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Gary
Gary
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Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra. Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced enveloping algebra for $gl_n$. If M is an indecomposable module for $U_\chi(gl_n)$, then is M still indecomposable as a module for $U_\chi(sl_n)$? I think it is still true. But I think it is not easy to prove it. Need Help.

Actually, I am considering that the similar question comes out naturally for a more general setting. Let A be a finite dimensional algebra over k, B is a sub algebra. Then can we find a condition for B such that all indecomposable (finite dimensional ) module for A is still indecomposable (finite dimensional ) module when restricted to B?

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra. Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced enveloping algebra for $gl_n$. If M is an indecomposable module for $U_\chi(gl_n)$, then is M still indecomposable as a module for $U_\chi(sl_n)$? I think it is still true. But I think it is not easy to prove it. Need Help.

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra. Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced enveloping algebra for $gl_n$. If M is an indecomposable module for $U_\chi(gl_n)$, then is M still indecomposable as a module for $U_\chi(sl_n)$? I think it is still true. But I think it is not easy to prove it. Need Help.

Actually, I am considering that the similar question comes out naturally for a more general setting. Let A be a finite dimensional algebra over k, B is a sub algebra. Then can we find a condition for B such that all indecomposable (finite dimensional ) module for A is still indecomposable (finite dimensional ) module when restricted to B?

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Gary
Gary
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  • 3

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra. Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced enveloping algebra for $gl_n$. If M is an indecomposable module for $U_\chi(gl_n)$, then is M still indecomposable as a module for $U_\chi(sl_n)$? I think it is still true. But I think it is not easy to prove it. Need Help. Emergency!

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra. Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced enveloping algebra for $gl_n$. If M is an indecomposable module for $U_\chi(gl_n)$, then is M still indecomposable as a module for $U_\chi(sl_n)$? I think it is still true. But I think it is not easy to prove it. Need Help. Emergency!

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra. Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced enveloping algebra for $gl_n$. If M is an indecomposable module for $U_\chi(gl_n)$, then is M still indecomposable as a module for $U_\chi(sl_n)$? I think it is still true. But I think it is not easy to prove it. Need Help.

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Gary
Gary
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indecomposable modules restricted from $gl_n$ to $sl_n$

Let K be an algebraic closed field, $gl_n$ be the general linear Lie algebra over K, and $sl_n$ be the special linear Lie algebra. Let $\chi\in gl_n^*$. Let $U_\chi(gl_n)$ be the corresponding reduced enveloping algebra for $gl_n$. If M is an indecomposable module for $U_\chi(gl_n)$, then is M still indecomposable as a module for $U_\chi(sl_n)$? I think it is still true. But I think it is not easy to prove it. Need Help. Emergency!