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Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 3, pp. 713-761
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Let $\mathcal {R}$ be a commutative ring, $\mathcal {G}$ be a generalized matrix algebra over $\mathcal {R}$ with weakly loyal bimodule and $\mathcal {Z}(\mathcal {G})$ be the center of $\mathcal {G}$. Suppose that $\mathfrak {q}\colon \mathcal {G}\times \mathcal {G} \rightarrow \mathcal {G}$ is an \hbox {$\mathcal {R}$-bilinear} mapping and that $\mathfrak {T}_{\mathfrak {q}}\colon \mathcal {G}\rightarrow \mathcal {G}$ is a trace of $\mathfrak {q}$. The aim of this article is to describe the form of $\mathfrak {T}_{\mathfrak {q}}$ satisfying the centralizing condition $[\mathfrak {T}_{\mathfrak {q}}(x), x]\in \mathcal {Z(G)}$ (and commuting condition $[\mathfrak {T}_{\mathfrak {q}}(x), x]=0$) for all $x\in \mathcal {G}$. More precisely, we will revisit the question of when the centralizing trace (and commuting trace) $\mathfrak {T}_{\mathfrak {q}}$ has the so-called proper form from a new perspective. Using the aforementioned trace function, we establish sufficient conditions for each Lie-type isomorphism of $\mathcal {G}$ to be almost standard. As applications, centralizing (commuting) traces of bilinear mappings and Lie-type isomorphisms on full matrix algebras and those on upper triangular matrix algebras are totally determined.
Let $\mathcal {R}$ be a commutative ring, $\mathcal {G}$ be a generalized matrix algebra over $\mathcal {R}$ with weakly loyal bimodule and $\mathcal {Z}(\mathcal {G})$ be the center of $\mathcal {G}$. Suppose that $\mathfrak {q}\colon \mathcal {G}\times \mathcal {G} \rightarrow \mathcal {G}$ is an \hbox {$\mathcal {R}$-bilinear} mapping and that $\mathfrak {T}_{\mathfrak {q}}\colon \mathcal {G}\rightarrow \mathcal {G}$ is a trace of $\mathfrak {q}$. The aim of this article is to describe the form of $\mathfrak {T}_{\mathfrak {q}}$ satisfying the centralizing condition $[\mathfrak {T}_{\mathfrak {q}}(x), x]\in \mathcal {Z(G)}$ (and commuting condition $[\mathfrak {T}_{\mathfrak {q}}(x), x]=0$) for all $x\in \mathcal {G}$. More precisely, we will revisit the question of when the centralizing trace (and commuting trace) $\mathfrak {T}_{\mathfrak {q}}$ has the so-called proper form from a new perspective. Using the aforementioned trace function, we establish sufficient conditions for each Lie-type isomorphism of $\mathcal {G}$ to be almost standard. As applications, centralizing (commuting) traces of bilinear mappings and Lie-type isomorphisms on full matrix algebras and those on upper triangular matrix algebras are totally determined.
DOI : 10.21136/CMJ.2019.0507-17
Classification : 15A78, 16R60, 16W10
Keywords: generalized matrix algebra; commuting trace; centralizing trace; Lie isomorphism; Lie triple isomorphism 
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     title = {Centralizing traces and {Lie-type} isomorphisms on generalized matrix algebras: a new perspective},
     journal = {Czechoslovak Mathematical Journal},
     pages = {713--761},
     year = {2019},
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Liang, Xinfeng; Wei, Feng; Fošner, Ajda. Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 3, pp. 713-761. doi: 10.21136/CMJ.2019.0507-17

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