The Brauer category and invariant theory

Gustav I. Lehrer; R. B. Zhang

Displaying similar documents to “The Brauer category and invariant theory”

$n$ -angulated quotient categories induced by mutation pairs

Zengqiang Lin (2015)

Czechoslovak Mathematical Journal

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Geiss, Keller and Oppermann (2013) introduced the notion of $n$ -angulated category, which is a “higher dimensional” analogue of triangulated category, and showed that certain $(n - 2)$ -cluster tilting subcategories of triangulated categories give rise to $n$ -angulated categories. We define mutation pairs in $n$ -angulated categories 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...

Yetter-Drinfeld-Long bimodules are modules

Daowei Lu, Shuan Hong Wang (2017)

Czechoslovak Mathematical Journal

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Let $H$ be a finite-dimensional bialgebra. In this paper, we prove that the category $ℒℛ (H)$ of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category $_{H \otimes H^{*}}^{H \otimes H^{*}} 𝒴𝒟$ over the tensor product bialgebra $H \otimes H^{*}$ as monoidal categories. Moreover if $H$ is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.

The categories of presheaves containing any category of algebras

V. Trnková, J. Reiterman

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ContentsIntroduction.................................................................................................................................................. 5I. Preliminaries........................................................................................................................................... 6II. Main theorem.......................................................................................................................................... 8III. The...

Schur-Finite Motives and Trace Identities

Alessio Del Padrone, Carlo Mazza (2009)

Bollettino dell'Unione Matematica Italiana

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We provide a sufficient condition that ensures the nilpotency of endomorphisms universally of trace zero of Schur-finite objects in a category of homological type, i.e., a $\mathbb{Q}$ -linear $\otimes$ -category with a tensor functor to super vector spaces. We present some applications in the category of motives, where our result generalizes previous results about finite-dimensional objects, in particular by Kimura. We also present some facts which suggest that this might be the best generalization possible...

Two results of $n$ -exangulated categories

Jian He, Jing He, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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M. Herschend, Y. Liu, H. Nakaoka introduced $n$ -exangulated categories, which are a simultaneous generalization of $n$ -exact categories and $(n + 2)$ -angulated categories. This paper consists of two results on $n$ -exangulated categories: (1) we give an equivalent characterization of axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an $n$ -exangulated category.

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer $n$ , let $A_{n}$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $ℳ_{n + 1}$ and $ℱ_{n}$ , where $ℳ_{n + 1}$ and $ℱ_{n}$ are the two extriangulated categories corresponding to the representation category of $A_{n + 1}$ and the morphism category of projective representations of $A_{n}$ , respectively. As a...

On the structure of triangulated categories with finitely many indecomposables

Claire Amiot (2007)

Bulletin de la Société Mathématique de France

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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$ . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $𝒯$ is of the form $ℤ Δ / G$ where $Δ$ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $ℤ Δ$ . Then we prove that for ‘most’ groups $G$ , the category $𝒯$ is standard, ...

Epics in the category of $T_{2}$ k-groups need not have dense range

W. F. Lamartin (1976)

Colloquium Mathematicae

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Derived endo-discrete artin algebras

Raymundo Bautista (2006)

Colloquium Mathematicae

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Let Λ be an artin algebra. We prove that for each sequence ${(h_{i})}_{i \in ℤ}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X \in^{b} (Λ)$ , the bounded derived category of Λ, with $l e n g t h_{E (X)} H^{i} (X) = h_{i}$ for all i ∈ ℤ and E(X) the endomorphism ring of X in $^{b} (Λ)$ if and only if $^{b} (M o d Λ)$ , the bounded derived category of the category $M o d Λ$ of all left Λ-modules, has no generic objects in the sense of [4].

On category $𝒪$ for cyclotomic rational Cherednik algebras

Iain G. Gordon, Ivan Losev (2014)

Journal of the European Mathematical Society

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We study equivalences for category $𝒪_{p}$ of the rational Cherednik algebras $𝐇_{p}$ of type $G_{ℓ} (n) = {(μ_{ℓ})}^{n} ⋊ 𝔖_{n}$ : a highest weight equivalence between $𝒪_{p}$ and $𝒪_{σ (p)}$ for $σ \in 𝔖_{ℓ}$ and an action of $𝔖_{ℓ}$ on an explicit non-empty Zariski open set of parameters $p$ ; a derived equivalence between $𝒪_{p}$ and $𝒪_{p^{'}}$ whenever $p$ and $p^{'}$ have integral difference; a highest weight equivalence between $𝒪_{p}$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$ . As a consequence, we confirm special cases...

How to construct a Hovey triple from two cotorsion pairs

James Gillespie (2015)

Fundamenta Mathematicae

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Let be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs $(, \tilde{})$ and $(\tilde{},)$ in satisfying $\tilde{} \subseteq$ and $\cap \tilde{} = \tilde{} \cap$ . We show how to construct a (necessarily unique) abelian model structure on with (resp. $\tilde{}$ ) as the class of cofibrant (resp. trivially cofibrant) objects, and (resp. $\tilde{}$ ) as the class of fibrant (resp. trivially fibrant) objects.

Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Justyna Kosakowska, Daniel Simson (2008)

Colloquium Mathematicae

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Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor $ℍ_{C} : C - C o m o d \to H_{C} - C o m o d$ that restricts to a representation equivalence $ℍ_{C} : C - c o m o d \to H_{C} - c o m o d_{s p}^{•}$ , where $H_{C}$ is a coradical square complete hereditary...

Generic representations of orthogonal groups: projective functors in the category $ℱ_{q u a d}$

Christine Vespa (2008)

Fundamenta Mathematicae

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We continue the study of the category of functors $ℱ_{q u a d}$ , associated to ₂-vector spaces equipped with a nondegenerate quadratic form, initiated in J. Pure Appl. Algebra 212 (2008) and Algebr. Geom. Topology 7 (2007). We define a filtration of the standard projective objects in $ℱ_{q u a d}$ ; this refines to give a decomposition into indecomposable factors of the first two standard projective objects in $ℱ_{q u a d}$ : $P_{H ₀}$ and $P_{H ₁}$ . As an application of these two decompositions, we give a complete description of the polynomial...

$A F$ -algebras and topology of mapping tori

Igor Nikolaev (2015)

Czechoslovak Mathematical Journal

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The paper studies applications of $C^{*}$ -algebras in geometric topology. Namely, a covariant functor from the category of mapping tori to a category of $A F$ -algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding $A F$ -algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension $2$ , $3$ and $4$ . In conclusion, we consider two numerical examples illustrating our main results.

The correspondence between Barsotti-Tate groups and Kisin modules when $p = 2$

Tong Liu (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a finite extension over $ℚ_{2}$ and $𝒪_{K}$ the ring of integers. We prove the equivalence of categories between the category of Kisin modules of height 1 and the category of Barsotti-Tate groups over $𝒪_{K}$ .

On the Definition of $2$ -Category of $2$ -Knots

V. M. Kharlamov, V. G. Turaev (1993)

Recherche Coopérative sur Programme n°25

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Base change for Picard-Vessiot closures

Andy R. Magid (2011)

Banach Center Publications

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The differential automorphism group, over F, Π₁(F₁) of the Picard-Vessiot closure F₁ of a differential field F is a proalgebraic group over the field $C_{F}$ of constants of F, which is assumed to be algebraically closed of characteristic zero, and its category of $C_{F}$ modules is equivalent to the category of differential modules over F. We show how this group and the category equivalence behave under a differential extension E ⊃ F, where $C_{E}$ is also algebraically closed.

Limits and colimits in certain categories of spaces of continuous functions

Marvin W. Grossman

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CONTENTSIntroduction................................................................................................................................................................................5§ 1. Notation and preliminaries.............................................................................................................................................6§ 2. Epimorphisms and monomorphisms.........................................................................................................................7§...

The Roquette category of finite $p$ -groups

Serge Bouc (2015)

Journal of the European Mathematical Society

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Let $p$ be a prime number. This paper introduces the Roquette category $ℛ_{p}$ of finite $p$ -groups, which is an additive tensor category containing all finite $p$ -groups among its objects. In $ℛ_{p}$ , every finite $p$ -group $P$ admits a canonical direct summand $\partial P$ , called the edge of $P$ . Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$ -groups, and the tensor structure of $ℛ_{p}$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors...

Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

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Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{s g}, I^{s g})$ and $(Q^{g}, I^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K Q^{s g} / ⟨ I^{s g} ⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K Q^{g} / ⟨ I^{g} ⟩$ . As a corollary, we find that $g l d i m K Q^{s g} / ⟨ I^{s g} ⟩ < \infty$ if and only if $g l d i m K Q / ⟨ I ⟩ < \infty$ if and only if $g l d i m K Q^{g} / ⟨ I^{g} ⟩ < \infty$ .

Crystal bases for the quantum queer superalgebra

Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, Myungho Kim (2015)

Journal of the European Mathematical Society

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In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_{q} (𝔮 (n))$ . We define the notion of crystal bases and prove the tensor product rule for $U_{q} (𝔮 (n))$ -modules in the category $𝒪_{int}^{\geq 0}$ . Our main theorem shows that every $U_{q} (𝔮 (n))$ -module in the category $𝒪_{int}^{\geq 0}$ has a unique crystal basis.

Relating quantum and braided Lie algebras

X. Gomez, S. Majid (2003)

Banach Center Publications

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We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if $_{Γ}$ is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space $k \oplus_{Γ}$ is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra $U (_{Γ})$ is a bialgebra in the category of A-comodules.

Right coideal subalgebras of $U_{q}^{+} ({𝔰𝔬}_{2 n + 1})$

V. K. Kharchenko (2011)

Journal of the European Mathematical Society

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We give a complete classification of right coideal subalgebras that contain all grouplike elements for the quantum group $U_{q}^{+} ({𝔰𝔬}_{2 n + 1})$ provided that $q$ is not a root of 1. If $q$ has a finite multiplicative order $t > 4$ ; this classification remains valid for homogeneous right coideal subalgebras of the Frobenius–Lusztig kernel $u_{q}^{+} ({𝔰𝔬}_{2 n + 1})$ . In particular, the total number of right coideal subalgebras that contain the coradical equals $(2 n)!!$ ; the order of the Weyl group defined by the root system of type $B_{n}$ .

On almost complex structures from classical linear connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $ℳ f_{m}$ be the category of $m$ -dimensional manifolds and local diffeomorphisms and let $T$ be the tangent functor on $ℳ f_{m}$ . Let $𝒱$ be the category of real vector spaces and linear maps and let $𝒱_{m}$ be the category of $m$ -dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F : 𝒱_{m} \to 𝒱$ admitting $ℳ f_{m}$ -natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$ -dimensional manifolds $M$ into almost complex structures $\tilde{J} (\nabla)$ on $F (T) M = ⋃_{x \in M} F (T_{x} M)$ .

Rational symplectic field theory over $ℤ^{2}$ for exact Lagrangian cobordisms

Tobias Ekholm (2008)

Journal of the European Mathematical Society

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We construct a version of rational Symplectic Field Theory for pairs $(X, L)$ , where $X$ is an exact symplectic manifold, where $L \subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X, L)$ a $ℤ$ -graded chain complex of vector spaces over $ℤ_{2}$ , filtered with $k$ filtration levels. The corresponding $k$ -level spectral sequence is invariant under deformations of $(X, L)$ and has the following property: if $(X, L)$ is obtained by...

Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations

Daniel Simson (2007)

Colloquium Mathematicae

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We prove that the study of the category C-Comod of left comodules over a K-coalgebra C reduces to the study of K-linear representations of a quiver with relations if K is an algebraically closed field, and to the study of K-linear representations of a K-species with relations if K is a perfect field. Given a field K and a quiver Q = (Q₀,Q₁), we show that any subcoalgebra C of the path K-coalgebra K◻Q containing $K^{◻} Q ₀ \oplus K^{◻} Q ₁$ is the path coalgebra $K^{◻} (Q,)$ of a profinite bound quiver (Q,), and the category...

Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiří Rosický (2015)

Journal of the European Mathematical Society

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We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $𝒮$ of morphisms in a locally presentable category $𝒞$ of structures, the orthogonal class of objects...

Category $𝒪$ for quantum groups

Henning Haahr Andersen, Volodymyr Mazorchuk (2015)

Journal of the European Mathematical Society

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In this paper we study the BGG-categories $𝒪_{q}$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $𝒪$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $𝒪$ and for finite dimensional $U_{q}$ -modules we are able...

A note on model structures on arbitrary Frobenius categories

Zhi-wei Li (2017)

Czechoslovak Mathematical Journal

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We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category $ℱ$ such that the homotopy category of this model structure is equivalent to the stable category $\underset{̲}{ℱ}$ as triangulated categories. This seems to be well-accepted by experts but we were unable to find a complete proof for it in the literature. When $ℱ$ is a weakly idempotent complete (i.e., every split monomorphism is an inflation) Frobenius category, the model structure we constructed is an exact...