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I am reading about infinite dimensional lie algebras from the books Infinite-Dimensional Lie Algebras by Victor G. Kac and Lie Algebras of Finite and Affine Type by Roger Carter. Now I have gone through the basic construction rules of realaisation of a matrix $A$ and then constructing the auxiliary lie algebra $\hat {\mathfrak g}(A)$ and then finally the required algebra $\mathfrak g(A)$ which is called the Kac-Moody Algebra if $A$ is a Generalised Cartan Matrix (GCM).

Now I am not finding a general example of this construction. Please help me.

I have found an example for the symmetric $n \times n$ matrix $A$ over $\mathbb C$ with $A_{i,j} = 2 \forall i=j \text{ , } A_{i,i+1}=-1$ and $0$ in all other places (This is the Cartan Matrix for the graph type $A_n$). The algebra $\mathfrak g(A)$ here is nothing but the lie algebra $\mathfrak {sl}_{n+1}(\mathbb C)$.

But I am not finding a suitable example where rank of the matrix is strictly less than the order of a the matrix (Note that in the above matrix $A$ has full rank).

It will be very helpful for me if someone kindly show me such a generalised example. I can then understand the construction more clearly. If such an example can be found in any book or paper please give me the details. I will definitely have a look.

Please help me. Thnx in advance.

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  • $\begingroup$ What sort of description would you want? Unless you are already familiar with a lot of examples of infinite dimensional Lie algebras, I don't think you can expect to get something familiar. $\endgroup$ – Tobias Kildetoft Aug 7 '15 at 7:40
  • $\begingroup$ @TobiasKildetoft An example of matrix A with non full rank and the steps for the realisation of A and construction of the algebra $\mathfrak g(A)$ will be very helpful for me. I am really not familiar with much infinite dimensional lie algebras but related things I can definitely read to understand the example. $\endgroup$ – usermath Aug 7 '15 at 7:52
  • $\begingroup$ en.wikipedia.org/wiki/Affine_Lie_algebra Has a fairly short description for when $A$ is of affine type, which is the easiest to understand after the finite ones (I am not sure anyone really understand the indefinite ones). $\endgroup$ – Tobias Kildetoft Aug 7 '15 at 7:55
  • $\begingroup$ @TobiasKildetoft Thank you so much. I was trying to understand the definition from the page you mentioned. One question I have. The definition is given $[a\otimes t^n +\alpha c,b\otimes t^m +\beta c]= [a,b]\otimes t^{m+n}+\langle a,b\rangle n\delta_{m+n,0}c$ Is the definition i.e. RHS independent of $\alpha,\beta$? or am I missing something? Please help me to clarify the definition. $\endgroup$ – usermath Aug 13 '15 at 4:51

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