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Let $\mathfrak{g}$ be a complex simple Lie algebra and let $\{e,f,h\}$ be a $\mathfrak{sl}_2$-triple in $\mathfrak{g}$, i. e. we have commutation relations: $$ [e,f]=h,~[h,e]=2e,~[h,f]=-2f. $$ In particular, elements $\{e,f,h\}$ span a three-dimensional Lie subalgebra in $\mathfrak{g}$.

Question. Suppose that $\mathfrak{g}$ acts on some vector space $V$ and we know the actions of elements $e,f,h$ (so we know that $V$ is a representation of $\mathfrak{g}$). What "minimal" information we need in addition to this in order to restore the action of the whole algebra $\mathfrak{g}$? I think we can suppose that $e$, $f$, $h$ are regular elements of the algebra $\mathfrak{g}$. I am particularly interested in the case when $\mathfrak{g}=\mathfrak{sl}_n$.

Of course, this question may not have a precise answer, but any help/references would be appreciated.

Let $\mathfrak{g}$ be a complex simple Lie algebra and let $\{e,f,h\}$ be a $\mathfrak{sl}_2$-triple in $\mathfrak{g}$, i. e. we have commutation relations: $$ [e,f]=h,~[h,e]=2e,~[h,f]=-2f. $$ In particular, elements $\{e,f,h\}$ span a three-dimensional Lie subalgebra in $\mathfrak{g}$.

Question. Suppose that $\mathfrak{g}$ acts on some vector space $V$ and we know the actions of elements $e,f,h$ (so we know that $V$ is a representation of $\mathfrak{g}$). What "minimal" information we need in addition to this in order to restore the action of the whole algebra $\mathfrak{g}$? I think we can suppose that $e$, $f$, $h$ are regular elements of the algebra $\mathfrak{g}$. I am particularly interested in the case when $\mathfrak{g}=\mathfrak{sl}_n$

Of course, this question may not have a precise answer, but any help/references would be appreciated.

Let $\mathfrak{g}$ be a complex simple Lie algebra and let $\{e,f,h\}$ be a $\mathfrak{sl}_2$-triple in $\mathfrak{g}$, i. e. we have commutation relations: $$ [e,f]=h,~[h,e]=2e,~[h,f]=-2f. $$ In particular, elements $\{e,f,h\}$ span a three-dimensional Lie subalgebra in $\mathfrak{g}$.

Question. Suppose that $\mathfrak{g}$ acts on some vector space $V$ and we know the actions of elements $e,f,h$ (so we know that $V$ is a representation of $\mathfrak{g}$). What "minimal" information we need in addition to this in order to restore the action of the whole algebra $\mathfrak{g}$? I think we can suppose that $e$, $f$, $h$ are regular elements of the algebra $\mathfrak{g}$. I am particularly interested in the case when $\mathfrak{g}=\mathfrak{sl}_n$.

Of course, this question may not have a precise answer, but any help/references would be appreciated.

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Let $\mathfrak{g}$ be a complex semisimplesimple Lie algebra and let $\{e,f,h\}$ be a $\mathfrak{sl}_2$-triple in $\mathfrak{g}$, i. e. we have commutation relations: $$ [e,f]=h,~[h,e]=2e,~[h,f]=-2f. $$ In particular, elements $\{e,f,h\}$ span a three-dimensional Lie subalgebra in $\mathfrak{g}$.

Question. Suppose that $\mathfrak{g}$ acts on some vector space $V$ and we know the actions of elements $e,f,h$ (so we know that $V$ is a representation of $\mathfrak{g}$). What "minimal" information we need in addition to this in order to restore the action of the whole algebra $\mathfrak{g}$? I think we can suppose that $e$, $f$, $h$ are regular elements of the algebra $\mathfrak{g}$. I am particularly interested in the case when $\mathfrak{g}=\mathfrak{sl}_n$

Of course, this question may not have a precise answer, but any help/references would be appreciated.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\{e,f,h\}$ be a $\mathfrak{sl}_2$-triple in $\mathfrak{g}$, i. e. we have commutation relations: $$ [e,f]=h,~[h,e]=2e,~[h,f]=-2f. $$ In particular, elements $\{e,f,h\}$ span a three-dimensional Lie subalgebra in $\mathfrak{g}$.

Question. Suppose that $\mathfrak{g}$ acts on some vector space $V$ and we know the actions of elements $e,f,h$ (so we know that $V$ is a representation of $\mathfrak{g}$). What "minimal" information we need in addition to this in order to restore the action of the whole algebra $\mathfrak{g}$? I think we can suppose that $e$, $f$, $h$ are regular elements of the algebra $\mathfrak{g}$.

Of course, this question may not have a precise answer, but any help/references would be appreciated.

Let $\mathfrak{g}$ be a complex simple Lie algebra and let $\{e,f,h\}$ be a $\mathfrak{sl}_2$-triple in $\mathfrak{g}$, i. e. we have commutation relations: $$ [e,f]=h,~[h,e]=2e,~[h,f]=-2f. $$ In particular, elements $\{e,f,h\}$ span a three-dimensional Lie subalgebra in $\mathfrak{g}$.

Question. Suppose that $\mathfrak{g}$ acts on some vector space $V$ and we know the actions of elements $e,f,h$ (so we know that $V$ is a representation of $\mathfrak{g}$). What "minimal" information we need in addition to this in order to restore the action of the whole algebra $\mathfrak{g}$? I think we can suppose that $e$, $f$, $h$ are regular elements of the algebra $\mathfrak{g}$. I am particularly interested in the case when $\mathfrak{g}=\mathfrak{sl}_n$

Of course, this question may not have a precise answer, but any help/references would be appreciated.

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Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\{e,f,h\}$ be a $\mathfrak{sl}_2$-triple in $\mathfrak{g}$, i. e. we have commutation relations: $$ [e,f]=h,~[h,e]=2e,~[h,f]=-2f. $$ In particular, elements $\{e,f,h\}$ span a three-dimensional Lie subalgebra in $\mathfrak{g}$.

Question. Suppose that $\mathfrak{g}$ acts on some vector space $V$ and we know the actions of elements $e,f,h$ (so we know that $V$ is a representation of $\mathfrak{g}$). What "minimal" information we need in addition to this in order to restore the action of the whole algebra $\mathfrak{g}$? I think we can suppose that $e$, $f$, $h$ are regular elements of the algebra $\mathfrak{g}$.

Of course, this question may not have a precise answer, but any help/references would be appreciated.

Let $\mathfrak{g}$ be a semisimple Lie algebra and let $\{e,f,h\}$ be a $\mathfrak{sl}_2$-triple in $\mathfrak{g}$, i. e. we have commutation relations: $$ [e,f]=h,~[h,e]=2e,~[h,f]=-2f. $$ In particular, elements $\{e,f,h\}$ span a three-dimensional Lie subalgebra in $\mathfrak{g}$.

Question. Suppose that $\mathfrak{g}$ acts on some vector space $V$ and we know the actions of elements $e,f,h$. What "minimal" information we need in addition to this in order to restore the action of the whole algebra $\mathfrak{g}$? I think we can suppose that $e$, $f$, $h$ are regular elements of the algebra $\mathfrak{g}$.

Of course, this question may not have a precise answer, but any help/references would be appreciated.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\{e,f,h\}$ be a $\mathfrak{sl}_2$-triple in $\mathfrak{g}$, i. e. we have commutation relations: $$ [e,f]=h,~[h,e]=2e,~[h,f]=-2f. $$ In particular, elements $\{e,f,h\}$ span a three-dimensional Lie subalgebra in $\mathfrak{g}$.

Question. Suppose that $\mathfrak{g}$ acts on some vector space $V$ and we know the actions of elements $e,f,h$ (so we know that $V$ is a representation of $\mathfrak{g}$). What "minimal" information we need in addition to this in order to restore the action of the whole algebra $\mathfrak{g}$? I think we can suppose that $e$, $f$, $h$ are regular elements of the algebra $\mathfrak{g}$.

Of course, this question may not have a precise answer, but any help/references would be appreciated.

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