D型Fock空间与量子对称对

Type D Fock Spaces and Quantum Symmetric Pairs

  • 摘要: Misra和Miwa通过Fock空间构造了量子仿射代数 U_v(\widehat \mathfraks\mathfrakl_\ell ) 的自然表示. 这将 U_q(\mathfraks\mathfrakl_N) 的Weyl模与 U_v(\widehat \mathfraks\mathfrakl_\ell ) 在Fock空间上的作用联系起来,其中q是单位根. 对于D型量子群 U_q(\mathfraks\mathfrako_2N) ,通过推广A型Fock空间上的仿射量子群作用,构造了一个从有限维 U_q(\mathfraks\mathfrako_2N) 模的Grothendieck群到广义Fock空间的嵌入,并且定义了仿射量子对称对在这些空间上的作用. 由此将仿射量子对称对的作用与单位根处的量子群的邻接原理和张量积分解联系起来,其中 U_q(\mathfraks\mathfrako_2N) 有限维模范畴中的张量积分解与A型量子群 U_q(\mathfraks\mathfrakl_N) 的有限维模范畴中的张量积分解类似.

     

    Abstract: A celebrated construction of the natural representation of the quantized affine algebra U_v(\widehat \mathfraks\mathfrakl_\ell ) is the Fock space by Misra and Miwa. This relates the classes of Weyl modules for U_q(\mathfraks\mathfrakl_N) where q is a root of unity to the action of U_v(\widehat \mathfraks\mathfrakl_\ell ) on the Fock space. For quantum group U_q(\mathfraks\mathfrako_2N) of type D, an embedding of the Grothendieck group of finite dimensional U_q(\mathfraks\mathfrako_2N) -modules into generalized Fock spaces was constructed and an action of an affine quantum symmetric pair on these spaces was defined. This relates the action of the affine quantum symmetric pair to the linkage principal for quantum groups at a root of unity and tensor product multiplicities in analogy to that for finite dimensional module categories of the quantum group U_q(\mathfraks\mathfrakl_N) .

     

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