论文标题
$ \ mathfrak {gl}（\ infty）$ - 模块的最高权重类别
Highest weight categories of $\mathfrak{gl}(\infty)$-modules
论文作者
论文摘要
我们研究了$ \ mathfrak {gl}（\ infty）$类似于类别$ \ mathcal o $的一个类别模块。我们修复了足够的cartan，borel和levi-type subalgebras $ \ mathfrak h，\ mathfrak b $和$ \ mathfrak l $带有$ \ mathfrak l \ mathfrak l \ cong \ cong \ mathfrak {gl}（gl infty）（\ inftty）^n $ l} {\ mathfrak {gl}（\ infty）} $是$ \ mathfrak h $ - semisimple的类别，$ \ mathfrak n $ -nilpotent模块满足了$ \ mathfrak l $ -modules。我们的主要结果是，这些是Cline，Parshall和Scott的最高体重类别。我们计算标准对象的简单多重性和Injective对象中的标准乘数，并表明BGG互惠形式在$ \ Mathcal O _ {\ Mathsf {\ Mathsf {la}}^{\ Mathfrak l} {\ Mathfrak l} {\ Mathfrak {\ Mathfrak {gl}（GL}（gl iffty）} $。我们还提供了$ \ Mathcal O _ {\ Mathsf {la}}}^{\ Mathfrak l} {\ Mathfrak {gl}（\ infty）} $的分解。
We study a category of modules over $\mathfrak{gl}(\infty)$ analogous to category $\mathcal O$. We fix adequate Cartan, Borel and Levi-type subalgebras $\mathfrak h, \mathfrak b$ and $\mathfrak l$ with $\mathfrak l \cong \mathfrak{gl}(\infty)^n$, and define $\mathcal O_{\mathsf{LA}}^{\mathfrak l}{\mathfrak{gl}(\infty)}$ to be the category of $\mathfrak h$-semisimple, $\mathfrak n$-nilpotent modules that satisfy a large annihilator condition as $\mathfrak l$-modules. Our main result is that these are highest weight categories in the sense of Cline, Parshall and Scott. We compute the simple multiplicities of standard objects and the standard multiplicities in injective objects, and show that a form of BGG reciprocity holds in $\mathcal O_{\mathsf{LA}}^{\mathfrak l}{\mathfrak{gl}(\infty)}$. We also give a decomposition of $\mathcal O_{\mathsf{LA}}^{\mathfrak l}{\mathfrak{gl}(\infty)}$ into irreducible blocks.
