| Title:
|
The group of commutativity preserving maps on strictly upper triangular matrices (English) |
| Author:
|
Wang, Dengyin |
| Author:
|
Zhu, Min |
| Author:
|
Rou, Jianling |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
335-350 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathcal {N}=N_n(R)$ be the algebra of all $n\times n$ strictly upper triangular matrices over a unital commutative ring $R$. A map $\varphi $ on $\mathcal {N}$ is called preserving commutativity in both directions if $xy=yx\Leftrightarrow \varphi (x)\varphi (y)=\varphi (y)\varphi (x)$. In this paper, we prove that each invertible linear map on $\mathcal {N}$ preserving commutativity in both directions is exactly a quasi-automorphism of $\mathcal {N}$, and a quasi-automorphism of $\mathcal {N}$ can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings. (English) |
| Keyword:
|
commutativity preserving map |
| Keyword:
|
automorphism |
| Keyword:
|
commutative ring |
| MSC:
|
13C10 |
| MSC:
|
15A04 |
| MSC:
|
15A27 |
| MSC:
|
15A99 |
| MSC:
|
17C30 |
| idZBL:
|
Zbl 06391498 |
| idMR:
|
MR3277740 |
| DOI:
|
10.1007/s10587-014-0105-x |
| . |
| Date available:
|
2014-11-10T09:32:09Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144002 |
| . |
| Reference:
|
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| Reference:
|
[2] Cao, Y., Chen, Z., Huang, C.: Commutativity preserving linear maps and Lie automorphisms of strictly triangular matrix space.Linear Algebra Appl. 350 41-66 (2002). Zbl 1007.15007, MR 1906746 |
| Reference:
|
[3] Cao, Y., Tan, Z.: Automorphisms of the Lie algebra of strictly upper triangular matrices over a commutative ring.Linear Algebra Appl. 360 105-122 (2003). Zbl 1015.17017, MR 1948476 |
| Reference:
|
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| Reference:
|
[5] Omladič, M.: On operators preserving commutativity.J. Funct. Anal. 66 105-122 (1986). Zbl 0587.47051, MR 0829380, 10.1016/0022-1236(86)90084-4 |
| Reference:
|
[6] Šemrl, P.: Non-linear commutativity preserving maps.Acta Sci. Math. 71 781-819 (2005). MR 2206609 |
| Reference:
|
[7] Wang, D., Chen, Z.: Invertible linear maps on simple Lie algebras preserving commutativity.Proc. Am. Math. Soc. 139 3881-3893 (2011). Zbl 1258.17014, MR 2823034, 10.1090/S0002-9939-2011-10834-7 |
| Reference:
|
[8] Watkins, W.: Linear maps that preserve commuting pairs of matrices.Linear Algebra Appl. 14 29-35 (1976). Zbl 0329.15005, MR 0480574, 10.1016/0024-3795(76)90060-4 |
| . |
