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رابطۀ الحاقی بین خود-تابعگونهای Hom و تانسور رستۀ (دو-) مدول-های جبرهای هم-مدولی روی یک جبر شبه-هاپف
Adjunctions Between Hom and Tensor as Endofunctors of (bi-)Module Category of Comodule Algebras Over a Quasi-Hopf Algebra
جبر
alg
علمی پژوهشی بنیادی
S
<span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">فرض کنید </span></span><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">H</span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> یک جبر شبه-هوپف روی حلقه جابهجایی </span></span><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">k</span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> و </span></span><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">A</span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> یک جبر هم-مدولی روی </span></span><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">H</span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> باشد. در این مقاله نشان میدهیم که گرچه رسته دو-مدول­ها، </span></span><sub><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">A</span></span></span></sub><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">M<sub>A</sub></span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">، لزوماً یک رستۀ تکواره­ای نیست، با این وجود هم-عمل عمل رستۀ </span></span><sub><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">H</span></span></span></sub><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">M<sub>H</sub></span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> روی </span></span> <sub><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">A</span></span></span></sub><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">M<sub>A</sub></span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">را باعث شده و از این رهگذر نسخه­های مناسبی از خود-تابعگون­های تانسور و </span></span><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">Hom</span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> از رستۀ </span></span> <sub><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">A</span></span></span></sub><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">M<sub>A</sub></span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">را معرفی کرده و الحاقی بین این خود-تابعگون­ها را توصیف می­کنیم. همچنین یکهها و هم-یکههای وابسته به آنها را صریحاً محاسبه میکنیم.<a href="./files/site1/files/61/%D9%81%D8%B1%D8%B6_%DA%A9%D9%86%DB%8C%D8%AF_H_%DB%8C%DA%A9_%D8%AC%D8%A8%D8%B1_%D8%B4%D8%A8%D9%87.pdf">./files/site1/files/61/%D9%81%D8%B1%D8%B6_%DA%A9%D9%86%DB%8C%D8%AF_H_%DB%8C%DA%A9_%D8%AC%D8%A8%D8%B1_%D8%B4%D8%A8%D9%87.pdf</a></span></span>
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<strong>Introduction</strong><br>
Over a commutative ring k, it is well known from the classical module theory that the tensor-endofunctor of<img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="33" > is left adjoint to the Hom-endofunctor. The unit and counit of this adjunction is obtained trivially.<br>
For a k-bialgebra (H, 𝝻, 𝝸, 𝞓, 𝞮) the category of (H,H)-bimodules is a monoidal category: the tensor product M<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="40" >of two arbitrary (H,H)-bimodules M and N is again an (H,H)-bimodule in which the bimodule structure of M<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="36" > is defined diagonally using the comultiplication. The associativity constraint of this category is formally trivial as in the category <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="33" > and it is followed from the coassociativity of the comultiplication. An antipode is an algebra anti-homomorphism S;H→H which is the inverse of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="29" >with respect to the convolution product in <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="57" >. A Hopf algebra is a bialgebra together with an antipode.<br>
As generalizations of the concepts bialgebra and Hopf algebra, V. G. Drinfeld introduced the concepts quasi-bialgebra and quasi-Hopf algebra respectively. A quasi-bialgebra over a commutative ring k is an associative algebra H with unit together with a comultiplication: H<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="12" >H<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="36" > and a counit: H<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="12" >k satisfying all axioms of bialgebras except the coassociativity of 𝞓. However, the non-coassociativity of has been controlled by a normalized 3-cocycle 𝞍∊ H<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="76" > in such a way that the category <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="42" > of (H,H)-bimodules is monoidal. In this case, the associativity constraint of the category is not the trivial one and it depends on the element 𝞍 ∊ H<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="76" >. However, we can yet consider tensor functors V<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image020.gif" width="33" > and -<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image022.gif" width="35" > as endofunctors of <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="42" >. A quasi-antipode has been defined as a generalization of antipode. A quasi-Hopf algebra is a quasi-bialgebra together with a quasi-antipode (S,α,β).<br>
Let (H,𝝻,𝝸,𝞓,𝞮,S,α,β) be a quasi-Hopf algebra with a bijective quasi-antipode S. Then it has been shown that the tensor endofunctors V<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image020.gif" width="33" > and -<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image022.gif" width="35" > of <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="42" > have right adjoints which are described in terms of Hom-functors. This means that <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="42" >is a biclosed monoidal category.<br>
Over a Hopf algebra H, the category <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image024.gif" width="36" >of left H-comodules is monoidal and algebras and coalgebras can be defined inside this category. In this way, a left H-comdule algebra is defined as an algebra in the monoidal category <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image024.gif" width="36" >of left H-comodules. However, if H is a quasi-bialgebra or even a quasi-Hopf algebra, because of non-coassociativity of comultiplication, we can not define an H-comodule algebra in this categorical language. To solve this problem, F. Hausser and F. Nill defined an H-comodule algebra in a formal way as a generalization of the quasi-bialgebra H and they considered some categories related to an H-comodule algebra such as the category of two-sided Hopf modules.<br>
In this article, the bimodule category <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image026.gif" width="36" > of a comodule algebra A over a quasi-Hopf algebra H is considered which is not necessarily monoidal. However, we define varieties of Tensor and Hom-endofunctors of this category and state Hom-tensor adjunctions between suitable pairs of these functors. In each case, we compute the unit and counit of adjunction explicitly.<br>
<strong>Material and methods</strong><br>
First we consider the category <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image028.gif" width="36" >of left B-modules, where B is a left comodule algebra over a quasi-Hopf algebra H and we note that the left action of <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image030.gif" width="36" >on <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image028.gif" width="36" >yields some varieties of Tensor and Hom-endofunctors of <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image028.gif" width="36" > and we observe that every Tensor functor defined in this way has a right adjoint which is described as a Hom-functor. Next we extend this idea for the bimodule category<img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image034.gif" width="39" >.<br>
<strong>Results and discussion</strong><br>
First we note that although bimodule category <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image026.gif" width="36" >of a comodule algebra A over a quasi-Hopf algebra H is not monoidal, the coaction of H on A yields an action of the bimodule category <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="42" > (which is monoidal) on this bimodule category. This action, in turn, allows us to define Tensor and Hom-functors as endofunctors of the bimodule category<img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image026.gif" width="36" >.<br>
In any case we obtain Tensor and Hom-endofunctors with the bimodule structure defined diagonally using the coation of H on A and the quasi-antipode (S,α, β) of H. After that we state Hom-Tensor adjunction between corresponding pairs of Hom and Tensor endofunctors. The units and counits of adjunctions are not trivial as in the Hopf algebra case and they strongly depend on the invariants of the comodule algebra A and the quasi-antipode (S,α, β).<br>
<strong>Conclusion</strong><br>
The following conclusions were drawn from this research.
<ul>
<li>Let H be a quasi-Hopf algebra with the quasi-antipod (S,α,β), (B,𝝀,<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image038.gif" width="23" > a left H-comodule algebra and V be an (H,H)-bimodule. Then the pair</li>
</ul>
<img alt="" height="27" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image040.gif" width="360" ><br>
is an adjoint pair of endofuntors with unit and counit given by<br>
<img alt="" height="27" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image042.gif" width="456" ><br>
<img alt="" height="27" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image044.gif" width="552" >where <img alt="" height="33" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image046.gif" width="132" >and <img alt="" height="30" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image048.gif" width="126" >are elements in H⊗B whose components are given in terms of quasi-antipode (S,α, β) and components of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image050.gif" width="16" >.
<ul>
<li>Let H be a quasi-Hopf algebra with quasi-antipod (S,α,β), (A,ρ,<img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image052.gif" width="23" > a right H-comodule algebra and V be an (H,H)-bimodule. Then the pair</li>
</ul>
<img alt="" height="27" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image054.gif" width="372" ><br>
is an adjoint pair of endofuntors with unit and counit given by<br>
<img alt="" height="27" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image056.gif" width="456" ><br>
<img alt="" height="27" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image058.gif" width="558" >where <img alt="" height="33" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image060.gif" width="135" >and <img alt="" height="30" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image062.gif" width="126" >are elements in A⊗H whose components are given in terms of quasi-antipode (S,α, β) and components of <img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image064.gif" width="17" >.<a href="./files/site1/files/63/3(1).pdf">./files/site1/files/63/3(1).pdf</a>
جبر (شبه-) هوپف, جبر هم- مدولی, رسته تکوارهای, عمل یک رسته تکوارهای
(quasi-) Hopf algebra, Comodule algebra, Monoidal category, Action of monoidal category
347
362
http://mmr.khu.ac.ir/browse.php?a_code=A-10-567-1&slc_lang=fa&sid=1
Saeid
Bagheri
سعید
باقری
bagheri_saeid@yahoo.com
10031947532846004072
10031947532846004072
Yes
Faculty of Mathematical Sciences, Malayer University, Malayer, Iran
دانشگاه ملایر، دانشکدۀ علوم ریاضی