Root system of $\mathfrak{gl}_n$ and reduction to $\mathfrak{sl}_n$ root system

Asked Aug 03 '20 at 18:01

Active Aug 03 '20 at 18:01

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I know that the root system of $\mathfrak{sl}_n$ can be described as the vector space spanned by $\{a^1, a^2, \ldots, a^{n-1}\}$ with inner product given by the Killing form,

$$(a^i,a^j)= 2\delta_{i,j} - \delta_{i+1,j}-\delta_{i,j+1}.$$

However, I have trouble understanding how it relates to the root system of $\mathfrak{gl}_n$ which is of higher rank? What is a basis and the Killing form for the general linear lie algebra?

Secondly, if I start with a root lattice of $\mathfrak{gl}_n$ how does it reduce to the root lattice of $\mathfrak{sl}_n$?

asked Aug 03 '20 at 18:01

winawer

See this post for the Killing form. – Dietrich Burde Aug 03 '20 at 18:32
Recently discussed this in the comments to https://math.stackexchange.com/a/3759751/96384. What is your definition of "the root system of $\mathfrak{gl}_n$" though? – Torsten Schoeneberg Aug 03 '20 at 20:35
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Maybe this answer is helpful: https://math.stackexchange.com/questions/1448506/mathfrakgl-style-vs-mathfraksl-style-fundamental-weights-for-a-r/1449784#1449784l. – David Hill Aug 04 '20 at 01:42

Root system of $\mathfrak{gl}_n$ and reduction to $\mathfrak{sl}_n$ root system

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