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| Products of Involutions in Steinberg Group over Skew Fields |
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Citation： |
Jizhu NAN,Hong YOU.Products of Involutions in Steinberg Group over Skew Fields[J].Chinese Annals of Mathematics B,2007,28(2):253~264 |
| Page view： 1110
Net amount： 787 |
Authors： |
Jizhu NAN; Hong YOU |
Foundation： |
the Key Project of the Ministry of Education of China (No. 03060). |
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| Abstract： |
Consider the stable Steinberg group ${\rm St}(K)$ over a skew field $K$. An element $x$ is called an involution if $x^{2}=1$. In this paper, an involution is allowed to be the identity. The authors prove that an element $A$ of ${\rm GL}_{n}(K)$ up to conjugation can be represented as $BC$, where $B$ is lower triangular and $C$ is simultaneously upper triangular. Furthermore, $B$ and $C$ can be chosen so that the elements in the main diagonal of $B$ are $\beta _{1},\beta _{2},\cdots ,\beta _{n}$, and of $C$ are $\gamma _{1},\gamma _{2},\cdots ,\gamma _{n}c_{n}$, where $c_{n}\in \lbrack K^{\ast },K^{\ast }]$ and $\prod\limits^{n}_{j=1}\overline{\beta _{j}\gamma _{j}}=\det A$. It is also proved that every element $\delta $ in ${\rm St}(K)$ is a product of $10$ involutions. |
Keywords： |
Steinberg group, Involution, Skew field |
Classification： |
15A23, 20H25 |
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