رابطۀ الحاقی بین خود-تابعگون‌های hom و تانسور رستۀ (دو-) مدول-های جبرهای هم-مدولی روی یک جبر شبه-هاپف

فرض کنید h یک جبر شبه-هوپف روی حلقه جابه‌جایی k و a یک جبر هم-مدولی روی h باشد. در این مقاله نشان می‌دهیم که گرچه رسته دومدول ها، ama، لزوماً یک رستۀ تکواره ای نیست، با این وجود هم-عمل عمل رستۀ hmh روی amaرا باعث شده و از این رهگذر نسخه های مناسبی از خود-تابعگون های تانسور و hom از رستۀ amaرا معرفی کرده و الحاقی بین این خودتابعگون ها را توصیف می کنیم. هم‌چنین یکه‌ها و هم-یکه‌های وابسته به آنها را صریحاً محاسبه می‌کنیم.

Adjunctions Between Hom and Tensor as Endofunctors of (bi-)Module Category of Comodule Algebras Over a Quasi-Hopf Algebra

IntroductionOver a commutative ring k, it is well known from the classical module theory that the tensorendofunctor of is left adjoint to the Homendofunctor. The unit and counit of this adjunction is obtained trivially.For a kbialgebra (H, 𝝻, 𝝸, 𝞓, 𝞮) the category of (H,H)bimodules is a monoidal category: the tensor product Mof two arbitrary (H,H)bimodules M and N is again an (H,H)bimodule in which the bimodule structure of M is defined diagonally using the comultiplication. The associativity constraint of this category is formally trivial as in the category and it is followed from the coassociativity of the comultiplication. An antipode is an algebra antihomomorphism S;H rarr;H which is the inverse of with respect to the convolution product in . A Hopf algebra is a bialgebra together with an antipode.As generalizations of the concepts bialgebra and Hopf algebra, V. G. Drinfeld introduced the concepts quasibialgebra and quasiHopf algebra respectively. A quasibialgebra over a commutative ring k is an associative algebra H with unit together with a comultiplication: HH and a counit: Hk satisfying all axioms of bialgebras except the coassociativity of 𝞓. However, the noncoassociativity of has been controlled by a normalized 3cocycle 𝞍∊ H in such a way that the category 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. However, we can yet consider tensor functors V and as endofunctors of . A quasiantipode has been defined as a generalization of antipode. A quasiHopf algebra is a quasibialgebra together with a quasiantipode (S, alpha;, beta;).Let (H,𝝻,𝝸,𝞓,𝞮,S, alpha;, beta;) be a quasiHopf algebra with a bijective quasiantipode S. Then it has been shown that the tensor endofunctors V and of have right adjoints which are described in terms of Homfunctors. This means that is a biclosed monoidal category.Over a Hopf algebra H, the category of left Hcomodules is monoidal and algebras and coalgebras can be defined inside this category. In this way, a left Hcomdule algebra is defined as an algebra in the monoidal category of left Hcomodules. However, if H is a quasibialgebra or even a quasiHopf algebra, because of noncoassociativity of comultiplication, we can not define an Hcomodule algebra in this categorical language. To solve this problem, F. Hausser and F. Nill defined an Hcomodule algebra in a formal way as a generalization of the quasibialgebra H and they considered some categories related to an Hcomodule algebra such as the category of twosided Hopf modules.In this article, the bimodule category of a comodule algebra A over a quasiHopf algebra H is considered which is not necessarily monoidal. However, we define varieties of Tensor and Homendofunctors of this category and state Homtensor adjunctions between suitable pairs of these functors. In each case, we compute the unit and counit of adjunction explicitly.Material and methodsFirst we consider the category of left Bmodules, where B is a left comodule algebra over a quasiHopf algebra H and we note that the left action of on yields some varieties of Tensor and Homendofunctors of and we observe that every Tensor functor defined in this way has a right adjoint which is described as a Homfunctor. Next we extend this idea for the bimodule category.Results and discussionFirst we note that although bimodule category of a comodule algebra A over a quasiHopf algebra H is not monoidal, the coaction of H on A yields an action of the bimodule category (which is monoidal) on this bimodule category. This action, in turn, allows us to define Tensor and Homfunctors as endofunctors of the bimodule category.In any case we obtain Tensor and Homendofunctors with the bimodule structure defined diagonally using the coation of H on A and the quasiantipode (S, alpha;, beta;) of H. After that we state HomTensor 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 quasiantipode (S, alpha;, beta;).ConclusionThe following conclusions were drawn from this research.Let H be a quasiHopf algebra with the quasiantipod (S, alpha;, beta;), (B,𝝀, a left Hcomodule algebra and V be an (H,H)bimodule. Then the pairis an adjoint pair of endofuntors with unit and counit given bywhere and are elements in H o×B whose components are given in terms of quasiantipode (S, alpha;, beta;) and components of .Let H be a quasiHopf algebra with quasiantipod (S, alpha;, beta;), (A, rho;, a right Hcomodule algebra and V be an (H,H)bimodule. Then the pairis an adjoint pair of endofuntors with unit and counit given bywhere and are elements in A o×H whose components are given in terms of quasiantipode (S, alpha;, beta;) and components of ../files/site1/files/63/3(1).pdf