Department of Mathematics, University of California, Santa Barbara, CA 93106; and.
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803
Proc Natl Acad Sci U S A. 2014 Jul 8;111(27):9696-703. doi: 10.1073/pnas.1313071111. Epub 2014 Jun 30.
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein-Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405-455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337-397] for the case of symmetric Kac-Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1-52] associated with double Bruhat cells coincide with the corresponding cluster algebras.
一个丛代数理论的主要方向是在李理论中出现的各种族的 variety 的（量子化的）坐标环上构造（量子）丛代数结构。我们证明了在一个非常大的公理化定义的非交换代数类中所有的代数都具有典范的量子丛代数结构。此外，它们与相应的上量子丛代数一致。我们还为一大类 Poisson 幂零代数建立了这些结果的类比。许多重要的坐标环族都包含在我们所涵盖的类中，这导致了一般结果在上述类型的问题中的广泛应用。作为结果，我们证明了 Berenstein-Zelevinsky 猜想[Berenstein A, Zelevinsky A (2005) Adv Math 195:405-455]对于双 Bruhat 胞腔的量子化坐标环，并且在所有量子单群上构造量子丛代数结构，扩展了 Geiß 等人的定理[Geiß C, et al. (2013) Selecta Math 19:337-397]对于对称 Kac-Moody 群的情况。此外，我们证明了与双 Bruhat 胞腔相关联的 Berenstein 等人的上丛代数[Berenstein A, et al. (2005) Duke Math J 126:1-52]与相应的丛代数一致。
