subalgebra of a matrix algebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:45:46Z http://mathoverflow.net/feeds/question/84591 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84591/subalgebra-of-a-matrix-algebra subalgebra of a matrix algebra Miguel 2011-12-30T14:33:29Z 2011-12-30T21:18:56Z <p>Let $K$ be an algebraic closed field and $M_n(K)$ the $K$-algebra of all matrices $n\times n$ over $K$. If $L$ and $M$ are two commutative isomorphic subalgebras of $M_n(K)$ it is true that there exista a regular matrix $S\in M_n(K)$ such that $SLS^{-1}=M$. That is the isomorphism is inner? </p> http://mathoverflow.net/questions/84591/subalgebra-of-a-matrix-algebra/84600#84600 Answer by Denis Serre for subalgebra of a matrix algebra Denis Serre 2011-12-30T17:06:23Z 2011-12-30T21:18:56Z <p>Two isomorphic subalgebra of $M_n(K)$ <strong>do not need to be conjugated</strong>. The following example is taken from Exercise 161 of my web site <a href="http://www.umpa.ens-lyon.fr/~serre/DPF/exobis.pdf" rel="nofollow">http://www.umpa.ens-lyon.fr/~serre/DPF/exobis.pdf</a></p> <p>Set $n=p+q$ with $q>p>0$. Then define $\mathcal A$ as the subset of $M_n(k)$ made of the matrices with block form $$\left(\begin{array}{cc} 0_p &amp; 0_{p\times q} \\ A &amp; 0_q \end{array}\right).$$ Likewise, ${\cal B}$ is made of the matrices $$\left(\begin{array}{cc} 0_q &amp; 0_{q\times p} \\ B &amp; 0_p \end{array}\right).$$ Both $\cal A$ and $\cal B$ are subalgebras of $M_n(k)$, with dimension $pq$ and the property that $MN=0_n$ for every two elements (of the same algebra). They are obviously isomorphic, because the algebra structure is trivial. But ${\cal A}$ and $\cal B$ are not conjugated in $M_n(k)$. However $\cal B$ is conjugated to ${\cal A}^T$ in $M_n(k)$.</p>