Some Remarks on Algebras Over an Algebraically Closed Field

Nesbitt, Cecil J.; Scott, Winston M.

doi:10.2307/1968979

Cited by 18 publications

(4 citation statements)

References 12 publications

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“…A-mod → A b -mod, X → eX is an equivalence. The following lemma is well-known [NS43], but we give a proof from the viewpoint of Definition 6.6.…”

Section: Reynolds Ideal and Its Generalizationmentioning

confidence: 99%

Further Results on the Structure of (Co)Ends in Finite Tensor Categories

Shimizu

2019

Appl Categor Struct

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Let C be a finite tensor category, and let M be an exact left Cmodule category. The action of C on M induces a functor ρ : C → Rex(M), where Rex(M) is the category of k-linear right exact endofunctors on M.Our key observation is that ρ has a right adjoint ρ ra given by the endAs an application, we establish the following results: (1) We give a description of the composition of the induction functor C * M → Z(C * M ) and Schauenburg's equivalence Z(C * M ) ≈ Z(C).(2) We introduce the space CF(M) of 'class functions' of M and initiate the character theory for pivotal module categories.(3) We introduce a filtration for CF(M) and discuss its relation with some ringtheoretic notions, such as the Reynolds ideal and its generalizations. (4) We show that Ext • C (1, ρ ra (id M )) is isomorphic to the Hochschild cohomology of M. As an application, we show that the modular group acts projectively on the Hochschild cohomology of a modular tensor category. 4 K. SHIMIZU 2. Preliminaries 2.1. Ends and coends. For basic theory on categories, we refer the reader to the book of Mac Lane [ML98]. Let C and D be categories, and let S and T be functorsAn end of S is an object E ∈ D equipped with a dinatural transformation π : E → S that is universal in a certain sense (here the object E is regarded as a constant functor from C op × C to D). Dually, a coend of T is an object C ∈ D equipped with a 'universal' dinatural transformation from T to C. An end of S and a coend of T are denoted by X∈C S(X, X) and X∈C T (X, X), respectively.A (co)end does not exist in general. We note the following useful criteria for the existence of (co)ends. Suppose that C is essentially small. Let C, D and S be as above. Since the category Set of all sets is complete, the end S ♮ (W ) := X∈C Hom D (W, S(X, X)) exists for each object W ∈ D. By the parameter theorem for ends [ML98, XI.7], we extend the assignment W → S ♮ (W ) to the contravariant functor S ♮ : D → Set. The following lemma is the dual of [Shi17c, Lemma 3.1]. Lemma 2.1. An end of S exists if and only if S ♮ is representable.We also note the following lemma: Lemma 2.2. Let A, B and V be categories, and let L : A → B, R : B → A and H : B op × A → V be functors. Suppose that L is left adjoint to R. Then we have an isomorphismmeaning that if either one of these ends exists, then both exist and they are canonically isomorphic.This lemma is the dual of [BV12, Lemma 3.9]. For later use, we recall the construction of the canonical isomorphism (2.1). Let E and E ′ be the left and the right hand side of (2.1), respectively, and letbe the respective universal dinatural transformations. We assume that (L, R) is an adjoint pair with unit η : id D → RL and counit ε : LR → id C . By the universal property of E, there is a unique morphism α :for all objects V ∈ A. Similarly, by the universal property of E ′ , there is a unique morphism β : E → E ′ satisfying π ′ (W ) • β = π(R(W )) • H(id R(W ) , ε W ) for all objects W ∈ B. By the zigzag identities and the dinaturality of π and π ′ , one can verify that α and...

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“…A-mod → A b -mod, X → eX is an equivalence. The following lemma is well-known [NS43], but we give a proof from the viewpoint of Definition 6.6.…”

Section: Reynolds Ideal and Its Generalizationmentioning

confidence: 99%

Further Results on the Structure of (Co)Ends in Finite Tensor Categories

Shimizu

2019

Appl Categor Struct

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“…Recall that every finite dimensional algebra over an algebraically closed field is Morita equivalent to a basic algebra (see e.g. [NS43,Jac89,Ben95]). This allows us to track separately the combinatorial and homological data of an algebra.…”

Section: Andmentioning

confidence: 99%

Generalisations of Hecke algebras from Loop Braid Groups

Damiani,

Martin,

Rowell

2020

Preprint

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We introduce a generalisation LHn of the ordinary Hecke algebras informed by the loop braid group LBn and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that LHn has analogues of several of these properties. In particular we introduce a class of local (tensor space/functor) representations of the braid group derived from a meld of the (nonfunctor) Burau representation [Bur35] and the (functor) Rittenberg representations [MR92], here thus called Burau-Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau-Rittenberg representation extends to a loop Burau-Rittenberg representation. And this factors through LHn. Let SPn denote the corresponding (not necessarily proper) quotient algebra, k the ground ring, and t ∈ k the loop-Hecke parameter. We prove the following:(1) LHn is finite dimensional over a field.(2) The natural inclusion LBn ֒→ LBn+1 passes to an inclusion SPn ֒→ SPn+1.(3) Over k = C, SPn/rad is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. (Specifically SPn/rad ∼ = CSn/e 1 (2,2) where Sn is the symmetric group and e 1 (2,2) is a λ = (2, 2) primitive idempotent.) (4) We determine the other fundamental invariants of SPn representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. (5) the structure of SPn is independent of the parameter t, except for t = 1. (6) For t 2 = 1 then LHn ∼ = SPn at least up to rank n = 7 (for t = −1 they are not isomorphic for n > 2; for t = 1 they are not isomorphic for n > 1). Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.

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“…Further, coupling Hochschild's theorem with an earlier theorem of Nesbitt and Scott [48], which in current language asserts that every finite-dimensional algebra is Morita equivalent to one which is commutative modulo its radical, one is led to try to represent each finite-dimensional algebra in terms of a finite directed graph, called the quiver of the algebra. When an algebra is commutative modulo its radical, the tensor algebra is the path algebra built from the quiver.…”

Section: (Z) Maps F Into E That Is Bounded and Analytic In The Uppermentioning

confidence: 99%