论文标题
量化Weyl代数，双中央属性和$ u_q（\ Mathfrak {gl} _n）$的新的第一个基本定理
Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_q(\mathfrak{gl}_n)$
论文作者
论文摘要
令$ \ MATHCAL P：= \ MATHCAL P_ {M \ times n} $表示$ M \ times n $矩阵的量化坐标环，配备了量化的包裹代数$ _Q（\ Mathfrak {\ mathfrak {gl} _m）$ u_q（$ u_q（$ u_q）的自然作用。令$ \ mathcal l $和$ \ Mathcal r $表示$ u_q（\ Mathfrak {gl} _m）$和$ u_q（\ Mathfrak {gl} _n）$ in $ \ mathrm {Endrm {end}（分别分别）。我们定义$ \ m artrm {end}（\ Mathcal p）$内部多项式差异操作员的代数的$ q $ - 单词，此后用$ \ nathcal {pd {pd} $表示，我们证明了$ \ mathcal l \ cap \ cap \ cap \ cap \ mathcal cop {pd} \ Mathcal {pd} $是$ \ Mathcal {pd} $内部的共同集中器。使用它，我们为$ u_q（\ mathfrak {gl} _n）$建立了不变理论的新的第一个基本定理。我们还根据$ u_q（\ mathfrak {gl} _m）$ u_q（\ mathfrak {gl} _m）$ and $ u_q（\ u_q（\ mathfrak {\ mathfrak {gl gl} _n）$，我们还用$ q $ - 确定的$ q $确定剂来计算与$ \ mathcal {pd} $的发电机。
Let $\mathcal P:=\mathcal P_{m\times n}$ denote the quantized coordinate ring of the space of $m\times n$ matrices, equipped with natural actions of the quantized enveloping algebras $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$. Let $\mathcal L$ and $\mathcal R$ denote the images of $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$ in $\mathrm{End}(\mathcal P)$, respectively. We define a $q$-analogue of the algebra of polynomial-coefficient differential operators inside $\mathrm{End}(\mathcal P)$, henceforth denoted by $\mathcal{PD}$, and we prove that $\mathcal L\cap \mathcal{PD}$ and $\mathcal{R}\cap \mathcal{PD}$ are mutual centralizers inside $\mathcal{PD}$. Using this, we establish a new First Fundamental Theorem of invariant theory for $U_q(\mathfrak{gl}_n)$. We also compute explicit formulas in terms of $q$-determinants for generators of the intersections with $\mathcal{PD}$ of the images of the Cartan subalgebras of $U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$.
