by Jespers, Eric; Okniński, Jan

A monoid S generated by {x1,. . .,xn} is said to be of (left) I-type if there exists a map v from the free Abelian monoid FaMn of rank n generated by {u1,. . .,un} to S so that for all a∈FaMn one has {v(u1a),. . .,v(una)}={x1v(a),. . .,xnv(a)}. Then S has a group of fractions, which is called a group of (left) I-type. These monoids first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh.

In this paper we show that monoids and groups of left I-type can be characterized as natural submonoids and groups of semidirect products of the free Abelian group Fan and the symmetric group of degree n. It follows that these notions are left–right symmetric. As a consequence we determine many aspects of the algebraic structure of such monoids and groups. In particular, they can often be decomposed as products of monoids and groups of the same type but on less generators and many such groups are poly-infinite cyclic. We also prove that the minimal prime ideals of a monoid S of I-type, and of the corresponding monoid algebra, are principal and generated by a normal element. Further, via left–right divisibility, we show that all semiprime ideals of S can be described. The latter yields an ideal chain of S with factors that are semigroups of matrix type over cancellative semigroups.

DOI: 10.1007/s10468-005-0342-7
Print publication date: 12/1/2005
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by Leroy, André; Matczuk, Jerzy

Let R be a ring, σ an injective endomorphism of R and δ a σ-derivation of R. We prove that if R is semiprime left Goldie then the same holds for the Ore extension R[x;σ,δ] and both rings have the same left uniform dimension.

DOI: 10.1007/s10468-005-0707-y
Print publication date: 12/1/2005
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by Buan, Aslak Bakke; Solberg, Øyvind

The set of pure-injective cotilting modules over an artin algebra is shown to have a monoid structure. This monoid structure does not restrict down to a monoid structure on the finitely generated cotilting modules in general, but it does whenever the algebra is of finite representation type. Pure-injective cotilting modules are also constructed from any set of finitely generated cotilting modules with bounded injective dimension.

DOI: 10.1007/s10468-005-0341-8
Print publication date: 12/1/2005
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by Naito, Satoshi; Sagaki, Daisuke

In our previous work, we introduced a bijection between the elements of the crystal base of the negative (resp. positive) part of the quantized universal enveloping algebra
$U_{q}(\mathfrak{g})$
of a Kac–Moody algebra
$\mathfrak{g}$
that are fixed by a diagram automorphism and the elements of the crystal base of the negative (resp. positive) part of the quantized universal enveloping algebra
$U_{q}(\breve {\mathfrak{g}})$
of the orbit Lie algebra
$\breve {\mathfrak{g}}$
of
$\mathfrak{g}$
. In this paper, we prove that this bijection commutes with the *-operation. As an application of this result we show that there exists a canonical bijection between the elements ℬ0(λ) of the crystal base ℬ(λ) of an extremal weight module of extremal weight λ over
$U_{q}(\mathfrak{g})$
that are fixed by a diagram automorphism and the elements of the crystal base
$\breve {\mathcal{B}}(\widehat {\lambda})$
of an extremal weight module of extremal weight
$\widehat {\lambda}$
over
$U_{q}(\breve {\mathfrak{g}})$
, if the crystal graph of
$\breve {\mathcal{B}}(\widehat {\lambda})$
is connected.

DOI: 10.1007/s10468-005-0234-x
Print publication date: 12/1/2005
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by Boltje, Robert; Külshammer, Burkhard

We prove a strengthened statement of Dress's induction theorem for the Green ring of modular representations of a finite group using the theory of the multiplicity module of an indecomposable modular representation. Moreover, we construct an integral canonical induction formula for the Green ring for indecomposable representations with normal vertex.

DOI: 10.1007/s10468-005-8296-3
Print publication date: 12/1/2005
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by Iyama, Osamu

We study the relationship between representation theoretic properties and homological properties of orders. We show that there is a close relationship among Auslander orders, τ-categories and Auslander regular rings. As an application, we give a combinatorial characterization of finite Auslander–Reiten quivers of orders.

DOI: 10.1007/s10468-005-0970-y
Print publication date: 12/1/2005
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by Bautista, Raymundo; Zuazua, Rita

Let Λ be a finite-dimensional algebra over an algebraically closed field k. We denote by mod Λ the category of finitely generated left Λ-modules. Consider the family ℱ(u) of the indecomposables M∈mod Λ such that
$\mathrm{dim}_{k}\,\mathrm{Hom}_{\Lambda}(M,\mathrm{Dtr}\,M)/\mathcal{S}(M,\mathrm{Dtr}\,M)=u$
, where
$\mathcal{S}(M,\mathrm{Dtr}\,M)$
is the subspace of morphisms which factorize through semisimple modules. If P,Q are projectives in mod Λ, ℱ(u)(P,Q) is the family of those modules M∈ℱ(u) such that a minimal projective presentation is of the formfM: P→Q. We prove that if Λ is of tame representation type then each ℱ(P,Q) has only a finite number of isomorphism classes or is parametrized by μ(u,P,Q) one-parameter families. We give an upper bound for this number in terms of u,P and Q. Then we give some sufficient conditions for tame of polynomial growth type. For the proof we consider similar results for bocses.

DOI: 10.1007/s10468-004-6938-5
Print publication date: 12/1/2005
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