The $q$-analogue of the Quantum Theory of Angular Momentum: a review from special functions

量子球上q-特殊多项式和q-积分的非对易模拟

Noncommutative analogues of q-special polynomials and q-integral on a quantum sphere

q-勒让德多项式可以看作是量子双陪集$U(1)\setminus SU_q(2)/U(1)$中的一些特殊的“函数“。它们形成一个变量的多项式族（取决于参数$q$）。通过引入一个双参数多项式族，得到了它们的进一步推广。如果前一个族产生于一个在某种意义上是“q-交换”的代数，则后一个族与它的非交换对应物有关。我们还介绍了一个双参数变形的不变积分的球。

量子MacMahon主定理的特殊化和推广

Specializations and Extensions of the quantum MacMahon Master Theorem

我们研究的一些专门化和扩展的量子版本的MacMahon主定理由Garoufaldet，乐和Zeilberger。特别地，我们得到了Cartier-Foata非交换版本的（t，q）-模拟和上下文代数的半强（t，q）-模拟。

A quantum analogue of generic bases for affine cluster algebras

We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types. Under the specialization of $q$ and coefficients to 1, these bases are generic bases of finite and affine cluster algebras.

An identification of Lusztig's modified forms of quantum algebras and their analogues

We introduce a multi-parameter twisted Hopf algebra associated with a root datum and show that its modified form is isomorphic to Lusztig's modified quantum algebra under certain restrictions on the parameters. By taking various specializations of the parameters, we obtain similar results for two/multi-parameter quantum algebras and Kang-Kashiwara-Oh's multi-parameter quantum superalgebras. As a consequence, the categories of weight modules of these algebras are identical.

WZW共形域论中融合规则的带状表和q-模拟

Ribbon tableaux and q-analogues of fusion rules in WZW conformal field theories

从已知的普通SU（n）张量积多重性的$q$ -类似物出发，引入了与SU（n）相关的WZW共形域论的融合系数的$q$ -类似物.我们猜想这些多项式的组合解释，这可以证明在特殊情况下。这使我们能够以一种简单的方式得出各种分支函数，最简单的是Virasoro代数的最小酉级数的字符。我们还获得了非交换θ函数空间的维数的$q$ -类似物。

The Grothendieck Group of a Quantum Projective Space Bundle

We compute the Grothendieck group K_0 of non-commutative analogues of quantum projective space bundles. Our results specialize to give the Grothendieck groups of non-commutative analogues of projective spaces, and specialize to recover the Grothendieck group of a usual projective space bundle over a regular noetherian separated scheme. As an application we develop an intersection theory for the quantum ruled surfaces defined by Van den Bergh.