Triangular decomposition of the composition algebra of the Kronecker algebra
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 algebra, for any preprojective module \(P\) and for any preinjective module \(Q\) the elements \([P]\) and \([Q]\) of \({\mathcal H}(R)\) belong to \({\mathcal C}(R)\). Moreover for any vector \(d=(d_1,\ldots,d_n)\) of \(\mathbb{Z}^n\) the element \(r_d=\sum_{[M]}[M]\) of \({\mathcal H}(R)\) also belongs to \({\mathcal C}(R)\). One of the main results of the paper asserts that if \(R=\Gamma\) is the Kronecker \(K\)-algebra \(\Gamma=\left(\begin{smallmatrix} K &K^2\\ 0 &K\end{smallmatrix}\right)\) then \({\mathcal C}(\Gamma)={\mathcal P}\cdot{\mathcal T}\cdot{\mathcal Q}\), where \({\mathcal P}\) and \({\mathcal Q}\) are the subalgebras of \({\mathcal C}(\Gamma)\) generated by the preprojective modules and the preinjective modules, respectively. Moreover it is shown that \({\mathcal T}\) is the subalgebra of \({\mathcal C}(\Gamma)\) generated by the regular elements.
scientific article; zbMATH DE number 934245