Search

In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field $\mathbb F$. Given a tensor category $\mathcal{C}$, we have two structure invariants of $\mathcal{C}$: the Green ring (or the representation ring) $r(\mathcal{C})$ and the Auslander algebra $A(\mathcal{C})$ of $\mathcal{C}$. We show that a Krull-Schmit abelian tensor category $\mathcal{C}$ of finite rank is uniquely determined (up to tensor equivalences) by its two structure invariants and the associated associator system of $\mathcal{C}$. In fact, we can reconstruct the tensor category $\mathcal{C}$ from its two invariants and the associator system. More general, given a quadruple $(R, A, \phi, a)$ satisfying certain conditions, where $R$ is a $\mathbb{Z}_+$-ring of rank $n$, $A$ is a finite dimensional $\mathbb F$-algebra with a complete set of $n$ primitive orthogonal idempotents, $\phi$ is an algebra map from $A\otimes_{\mathbb F}A$ to an algebra $M(R, A)$ constructed from $A$ and $R$, and $a=\{a_{i,j,l}|1\leqslant i,j,l\leqslant n}$ is a family of ``invertible" matrices over $A$, we can construct a Krull-Schmidt and abelian tensor category $\mathcal C$ over $\mathbb{F}$ such that $R$ is the Green ring of $\mathcal C$ and $A$ is the Auslander algebra of $\mathcal C$. In this case, $\mathcal C$ has finitely many indecomposable objects (up to isomorphisms) and finite dimensional Hom-spaces. Moreover, we will give a necessary and sufficient condition for such two tensor categories to be tensor equivalent.

Motivated by ideas from string theory and quantum field theory new invariants of knots and 3-dimensional manifolds have been constructed from complex algebraic structures such as Hopf algebras (Reshetikhin and Turaev), monoidal categories with additional structure (Turaev and Yetter), and modular functors (Walker and Kontsevich). These constructions are very closely related. We take a unifying categorical approach based on a natural 2-dimensional generalization of a topological field theory in the sense of Atiyah and Segal, and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces. In particular, we show that any linear category over a field with an action of the surface category is semi-simple and Artinian., Accepted for publication in the Journal of the LMS, April 1996

We compute the Green ring of the Taft algebra $H_n(q)$, where $n$ is a positive integer greater than 1, and $q$ is an $n$-th root of unity. It turns out that the Green ring $r(H_n(q))$ of the Taft algebra $H_n(q)$ is a commutative ring generated by two elements subject to certain relations defined recursively. Concrete examples for $n=2,3,..., 8$ are given., To appear in Proc. AMS

We consider the Brauer group ${\rm BM}'(k,G)$ of a group $G$ (finite or infinite) over a commutative ring $k$ with identity. A split exact sequence $$1\longrightarrow {\rm Br}'(k)\longrightarrow {\rm BM}'(k,G)\longrightarrow {\rm Gal}(k,G) \longrightarrow 1$$ is obtained. This generalizes the Fr\"ohlich-Wall exact sequence from the case of a field to the case of a commutative ring, and generalizes the Picco-Platzeck exact sequence from the finite case of $G$ to the infinite case of $G$. Here ${\rm Br}'(k)$ is the Brauer-Taylor group of Azumaya algebras (not necessarily with unit). The method developed in this paper might provide a key to computing the equivariant Brauer group of an infinite quantum group., Comment: 13 pages

The topic of this work is the Brauer group of a commutative ring. Recent investigations have yielded generalizations of this invariant in several directions; herein we wish to propose another: that the Brauer group be viewed as a subgroup of the unit group of the “Brauer ring”. In justification, we should expect this ring to store and yield information about separable algebras which the Brauer group does not, and we should hope to be able to recover the Brauer group from purely ring theoretic properties. One purpose of the present paper is to describe the extent to which these goals are achieved. Another is to show that, in a categorical sense, the ring we shall describe is best possible. ¶ To obtain structural results for the Brauer ring, we will make use of the theory of Green-functors, especially of the Burnside ring. Our structure theory will be independent of the construction of the Brauer ring, thus giving promise for applications in other areas. So as to not get too lost in our categorical approach, we have omitted some straight-forward axiom checking proofs; they may be regarded as exercises. ¶ In this paper we will restrict our attention to the field case, leaving the Brauer ring of more general algebraic objects to another time. ¶ The author wishes to express his deep respect and thanks to Professor R. S. Pierce, for his endless insights in to the problems at hand, and for asking the fundamental questions on which this work is based.

Employing sheaf-theoretic techniques and idempotent lifting properties, the following extension of the Wedderburn Principal Theorem is shown: If A is any finitely generated algebra over an almost Dedekind domain R such that A modulo its Baer lower radical is separable over R, then A contains a separable subalgebra S which adds with the lower radical as R-modules to give A. Certain structural results concerning separable algebras over arbitrary Prufer domains are obtained. These parallel results already known for Dedekind domains. Finally, results relating the Hochschild and the weak global dimensions of an algebra with the weak global dimension of the ground ring are obtained. The ground rings include some Prufer domains and their generalizations. The purpose of this article is to examine the structure of commutative and non-commutative algebras over Prufer domains, and in particular over almost Dedekind domains. We shall prove for these last rings a generalization of the Wedderburn Principal Theorem. In § 1, it will be shown that every finitely generated, torsion-free, separable commutative algebra S over a Prufer domain R is again a Prufer domain. If R is an almost Dedekind domain, then so also is S. In § 2, similar results will be obtained for non-commutative separable algebras over these same types of ground ring. Moreover, since a finitely generated, torsion-free algebra over a Prufer domain is projective, one may write any separable algebra over a Prufer domain as a direct sum of the torsion ideal and the projective ideal. This decomposition can be "lifted" to any finitely generated algebra A which is separable modulo its lower radical. An application of sheaf-theoretic techniques as in [3] to this decomposition obtains the Wedderburnlike structure theorem for almost Dedekind domains. In § 3, we will give results relating the Hochschild dimension and the weak global dimension of a finitely generated, torsion-free algebra Received by the editors August 9,1973. AMS Subject Classification (1970): Primary 13F05, 16A16, 16A46; Secondary 13E10, 16A32, 16A60.