A categorification of U T.sl.1j1// and its tensor product representations
A categorification of U T.sl.1j1// and its tensor product representations
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U T.sl.1j1// 的分类及其张量积表示
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发表时间：
2013
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通讯作者：
Yin Tian
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作者：
Yin Tian
我们定义了量子超群Uq.sl.1j1//的一个变体，即超代数UT. sl.1j1//，以及它的表示式Vn 1（n>0）.构造了DG代数A，B和Rn族，并考虑了DG范畴DGP.A/，DGP.B/和DGP.Rn/，它们是由某些特殊投射模生成的DG A-，B -和Rn -模范畴的满DG子范畴.对它们的0阶同调范畴HP.A/，HP.B/和HP.Rn/进行了三角剖分，并给出了环，二次穿孔圆盘和n次穿孔圆盘的接触范畴的代数表示.它们的Grothendieck群分别同构于UT .sl.1j1//，UT .sl.1j1// ZUT .sl.1j1//和V n 1.我们把UT 1 j 1//上的乘法和余乘法归结为一个双函子HP.A/ HP.A/！HP.A/和一个函子HP.A/！HP.B/。V n 1上的UT .sl.1j1//-作用被提升到双函子HP.A/ HP.Rn/！HP.Rn/ .
We define the Hopf superalgebra UT .sl.1j1// , which is a variant of the quantum supergroup Uq.sl.1j1// , and its representations V n 1 for n>0 . We construct families of DG algebras A , B and Rn , and consider the DG categories DGP.A/ , DGP.B/ and DGP.Rn/ , which are full DG subcategories of the categories of DG A–, B – and Rn –modules generated by certain distinguished projective modules. Their 0th homology categories HP.A/ , HP.B/ and HP.Rn/ are triangulated and give algebraic formulations of the contact categories of an annulus, a twice punctured disk and an n times punctured disk. Their Grothendieck groups are isomorphic to UT .sl.1j1// , UT .sl.1j1// ZUT .sl.1j1// and V n 1 , respectively. We categorify the multiplication and comultiplication on UT .sl.1j1// to a bifunctor HP.A/ HP.A/! HP.A/ and a functor HP.A/! HP.B/ , respectively. The UT .sl.1j1//–action on V n 1 is lifted to a bifunctor HP.A/ HP.Rn/! HP.Rn/ .