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.










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    $begingroup$


    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.










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      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.










      share|cite|improve this question









      $endgroup$




      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






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      asked Jan 29 at 12:40









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