Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

Xinfeng Liang; Feng Wei; Ajda Fošner

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 3, page 713-761
  • ISSN: 0011-4642

Abstract

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Let ℛ be a commutative ring, 𝒢 be a generalized matrix algebra over ℛ with weakly loyal bimodule and 𝒵 ( 𝒢 ) be the center of 𝒢 . Suppose that 𝔮 : 𝒢 × 𝒢 → 𝒢 is an ℛ -bilinear mapping and that 𝔗 𝔮 : 𝒢 → 𝒢 is a trace of 𝔮 . The aim of this article is to describe the form of 𝔗 𝔮 satisfying the centralizing condition [ 𝔗 𝔮 ( x ) , x ] ∈ 𝒵 ( 𝒢 ) (and commuting condition [ 𝔗 𝔮 ( x ) , x ] = 0 ) for all x ∈ 𝒢 . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) 𝔗 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 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.

How to cite

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Liang, Xinfeng, Wei, Feng, and Fošner, Ajda. "Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective." Czechoslovak Mathematical Journal 69.3 (2019): 713-761. <http://eudml.org/doc/294848>.

@article{Liang2019,
abstract = {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 $\mathcal \{R\}$-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.},
author = {Liang, Xinfeng, Wei, Feng, Fošner, Ajda},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized matrix algebra; commuting trace; centralizing trace; Lie isomorphism; Lie triple isomorphism},
language = {eng},
number = {3},
pages = {713-761},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective},
url = {http://eudml.org/doc/294848},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Liang, Xinfeng
AU - Wei, Feng
AU - Fošner, Ajda
TI - Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 3
SP - 713
EP - 761
AB - 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 $\mathcal {R}$-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.
LA - eng
KW - generalized matrix algebra; commuting trace; centralizing trace; Lie isomorphism; Lie triple isomorphism
UR - http://eudml.org/doc/294848
ER -

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