathfrak{sl}_2)$. We compute the spectrum of the cubic element on finite-dimensional and Verma modules of~$U_q(\mathfrak{sl}_2)$ and the corresponding Dirac cohomology.
