A tower construction for the radical in Brauer's centralizer algebras
The Brauer centralizer algebras \({\mathcal A}_f^{(x)}\) are studied in the case that the multiplication constant \(x\) is a rational integer. If \(F_f\) is the set of all graphs with \(2f\) points and \(f\) lines in which every point has degree 1 (1-factors on \(2f\) points), then \({\mathcal A}_f^{(x)}\) is an \(\mathbb{R}\)-algebra with \(F_f\) as a basis. The set of all diagrams with at least \(k\) horizontal lines in the rows is an ideal of \({\mathcal A}^{(x)}_f\) denoted by \({\mathcal A}^{(x)}_f (k)\). A description of the radical of the quotient \({\mathcal A}^{(x)}_f (k)/{\mathcal A}^{(x)}_f (k+1)\) was obtained earlier by authors in terms of \((m,k)\)-partial 1-factors. The tower construction is a method to study towers of semisimple algebras. The first step in this paper is to modify the tower construction for the purpose of its application in the case when the algebras of the tower are not semisimple. Let \(R\) be a finite dimensional semisimple \(R\)-algebra with matrix ring decomposition \(R=\bigoplus_\mu R_\mu\), \(V_\mu\) is the corresponding irreducible representation of \(R\), \(M=\bigoplus g_\mu V\) is a left \(R\)-module and \(M'\) is a right \(R\)-module with a linear isomorphism \(M \cong M'\) (\(m \rightsquigarrow m'\)) such that \((rm)'=m'r'\). For every special \(\mathbb{R}\)-bilinear map \(\varepsilon:M \times M'\to R\) is defined the algebra \({\mathcal A}(R, M, \varepsilon)\) with \(\mathbb{R}\)-basis \(M'\otimes_RM\) and multiplication depending on \(\varepsilon\). The map \(\varepsilon\) is determined by the sequence of symmetric matrices \((H_\mu)\). The structure of the algebra \({\mathcal A} (R, M, \varepsilon)\) is described by the characteristic sequence \((H_\mu):{\mathcal A}(R, M, \varepsilon) \cong \bigoplus_\mu \Lambda (H_\mu)\), where \(\Lambda (H_\mu)\) is an \(\mathbb{R}\)-algebra with \(R\)-basis \(\mathbb{R}^d \otimes \mathbb{R}^d\) and multiplication defined by \(H_\mu\). The general construction is used in the situation when in the chain \({\mathcal A}^{(x)}_{f-2} \subseteq {\mathcal A}^{(x)}_{f-1} \subseteq {\mathcal A}^{(x)}_f\) the algebra \({\mathcal A}^{(x)}_{f-2}\) has a nontrivial radical \({\mathcal N}_{f-2} \neq 0\). The special procedure of transition from \(F_{f-2}\) to \(F_f\) \((v \rightsquigarrow v^{r1}_{ab})\) permits to obtain on the base of \(N_{f-2}\) an ideal \(N^{(1)}_f \subseteq N_f=\text{Rad} ({\mathcal A}^{(x)}_f)\) of \({\mathcal A}^{(x)}_f\). Further it is shown how to analyze the quotient \({\mathcal A}^{(x)}_f/N^{(1)}_f\) using the semisimple quotient \({\mathcal A}^{(x)}_{f-2}/N_{f-2}=R\). For \(M={\mathcal A}^{(x)}_{f-1} /N_{f-2} {\mathcal A}^{(x)}_{f-1}\) is constructed the algebra \({\mathcal A}(R, M, \varepsilon)\), where \(M'={\mathcal A}^{(x)}_{f-1} N_{f-2}\) and \(\varepsilon\) is defined via the canonical map \({\mathcal A}_{f-1} \otimes_{\mathbb{R}} {\mathcal A}'_{f-1} \to {\mathcal A}_{f-2}\). The isomorphism \(M' \otimes_R M \cong {\mathcal A}^{(x)}_f (1) /{\mathcal N}^{(1)}_f\) is proved. In the rest of the work are shown: 1) the idempotents of \({\mathcal A}^{(x)}_f\); 2) the matrices \((H_\mu)\) in the characteristic sequence of \(\varepsilon\); 3) the structure of the radical \(N_f = \text{Rad} ({\mathcal A}_f^{(x)})\). This article contains all necessary proofs, and it is excellently illustrated by examples and diagrams.
scientific article; zbMATH DE number 579549