by Kosakowska, Justyna

Let $K$ be an algebraically closed field and let $I$ be a finite partially ordered set of finite prinjective type. We study generic extensions of prinjective modules over the incidence $K$-algebra $KI$ of $I$. We prove that there exist generic extensions of prinjective $KI$-modules and describe properties of the monoid ${\user1{\mathcal{M}}}{\left( I \right)}$ of generic extensions.

DOI: 10.1007/s10468-006-9033-2
Online Date: 7/28/2006
Print publication date: 12/1/2006
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by Green, Edward L.; Zhang, Pu

Let $A$ be a finite-dimensional hereditary algebra of finite or tame representation type over a finite field, and let $M$ be a rigid $A$-module. Then the element $[M]$ in the Ringel–Hall algebra $\mathcal{H}{\left( A \right)}$ is an iterated skew commutator of the isoclasses of simple $A$-modules. This gives a new characterization of the rigidness of an indecomposable module over a tame hereditary algebra.

DOI: 10.1007/s10468-006-9010-9
Online Date: 7/26/2006
Print publication date: 12/1/2006
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by Kang, Seok-Jin; Lee, Hyeonmi

Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals $\mathcal{B}(\lambda)$ for quantum affine algebras of type $A_{\,n}^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $A_{\,2n-1}^{(2)}$, $A_{\,2n}^{(2)}$, and $D_{n+1}^{(2)}$. The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the construction of crystals.

DOI: 10.1007/s10468-006-9013-6
Online Date: 7/25/2006
Print publication date: 12/1/2006
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by Kennedy, Christopher

Deep matrix algebras $$\varepsilon {\left( {X,A} \right)}$$ based on a set $X$ over a ring $A$ are introduced and studied by McCrimmon when $X$ has infinite cardinality. Here, we construct a new $$\varepsilon {\left( {X,A} \right)}$$-module that is faithful for $X$ of any cardinality. For $A=\mathbb{K}$ a field of arbitrary characteristic and $2\le |X\,|<\infty$, $$\varepsilon {\left( {X,\mathbb{K}} \right)}$$ is studied in depth. The algebra $$\varepsilon {\left( {X,\mathbb{K}} \right)}$$ is shown to possess a unique proper non-zero ideal, isomorphic to $Mat_{\infty}(\mathbb{K})$. This leads to a new and interesting simple algebra, $${\user1{\mathcal{A}}}{\left( {X,\,\mathbb{K}} \right)}$$, the quotient of $$\varepsilon {\left( {X,\mathbb{K}} \right)}$$ by its unique ideal. Both $$\varepsilon {\left( {X,\mathbb{K}} \right)}$$ and the quotient algebra are shown to have centers isomorphic to $\mathbb{K}$. Via the new faithful representation, all automorphisms of $$\varepsilon {\left( {X,\mathbb{K}} \right)}$$ are shown to be inner in the sense of Definition 18.

DOI: 10.1007/s10468-006-9024-3
Online Date: 7/21/2006
Print publication date: 10/1/2006
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by Bashir, Robert

We find conditions on $\mathcal F$, a class of objects of a Grothendieck category, sufficient for the existence of $\mathcal F$-covers. The theory includes the existence of flat covers of modules.

DOI: 10.1007/s10468-006-9030-5
Online Date: 7/20/2006
Print publication date: 10/1/2006
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by Szántó, Csaba

We present formulas for the structure constants (Hall numbers) of the Hall algebra $+AFw-mathcal{H}(kK)$ associated to the Kronecker algebra. The formulas which in some cases involve the classical Hall polynomials $g^{+AFw-lambda}_{(r)+AFw-mu}$ enable us to determine every Hall number. Using again these formulas we construct new PBW-bases with simple structure constants for the composition algebra $+AFw-mathcal{C}(kK)$, making possible the definition of the generic composition algebra via Hall polynomials.

DOI: 10.1007/s10468-006-9019-0
Online Date: 7/19/2006
Print publication date: 10/1/2006
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by Kawai, Hiroaki; Sasaki, Hiroki

M. Linckelmann defined the cohomology algebras of blocks of finite groups. This note is an attempt to analyze an inclusion of cohomology algebras of blocks that corresponds under Brauer correspondence through transfer maps between the Hochschild cohomology algebras of the blocks.

DOI: 10.1007/s10468-006-9026-1
Online Date: 7/18/2006
Print publication date: 10/1/2006
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by Schmuedgen, Konrad; Wagner, Elmar

In this paper, we study and classify Hilbert space representations of cross product $\ast$-algebras of the quantized enveloping algebra ${\user1{\mathcal{U}}}_{q} {\left( {{\text{e}}_{{\text{2}}} } \right)}$ with the coordinate algebras ${\user1{\mathcal{O}}}{\left( {{\text{E}}_{q} {\left( 2 \right)}} \right)}$ of the quantum motion group and ${\user1{\mathcal{O}}}{\left( {\mathbb{C}_{q} } \right)}$ of the complex plane, and of the quantized enveloping algebra ${\user1{\mathcal{U}}}_{q} {\left( {{\text{su}}_{{1,1}} } \right)}$ with the coordinate algebras ${\user1{\mathcal{O}}}{\left( {{\text{SU}}_{q} {\left( {1,1} \right)}} \right)}$ of the quantum group $\mathrm{SU}_q(1,1)$ and ${\user1{\mathcal{O}}}{\left( {{\text{U}}_{p} } \right)}$ of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitly described.

DOI: 10.1007/s10468-006-9031-4
Online Date: 7/15/2006
Print publication date: 10/1/2006
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by Prest, Mike; Puninski, Gena

We classify the indecomposable pure injective modules over a wide class of $1$-domestic string algebras and calculate the Krull–Gabriel dimension of these algebras.

DOI: 10.1007/s10468-006-9028-z
Online Date: 7/12/2006
Print publication date: 8/1/2006
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by Caldero, Philippe; Chapoton, Frédéric; Schiffler, Ralf

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of these results, we answer several conjectures on the connection between cluster algebras and quiver representations.

DOI: 10.1007/s10468-006-9018-1
Online Date: 7/11/2006
Print publication date: 8/1/2006
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