arXiv 0810.34582008-10math.RTmath.RA

仿射李代数的诱导模

Induced Modules for Affine Lie Algebras

Vyacheslav Futorny, Iryna Kashuba

发表信息：: SIGMA 5 (2009), 026, 14 pages
DOI：: 10.3842/SIGMA.2009.026

摘要：

研究了仿射李代数上具有任意重数的非零中心荷的诱导模。对于仿射李代数${\mathfrak G}$的伪抛物子代数${\mathcal P}$，我们的主要结果建立了某类${\mathcal P}$诱导的${\mathfrak G}$ -模与其中心元具有内射作用的权${\mathcal P}$ -模之间的等价性.特别地，归纳函子保持不可约模。若${\mathcal P}$是一个具有有限维Levi因子的抛物子代数，则它定义了一个唯一的伪抛物子代数${\mathcal P}^{ps}$，${\mathcal P}\subset {\mathcal P}^{ps}$ .在这种情况下，${\mathcal P}$ -诱导模的结构完全由${\mathcal P}^{ps}$ -诱导模的结构决定。这些结果推广了V. Futorny，S. König，V. Mazorchuk [Forum Math. 13（2001），641-661]，B. Cox [Pacific J.Math.165（1994），269-294]和I.季米特洛夫、富托尔尼岛Penkov [Comm.Math.Phys.250（2004），47-63]。

英文摘要：

We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra ${\mathcal P}$ of an affine Lie algebra ${\mathfrak G}$, our main result establishes the equivalence between a certain category of ${\mathcal P}$-induced ${\mathfrak G}$-modules and the category of weight ${\mathcal P}$-modules with injective action of the central element of ${\mathfrak G}$. In particular, the induction functor preserves irreducible modules. If ${\mathcal P}$ is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra ${\mathcal P}^{ps}$, ${\mathcal P}\subset {\mathcal P}^{ps}$. The structure of ${\mathcal P}$-induced modules in this case is fully determined by the structure of ${\mathcal P}^{ps}$-induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S. König, V. Mazorchuk [Forum Math. 13 (2001), 641-661], B. Cox [Pacific J. Math. 165 (1994), 269-294] and I. Dimitrov, V. Futorny, I. Penkov [Comm. Math. Phys. 250 (2004), 47-63].

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