正nilradical的等变量子化与协变微分微分计算
Equivariant quantizations of the positive nilradical and covariant differential calculi
AI总结：
本文研究了正nilradical的等变量子化，并利用其构造量子旗流形上的协变微分计算。
摘要：
考虑将复半单李代数的正nilradical分解为$\mathfrak{n} = \mathfrak{n}_1 \oplus \cdots \oplus \mathfrak{n}_r$，其中每个$\mathfrak{n}_k$在适当的利维因子下是一个模。我们证明这种分解可以量子化为量子 enveloping 代数正部分的一个有限维子空间$\mathfrak{n}^q_k = \mathfrak{n}^q_1 \oplus \cdots \oplus \mathfrak{n}^q_r$，其中每个$\mathfrak{n}^q_k$在量化利维因子的左伴随作用下是一个模。进一步，我们证明$\mathbb{C} \oplus \mathfrak{n}^q$是一个左余理想，除了可能对应于某些例外李代数的组件外。最后，我们利用这些量子化来在量子旗流形上构造协变一阶微分计算，这些计算在某种意义上与上述分解兼容，与Heckenberger-Kolb在不可约情况中引入的计算一致。
英文摘要：
Consider a decomposition $\mathfrak{n} = \mathfrak{n}_1 \oplus \cdots \oplus \mathfrak{n}_r$ of the positive nilradical of a complex semisimple Lie algebra of rank $r$, where each $\mathfrak{n}_k$ is a module under an appropriate Levi factor. We show that this can be quantized as a finite-dimensional subspace $\mathfrak{n}^q_k = \mathfrak{n}^q_1 \oplus \cdots \oplus \mathfrak{n}^q_r$ of the positive part of the quantized enveloping algebra, where each $\mathfrak{n}^q_k$ is a module under the left adjoint action of a quantized Levi factor. Furthermore, we show that $\mathbb{C} \oplus \mathfrak{n}^q$ is a left coideal, with the possible exception of components corresponding to some exceptional Lie algebras. Finally we use these quantizations to construct covariant first-order differential calculi on quantum flag manifolds, compatible in a certain sense with the decomposition above, which coincide with those introduced by Heckenberger-Kolb in the irreducible case.
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