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0

$begingroup$

Let $A$ be a unital associative $mathbbC$ algebra and let $M$ be a 2-dimensional $A$-module with basis $leftmathbfe_1, mathbfe_2right$. $A$ has two generators $lefta_1, a_2right$ and they act on the basis of $M$ by
beginalign*
& a_1 cdot mathbfe_1 = -mathbfe_1 & a_2 cdot mathbfe_1 = mathbfe_1 + mathbfe_2\
& a_1 cdot mathbfe_2 = mathbfe_1 + mathbfe_2 & a_2 cdot mathbfe_2 = -mathbfe_2\
endalign*
Is it possible to find a basis $leftoverlinemathbfe_1, overlinemathbfe_2right$ of $M$ such that the action of the $a_i$ are still integer values, but are non-negative, i.e.
beginalign*
& a_1 cdot overlinemathbfe_1 = n_1 overlinemathbfe_1 + n_2 overlinemathbfe_2
& a_2 cdot overlinemathbfe_1 = m_1 overlinemathbfe_1 + m_2 overlinemathbfe_2\
& a_1 cdot overlinemathbfe_2 = n_3 overlinemathbfe_1 + n_4 overlinemathbfe_2
& a_2 cdot overlinemathbfe_2 = m_3 overlinemathbfe_1 + m_4 overlinemathbfe_2\
endalign*
where the $n_i$ and $m_i$ are in $mathbbZ$ and $n_i, m_i geq 0$ for $1 leq i leq 4$.

linear-algebra change-of-basis algebras

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asked Apr 8 at 1:00

AnfängerAnfänger

277110

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reputation from Anfänger ending ending at 2019-04-17 15:27:41Z">in 6 days.

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0

$begingroup$

Let $A$ be a unital associative $mathbbC$ algebra and let $M$ be a 2-dimensional $A$-module with basis $leftmathbfe_1, mathbfe_2right$. $A$ has two generators $lefta_1, a_2right$ and they act on the basis of $M$ by
beginalign*
& a_1 cdot mathbfe_1 = -mathbfe_1 & a_2 cdot mathbfe_1 = mathbfe_1 + mathbfe_2\
& a_1 cdot mathbfe_2 = mathbfe_1 + mathbfe_2 & a_2 cdot mathbfe_2 = -mathbfe_2\
endalign*
Is it possible to find a basis $leftoverlinemathbfe_1, overlinemathbfe_2right$ of $M$ such that the action of the $a_i$ are still integer values, but are non-negative, i.e.
beginalign*
& a_1 cdot overlinemathbfe_1 = n_1 overlinemathbfe_1 + n_2 overlinemathbfe_2
& a_2 cdot overlinemathbfe_1 = m_1 overlinemathbfe_1 + m_2 overlinemathbfe_2\
& a_1 cdot overlinemathbfe_2 = n_3 overlinemathbfe_1 + n_4 overlinemathbfe_2
& a_2 cdot overlinemathbfe_2 = m_3 overlinemathbfe_1 + m_4 overlinemathbfe_2\
endalign*
where the $n_i$ and $m_i$ are in $mathbbZ$ and $n_i, m_i geq 0$ for $1 leq i leq 4$.

linear-algebra change-of-basis algebras

share|cite|improve this question

asked Apr 8 at 1:00

AnfängerAnfänger

277110

$endgroup$

This question has an open bounty worth +50
reputation from Anfänger ending ending at 2019-04-17 15:27:41Z">in 6 days.

This question has not received enough attention.

add a comment | 

$begingroup$

Let $A$ be a unital associative $mathbbC$ algebra and let $M$ be a 2-dimensional $A$-module with basis $leftmathbfe_1, mathbfe_2right$. $A$ has two generators $lefta_1, a_2right$ and they act on the basis of $M$ by
beginalign*
& a_1 cdot mathbfe_1 = -mathbfe_1 & a_2 cdot mathbfe_1 = mathbfe_1 + mathbfe_2\
& a_1 cdot mathbfe_2 = mathbfe_1 + mathbfe_2 & a_2 cdot mathbfe_2 = -mathbfe_2\
endalign*
Is it possible to find a basis $leftoverlinemathbfe_1, overlinemathbfe_2right$ of $M$ such that the action of the $a_i$ are still integer values, but are non-negative, i.e.
beginalign*
& a_1 cdot overlinemathbfe_1 = n_1 overlinemathbfe_1 + n_2 overlinemathbfe_2
& a_2 cdot overlinemathbfe_1 = m_1 overlinemathbfe_1 + m_2 overlinemathbfe_2\
& a_1 cdot overlinemathbfe_2 = n_3 overlinemathbfe_1 + n_4 overlinemathbfe_2
& a_2 cdot overlinemathbfe_2 = m_3 overlinemathbfe_1 + m_4 overlinemathbfe_2\
endalign*
where the $n_i$ and $m_i$ are in $mathbbZ$ and $n_i, m_i geq 0$ for $1 leq i leq 4$.

linear-algebra change-of-basis algebras

share|cite|improve this question

asked Apr 8 at 1:00

AnfängerAnfänger

277110

$endgroup$

share|cite|improve this question

asked Apr 8 at 1:00

AnfängerAnfänger

277110

share|cite|improve this question

asked Apr 8 at 1:00

AnfängerAnfänger

277110

asked Apr 8 at 1:00

AnfängerAnfänger

277110

asked Apr 8 at 1:00

AnfängerAnfänger

277110