Homomorphisms from the unitary group to the general linear group over complex number field and applications

Chong-Guang Cao; Xian Zhang

Archivum Mathematicum (2002)

  • Volume: 038, Issue: 3, page 209-217
  • ISSN: 0044-8753

Abstract

top
Let M n be the multiplicative semigroup of all n × n complex matrices, and let U n and G L n be the n –degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from U n to G L m when n > m ≥ 1 or n = m ≥ 3 , and thereby determine multiplicative homomorphisms from U n to M m when n > m ≥ 1 or n = m ≥ 3 . This generalize Hochwald’s result in [Lin. Alg. Appl.  212/213:339-351(1994)]: if f : U n → M n is a spectrum–preserving multiplicative homomorphism, then there exists a matrix R in G L n such that f ( A ) = R A R for any A ∈ U n .

How to cite

top

Cao, Chong-Guang, and Zhang, Xian. "Homomorphisms from the unitary group to the general linear group over complex number field and applications." Archivum Mathematicum 038.3 (2002): 209-217. <http://eudml.org/doc/248947>.

@article{Cao2002,
abstract = {Let $M_n$ be the multiplicative semigroup of all $n\times n$ complex matrices, and let $U_n$ and $GL_n$ be the $n$–degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from $U_n$ to $GL_m$ when $n>m\ge 1$ or $n=m\ge 3$, and thereby determine multiplicative homomorphisms from $U_n$ to $M_m$ when $n>m\ge 1$ or $n=m\ge 3$. This generalize Hochwald’s result in [Lin. Alg. Appl.  212/213:339-351(1994)]: if $f:U_n\rightarrow M_n$ is a spectrum–preserving multiplicative homomorphism, then there exists a matrix $R$ in $GL_n$ such that $ f(A)=\{R\}AR$ for any $A\in U_n$.},
author = {Cao, Chong-Guang, Zhang, Xian},
journal = {Archivum Mathematicum},
keywords = {homomorphism; unitary group; general linear group; multiplicative homomorphisms; unitary groups; general linear groups},
language = {eng},
number = {3},
pages = {209-217},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Homomorphisms from the unitary group to the general linear group over complex number field and applications},
url = {http://eudml.org/doc/248947},
volume = {038},
year = {2002},
}

TY - JOUR
AU - Cao, Chong-Guang
AU - Zhang, Xian
TI - Homomorphisms from the unitary group to the general linear group over complex number field and applications
JO - Archivum Mathematicum
PY - 2002
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 038
IS - 3
SP - 209
EP - 217
AB - Let $M_n$ be the multiplicative semigroup of all $n\times n$ complex matrices, and let $U_n$ and $GL_n$ be the $n$–degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from $U_n$ to $GL_m$ when $n>m\ge 1$ or $n=m\ge 3$, and thereby determine multiplicative homomorphisms from $U_n$ to $M_m$ when $n>m\ge 1$ or $n=m\ge 3$. This generalize Hochwald’s result in [Lin. Alg. Appl.  212/213:339-351(1994)]: if $f:U_n\rightarrow M_n$ is a spectrum–preserving multiplicative homomorphism, then there exists a matrix $R$ in $GL_n$ such that $ f(A)={R}AR$ for any $A\in U_n$.
LA - eng
KW - homomorphism; unitary group; general linear group; multiplicative homomorphisms; unitary groups; general linear groups
UR - http://eudml.org/doc/248947
ER -

References

top
  1. Beasley L. B., Pullman N. L., Linear operators preserving idempotent matrices over fields, Linear Algebra Appl. 146 (1991), 7–20. (1991) Zbl0718.15004MR1083460
  2. Cao Chongguang, Zhang Xian, Multiplicative semigroup automorphisms of upper triangular matrices over rings, Linear Algebra Appl. 278, No. 1-3 (1998), 85–90. (1998) MR1637327
  3. Chen, Yu., Homomorphisms of two dimentional linear groups, Comm. Algebra 18 (1990), 2383–2396. (1990) MR1063142
  4. Dicks W., Hartley B., On homomorphisms between linear groups over division rings, Comm. Algebra 19 (1991), 1919–1943. (1991) MR1121114
  5. Hochwald S. H., Multiplicative maps on matrices that preserve spectrum, Linear Algebra Appl. 212/213 (1994), 339–351. (1994) MR1306985
  6. Hua, Luogeng, Wan, Zhexian, Classical groups, Science Technology Press, Shanghai, 1962 (Chinese). (1962) 
  7. Liu, Shaowu, Wang L., Homomorphisms between symplectic groups, Chinese Ann. Math. Ser. B 14 No. 3 (1993), 287–296. (1993) Zbl0797.20037MR1264302
  8. Liu, Shaowu, Automorphisms of singular symplectic groups over fields, Chinese Con. Math. 19 No. 2 (1998), 203–213. (19 N) MR1651628
  9. Petechuk V. M., Homomorphisms of linear groups over commutative rings, Mat. Zametki 46 No. 5 (1989), 50–61 (Russian); translation in Math. Notes 46. 5-6 (1989), 863–870. (1989) Zbl0702.20032MR1033420
  10. Petechuk V. M., Homomorphisms of linear groups over rings, Mat. Zametki 45 No. 2 (1989), 83–94 (Russian); translation in Math. Notes 45 1–2 (1989), 144–151. (1989) Zbl0682.20037MR1002522
  11. Zha, Jianguo, Determination of homomorphisms between linear groups of the same degree over division rings, J. London Math. Soc., II. Ser. 53 No. 3 (1996), 479–488. (1996) Zbl0858.20038MR1396712
  12. Zhang, Xian, Hu, Yahui, Cao, Chongguang, Multiplicative group automorphisms of invertible upper triangular matrices over fields, Acta Math. Sci. (English ed.) 20 4 (2000), 515–521. Zbl0985.20040MR1805020

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.