Quotient Category Research Articles

Defining relations for the orbit type strata of SU(2)-lattice gauge models

We consider an SU(2)-lattice gauge model in the tree gauge. Classically, this is a system with symmetries whose configuration space is a direct product of copies of SU(2), acted upon by diagonal inner automorphisms. We derive defining relations for the orbit type strata in the reduced classical phase space. The latter is realized as a certain quotient of a direct product of copies of the complexified group SL(2,\CC) (sometimes named the GIT-quotient because it provides a categorical quotient in the sense of geometric invariant theory). The relations derived can be used for the construction of the orbit type costratification of the Hilbert space of the quantum system in the sense of Huebschmann.

Some categorical aspects of coarse spaces and balleans

Coarse spaces [26] and balleans [23] are known to be equivalent constructions ([25]). The main subject of this paper is the category, Coarse, having as objects these structures, and its quotient category Coarse/∼. We prove that the category Coarse is topological and hence Coarse is complete and co-complete and one has a complete description of its epimorphisms and monomorphisms. In particular, Coarse has products and coproducts, quotients, etc., and Coarse is not balanced. A special attention is paid to investigate quotients in Coarse by introducing some particular classes of maps, i.e. (weakly) soft maps which allow one to explicitly describe when the quotient ball structure of a ballean is a ballean. A particular type of quotients, namely the adjunction spaces, is considered in detail in order to obtain a description of the epimorphisms in Coarse/∼, shown to be the bornologous maps with large image. The monomorphisms in Coarse/∼ are the coarse embeddings; consequently, the bimorphisms in Coarse/∼ are precisely the isomorphisms, i.e., Coarse/∼ is a balanced category.

Hopf dense Galois extensions with applications

Let H be a finite dimensional Hopf algebra, and let A be a left H-module algebra. Motivated by the study of the isolated singularities of AH and the endomorphism ring EndAH(A), we introduce the concept of Hopf dense Galois extensions in this paper. Hopf dense Galois extensions yield certain equivalences between the quotient categories over A and AH. A special class of Hopf dense Galois extensions consists of the so-called densely group graded algebras, which are weaker versions of strongly graded algebras. A weaker version of Dade's Theorem holds for densely group graded algebras. As applications, we recover the classical equivalence of the noncommutative projective scheme over a noetherian N-graded algebra A and its d-th Veronese subalgebra A(d) respectively. Hopf dense Galois extensions are also applied to the study of noncommutative graded isolated singularities.

Mutation pairs and quotient categories of Abelian categories

ABSTRACTThe notion of 𝒟-mutation pairs of subcategories in an abelian category is defined in this article. When (𝒵,𝒵) is a 𝒟-mutation pair in an abelian category 𝒜, the quotient category 𝒵∕𝒟 carries naturally a triangulated structure. Moreover, our result generalize the construction of the quotient triangulated category by Happel [10, Theorem 2.6]. Finally, we find a one-to-one correspondence between cotorsion pairs in 𝒜 and cotorsion pairs in the quotient category 𝒵∕𝒟, and study homological finiteness of subcategories in a mutation pair.

Right n-angulated categories arising from covariantly finite subcategories

ABSTRACTWe define right n-angulated categories, which are analogous to right triangulated categories. Let 𝒞 be an additive category and 𝒳 a covariantly finite subcategory of 𝒞. We show that under certain conditions, the quotient 𝒞∕[𝒳] is a right n-angulated category. This has immediate applications to n-angulated quotient categories.

Quotients of A2 * T2

AbstractWe study unitary quotients of the free product unitary pivotal category A2 * T2. We show that such quotients are parametrized by an integer n ≥ 1 and an 2n–th root of unity ω. We show that for n = 1, 2, 3, there is exactly one quotient and ω = 1. For 4 ≤ n ≤ 10, we show that there are no such quotients. Our methods also apply to quotients of T2 * T2, where we have a similar result.The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of A2 × T2 and T2 . T2, we anticipate that our technique can be extended to a general method for proving the nonexistence of planar algebras with a specified principal graph.During the preparation of this manuscript, we learnt of Liu's independent result on composites of A3 and A4 subfactor planar algebras (arxiv:1308.5691). In 1994, BischHaagerup showed that the principal graph of a composite of A3 and A4 must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for n ≥ 4.This is an abridged version of arxiv:1308.5723.

Relative singularity categories I: Auslander resolutions

Let R be an isolated Gorenstein singularity with a non-commutative resolution A=EndR(R⊕M). In this paper, we show that the relative singularity category ΔR(A) of A has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category Dsg(R) of Buchweitz and Orlov as a certain canonical quotient category. If R has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that Dsg(R) determines ΔR(Aus(R)), where Aus(R) is the corresponding Auslander algebra. The proofs of these results use dg algebras, A∞ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.

On lax-algebraic (co)nuclei

Quantic (co)nuclei provide a convenient technique for constructing quo- tients and subquantales of quantales. This paper shows its analogue for the lax- algebraic approach to topology of M. M. Clementino, D. Hofmann, and W. Tholen, based in a monad and a unital quantale. As a result, we get a machinery for constructing quotient categories and subcategories, which provides, in particular, several of the already dened ones by M. M. Clementino et al. We also get a repre- sentation theorem for the approach of M. M. Clementino et al.

Geometric complexity theory V: Efficient algorithms for Noether normalization

We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noether’s Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry. In particular, we show the following: (1) The categorical quotient for any finite dimensional representation V V of S L m SL_m , with constant m m , is explicit in characteristic zero. (2) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in the dimension of V V . (3) The categorical quotient of the space of r r -tuples of m × m m \times m matrices by the simultaneous conjugation action of S L m SL_m is explicit in any characteristic. (4) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in m m and r r in any characteristic p ∉ [ 2 , ⌊ m / 2 ⌋ ] p \not \in [2,\lfloor m/2 \rfloor ] . (5) NNL for every explicit variety in zero or large enough characteristic can be solved deterministically in quasi-polynomial time, assuming the hardness hypothesis for the permanent in geometric complexity theory. The last result leads to a geometric complexity theory approach to put NNL for every explicit variety in P.

Higher Auslander–Reiten sequences and t-structures

Let R be an artin algebra and C an additive subcategory of mod(R). We construct a t-structure on the homotopy category K−(C) and argue that its heart HC is a natural domain for higher Auslander–Reiten (AR) theory. In the paper [5] we showed that K−(mod(R)) is a natural domain for classical AR theory. Here we show that the abelian categories Hmod(R) and HC interact via various functors. If C is functorially finite then HC is a quotient category of Hmod(R). We illustrate our theory with two examples:When C is a maximal n-orthogonal subcategory Iyama developed a higher AR theory, see [10]. In this case we show that the simple objects of HC correspond to Iyama's higher AR sequences and derive his higher AR duality from the existence of a Serre functor on the derived category Db(HC).The category O of a complex semi-simple Lie algebra fits into higher AR theory in the situation when R is the coinvariant algebra of the Weyl group.

N-angulated quotient categories induced by mutation pairs

We define mutation pair in an n-angulated category and prove that given such a mutation pair, the corresponding quotient category carries a natural n-angulated structure. This result generalizes a theorem of Iyama-Yoshino in classical triangulated category. As an application, we obtain that the quotient category of a Frobenius n-angulated category is also an n-angulated category.

Abelian quotients of triangulated categories

We study abelian quotient categories A=T/J, where T is a triangulated category and J is an ideal of T. Under the assumption that the quotient functor is cohomological we show that it is representable and give an explicit description of the functor. We give technical criteria for when a representable functor is a quotient functor, and a criterion for when J gives rise to a cluster-tilting subcategory of T. We show that the quotient functor preserves the AR-structure. As an application we show that if T is a finite 2-Calabi–Yau category, then with very few exceptions J is a cluster-tilting subcategory of T.

The Grothendieck group of non-commutative non-noetherian analogues of [formula omitted] and regular algebras of global dimension two

Let V be a finite-dimensional positively-graded vector space. Let b∈V⊗V be a homogeneous element whose rank is dim⁡(V). Let A=TV/(b), the quotient of the tensor algebra TV modulo the 2-sided ideal generated by b. Let gr(A) be the category of finitely presented graded left A-modules and fdim(A) its full subcategory of finite dimensional modules. Let qgr(A) be the quotient category gr(A)/fdim(A). We compute the Grothendieck group K0(qgr(A)). In particular, if the reciprocal of the Hilbert series of A, which is a polynomial, is irreducible, then K0(qgr(A))≅Z[θ]⊂R as ordered abelian groups where θ is the smallest positive real root of that polynomial. When dimk⁡(V)=2, qgr(A) is equivalent to the category of coherent sheaves on the projective line, P1, or a stacky P1 if V is not concentrated in degree 1. If dimk⁡(V)≥3, results of Piontkovski and Minamoto suggest that qgr(A) behaves as if it is the category of “coherent sheaves” on a non-commutative, non-noetherian analogue of P1.

On the Ext-computability of Serre quotient categories

To develop a constructive description of Ext in categories of coherent sheaves over certain schemes, we establish a binatural isomorphism between the Ext-groups in Serre quotient categories A/C and a direct limit of Ext-groups in the ambient Abelian category A. For Ext1 the isomorphism follows if the thick subcategory C⊂A is localizing. For the higher extension groups we need further assumptions on C. With these categories in mind we cannot assume A/C to have enough projectives or injectives and therefore use Yoneda's description of Ext.

Spectra of tensor triangulated categories over category algebras

Let \({\mathcal{C}}\) be a finite EI category and k be a field. We consider the category algebra \({k\mathcal{C}}\). Suppose \({\sf{K}(\mathcal{C})=\sf{D}^b(k \mathcal{C}-\sf{mod})}\) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category, and we compute its spectrum in the sense of Balmer. When \({\mathcal{C}=G \propto \mathcal{P}}\) is a finite transporter category, the category algebra becomes Gorenstein, so we can define the stable module category \({\underline{\sf{CM}} k(G \propto \mathcal{P})}\), of maximal Cohen–Macaulay modules, as a quotient category of \({{\sf{K}}(G \propto \mathcal{P})}\). Since \({\underline{\sf{CM}} k(G\propto\mathcal{P})}\) is also tensor triangulated, we compute its spectrum as well. These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories.

A block extension of categorical results in the Green correspondence context

Abstract In [J. Pure Appl. Algebra 2 (1972), 371–393, Theorem 4.1], J. A. Green shows that the Green Correspondence in Finite Group Modular Representation Theory is a consequence of an equivalence between two quotient categories of appropriate subcategories in the Green Correspondence context. In [Adv. Math. 104 (1994), 297–314, Theorems 3.5, 3.6 and 3.7], M. Auslander and M. Kleiner prove a similar result. M. Linckelmann suggested that the quotient categories in these results are the same. Utilizing extensions of [The Representation Theory of Finite Groups, North-Holland, Amsterdam, 1982, III, Theorem 7.8] or [Representations of Finite Groups, Academic Press, San Diego, 1988, Chapter 5, Corollary 3.11], we extend these results to blocks of finite groups. In order to state and prove our results and to remain relatively self-contained, we follow the procedures of [Adv. Math. 104 (1994), 297–314] in the Green Correspondent context. This is presented in Section 1. In Section 2 we present our main results. In Section 3 we give a very short proof of a theorem of H. Fitting for 𝒪-algebras that is essential in the proof of basic results of J. A. Green, [J. Pure Appl. Algebra 2 (1972), 371–393, Lemma 3.9 and Theorem 3.10].

Holomorphic Flexibility Properties of the Space of Cubic Rational Maps

For each natural number $$d$$ , the space $$R_d$$ of rational maps of degree $$d$$ on the Riemann sphere has the structure of a complex manifold. The topology of these manifolds has been extensively studied. The recent development of Oka theory raises some new and interesting questions about their complex structure. We apply geometric invariant theory to the cases of degree 2 and 3, studying a double action of the Mobius group on $$R_d$$ . The action on $$R_2$$ is transitive, implying that $$R_2$$ is an Oka manifold. The action on $$R_3$$ has $${\mathbb C}$$ as a categorical quotient; we give an explicit formula for the quotient map and describe its structure in some detail. We also show that $$R_3$$ enjoys the holomorphic flexibility properties of strong dominability and $${\mathbb C}$$ -connectedness.

INVARIANT HILBERT SCHEMES AND DESINGULARIZATIONS OF QUOTIENTS BY CLASSICAL GROUPS

Let W be a finite-dimensional representation of a reductive algebraic group G. The invariant Hilbert scheme \( \mathcal{H} \) is a moduli space that classifies the G-stable closed subschemes Z of W such that the affine algebra k[Z] is the direct sum of simple G-modules with prescribed multiplicities. In this article, we consider the case where G is a classical group acting on a classical representation W and k[Z] is isomorphic to the regular representation of G as a G-module. We obtain families of examples where \( \mathcal{H} \) is a smooth variety, and thus for which the Hilbert–Chow morphism \( \gamma :\mathcal{H}\to W//G \) is a canonical desingularization of the categorical quotient.

Quotient categories, stability conditions, and birational geometry

This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. We prove that this category has homological dimension c. As an application, we describe the space of stability conditions on its derived category in the case c \(=\) 1. Moreover, we describe all exact equivalences between these quotient categories in this particular case, which is closely related to classification problems in birational geometry.

Quotients, automorphisms and differential operators

Let V be a G-module, where G is a complex reductive group. Let Z≔V//G denote the categorical quotient and let π : V→Z be the morphism dual to the inclusion 𝒪(V)G⊂𝒪(V). Let φ : Z→Z be an algebraic automorphism. Then one can ask whether there is an algebraic map Φ : V→V which lifts φ, that is, π(Φ(v))=φ(π(v)) for all v∈V. In Kuttler (Transform. Groups 16 (2011) 1115–1135), the case is treated where V=r𝔤 is a multiple of the adjoint representation of G. It is shown that, for r sufficiently large (often r⩾2 will do), any φ has a lift. We consider the case of general representations (satisfying some mild assumptions). It turns out that it is natural to consider holomorphic lifting of holomorphic automorphisms of Z, and we show that if a holomorphic φ and its inverse lift holomorphically, then φ has a lift Φ which is an automorphism such that Φ(gv)=σ(g)Φ(v), v∈V, g∈G, where σ is an automorphism of G. We reduce the lifting problem to the group of automorphisms of Z which preserve the natural grading of 𝒪(Z) ≃ 𝒪(V)G. Lifting does not always hold, but we show that it always does for representations of tori in which case algebraic automorphisms lift to algebraic automorphisms. We extend Kuttler's methods to show lifting in the case where V contains a copy of 𝔤.