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About the definition of Kac-Moody algebras
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I'm following Kac's book on Infinite dimensional Lie algebras and I have just seen the definition of a Kac-Moody algebra associated with a generalized Cartan matrix $A$.
Prior in the text, it was proven that the auxiliary lie algebra $tilde{mathfrak{g}}(A)$ (the first Lie algebra defined in the text by generators with Chevalley relations) has a a maximal ideal $mathfrak r$ with respect to the property $mathfrak i cap mathfrak h = 0.$ Then, the Kac-Moody algebra $mathfrak g(A)$ is just the Lie algebra: $mathfrak g(A) = tilde{mathfrak{g}}(A)/mathfrak r$.
What is the motivation behind the quocient? Why would one choose this maximal ideal? Why is this definition natural and "flourishing"? I would like to get more intuition behind how these concepts were developed. Thank you.
definition kac-moody-algebras
asked Jan 29 at 12:40
math.hmath.h
934617
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add a comment |
$begingroup$
I'm following Kac's book on Infinite dimensional Lie algebras and I have just seen the definition of a Kac-Moody algebra associated with a generalized Cartan matrix $A$.
Prior in the text, it was proven that the auxiliary lie algebra $tilde{mathfrak{g}}(A)$ (the first Lie algebra defined in the text by generators with Chevalley relations) has a a maximal ideal $mathfrak r$ with respect to the property $mathfrak i cap mathfrak h = 0.$ Then, the Kac-Moody algebra $mathfrak g(A)$ is just the Lie algebra: $mathfrak g(A) = tilde{mathfrak{g}}(A)/mathfrak r$.
What is the motivation behind the quocient? Why would one choose this maximal ideal? Why is this definition natural and "flourishing"? I would like to get more intuition behind how these concepts were developed. Thank you.
definition kac-moody-algebras
asked Jan 29 at 12:40
math.hmath.h
934617
$endgroup$
add a comment |
$begingroup$
I'm following Kac's book on Infinite dimensional Lie algebras and I have just seen the definition of a Kac-Moody algebra associated with a generalized Cartan matrix $A$.
Prior in the text, it was proven that the auxiliary lie algebra $tilde{mathfrak{g}}(A)$ (the first Lie algebra defined in the text by generators with Chevalley relations) has a a maximal ideal $mathfrak r$ with respect to the property $mathfrak i cap mathfrak h = 0.$ Then, the Kac-Moody algebra $mathfrak g(A)$ is just the Lie algebra: $mathfrak g(A) = tilde{mathfrak{g}}(A)/mathfrak r$.
What is the motivation behind the quocient? Why would one choose this maximal ideal? Why is this definition natural and "flourishing"? I would like to get more intuition behind how these concepts were developed. Thank you.
definition kac-moody-algebras
asked Jan 29 at 12:40
math.hmath.h
934617
$endgroup$
I'm following Kac's book on Infinite dimensional Lie algebras and I have just seen the definition of a Kac-Moody algebra associated with a generalized Cartan matrix $A$.
Prior in the text, it was proven that the auxiliary lie algebra $tilde{mathfrak{g}}(A)$ (the first Lie algebra defined in the text by generators with Chevalley relations) has a a maximal ideal $mathfrak r$ with respect to the property $mathfrak i cap mathfrak h = 0.$ Then, the Kac-Moody algebra $mathfrak g(A)$ is just the Lie algebra: $mathfrak g(A) = tilde{mathfrak{g}}(A)/mathfrak r$.
What is the motivation behind the quocient? Why would one choose this maximal ideal? Why is this definition natural and "flourishing"? I would like to get more intuition behind how these concepts were developed. Thank you.
definition kac-moody-algebras
definition kac-moody-algebras
asked Jan 29 at 12:40
math.hmath.h
934617
asked Jan 29 at 12:40
math.hmath.h
934617
asked Jan 29 at 12:40
math.hmath.h
934617
asked Jan 29 at 12:40
math.hmath.h
934617
asked Jan 29 at 12:40
math.hmath.h
934617
934617
add a comment |
add a comment |
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