Triangular decomposition of the composition algebra of the Kronecker algebra
type: Item
instance of: scholarly article
title: Triangular decomposition of the composition algebra of the Kronecker algebra
zbMATH Open document ID: 0880.16008
author: Pu Zhang
published in: Journal of Algebra
publication date: 1996-12-15T00:00:00Z
review text: Let \(K\) be a finite field and let \(R\) be a finite dimensional \(K\)-algebra. Denote by \(S_1,\ldots,S_n\) a complete set of pairwise non-isomorphic simple right \(R\)-modules. Given a finite right \(R\)-module \(M\) we denote by \(\mathbf{dim} M=(m_1,\dots,m_n)\) the dimension vector of \(M\) in \(\mathbb{Z}^n\), that is the coordinate \(m_j\) is the Jordan-Hölder multiplicity of \(S_j\) in \(M\). We recall that the Ringel-Hall \(\mathbb{Q}\)-algebra of \(R\) is the algebra \({\mathcal H}(R)\) over the field \(\mathbb{Q}\) of rationals whose basis is the set of the isomorphism classes \([X]\) of finite right \(R\)-modules, and the multiplication is defined by the formula \([N_1]\cdot[N_2]=\sum_{[M]}F^M_{N_1,N_2}[M]\), where \(F^M_{N_1,N_2}\) is the number of submodules \(N\) of \(M\) with \(N\cong N_1\) and \(M/N\cong N_2\). Note that \({\mathcal H}(R)\) is an associative algebra with the identity \([0]\). The composition algebra of \(R\) is defined to be the \(\mathbb{Q}\)-subalgebra \({\mathcal C}(R)\) of \({\mathcal H}(R)\) generated by the elements \([S_1],\ldots,[S_n]\). One can show that \({\mathcal H}(R)\) is a \(\mathbb{Z}^n\)-graded \(\mathbb{Q}\)-algebra and \({\mathcal C}(R)=\bigoplus_d{\mathcal C}(R)_d\) is a \(\mathbb{Z}^n\)-graded \(\mathbb{Q}\)-subalgebra of \({\mathcal H}(R)\), where \({\mathcal C}(R)_d\), with \(d=(d_1,\ldots,d_n)\), is the \(\mathbb{Q}\)-subspace of \({\mathcal H}(R)\) generated by all monomials \([S_{j_1}]\cdot[S_{j_2}]\cdots[S_{j_n}]\) such that the number of occurrences of \(j\) in the sequence \(\{j_1,\dots,j_n\}\) is just \(d_j\). For any vector \(d=(d_1,\ldots,d_n)\) of \(\mathbb{Z}^n\) we consider the element \(r_d=\sum_{[M]}[M]\) of \({\mathcal H}(R)\), where \(M\) runs through the isomorphism classes of regular \(R\)-modules \(M\) with \(\mathbf{dim} M=d\). We denote by \({\mathcal T}\) the subalgebra of \({\mathcal H}(R)\) generated by all elements \(r_d\). It is shown in the paper that in case \(R\) is a hereditary a…
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 934245
zbMATH Keywords: Ringel-Hall algebras
zbMATH Keywords: composition algebras
zbMATH Keywords: regular modules
zbMATH Keywords: preprojective modules
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: simple right modules
zbMATH Keywords: Jordan-Hölder multiplicities
zbMATH Keywords: graded algebras
zbMATH Keywords: hereditary algebras
zbMATH Keywords: preinjective modules
MaRDI profile type: Publication
OpenAlex ID: W1966237475
DOI: 10.1006/JABR.1996.0252
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description: scientific article; zbMATH DE number 934245