某些双参数表示列的泛量子维数
On universal quantum dimensions of certain two-parameter series of representations
摘要：
本文给出了单李代数任意个数伴随和$X_2$表示的Cartan积的量子维数的Vogel意义下的普适表达式。同样的公式神秘地给出了同一个李代数的其他一些表示在万有参数排列下的量子维数。我们列出这些代表例外代数和稳定版本的经典代数，当经典代数的秩是足够大的w.r.t.代表的权力。我们表明，通用公式可以有奇性的沃格尔平面上的一些代数，他们给出正确的答案时，限制在适当的线沃格尔平面上。我们注意到，同一不可约表示可以有几个通用公式，其（量子）维数，并讨论了这一现象的含义科恩-德曼方法计算的通用公式。
英文摘要：
We present the universal, in Vogel's sense, expression for the quantum dimension of Cartan product of an arbitrary number of adjoint and $X_2$ representations of simple Lie algebras. The same formula mysteriously gives quantum dimensions of some other representations of the same Lie algebra under permutations of universal parameters. We list these representations for exceptional algebras and stable versions for classical algebras, when the rank of the classical algebra is sufficiently large w.r.t. the powers of representations. We show that universal formulae can have singularities on Vogel's plane for some algebras and that they give correct answers when restricted on appropriate lines on Vogel's plane. We note that the same irreducible representation can have several universal formulae for its (quantum) dimension, and discuss the implication of this phenomena on the Cohen - de Man method of calculation of universal formulae.
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