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Let $I = \{1,2,\dots,n\}$ and $S \subset I$. The set $I$ will be indexing the simple roots and $S$ will be indexing the odd generators of a Lie superalgebra.

A real matrix $A=(a_{ij})_{i,j\in I}$ is said to be a generalized Cartan matrix if the following conditions are satisfied:

  1. $A$ is symmetric;

  2. $a_{ii}=2$ or $a_{ii}\leq 0$;

  3. $a_{ij}\leq 0$ if $i\neq j$;

  4. $\frac{2a_{ij}}{a_{ii}}\in\mathbb{Z}$ if $a_{ii}>0$;

  5. $\frac{2a_{ij}}{a_{ii}}\in2\mathbb{Z}$ if $a_{ii}>0$ and $i \in S$.

  6. $a_{ij}=0$ if and only if $a_{ji}=0$.

The Lie superalgebra $\mathfrak g = \mathfrak g(A)$ associated to the generalized Cartan matrix $A$ is the Lie superalgebra generated by $e_i, f_i, h_i, i \in I$ with the following defining relations:

  1. $[h_i, h_j]=0$ for $i,j\in I$,

  2. $[h_i, e_j]=a_{ij}e_j$, $[h_i, f_j]=-a_{ij}f_j$ for $i,j\in I$,

  3. $[e_i, f_j]=\delta_{ij}h_i$ for $i, j\in I$,

  4. $\deg h_i = 0, i \in I$,

  5. $\deg e_i = 0 = \deg f_i$ if $i \notin S$,

  6. $\deg e_i = 1 = \deg f_i$ if $i \in S$,

  7. $(\text{ad }e_i)^{1-\frac{2a_{ij}}{a_{ii}}}e_j=0 = (\text{ad }f_i)^{1-\frac{2a_{ij}}{a_{ii}}}f_j$ if $a_{ii} > 0$ and $i \ne j$,

  8. $(\text{ad }e_i)^{1-\frac{a_{ij}}{a_{ii}}}e_j=0 = (\text{ad }f_i)^{1-\frac{a_{ij}}{a_{ii}}}f_j$ if $i \in S$ and $a_{ii}>0$ and $i \ne j$,

  9. $[e_i, e_j]= 0 = [f_i, f_j]$ if $a_{ij}=0$.

I have the following questions regarding the definition.

a) In the definition of the matrix $A$ what is the role of (5)? I have seen only condition (4) in the case of Lie algebras. Also why the powers in (7) and (8) are differed by a factor of 2?

b) How (5) in the definition of $A$ and (8) affects the structure of the root spaces of $\mathfrak g$?

c) What are some common differences between this structure and Lie algebras? For example which roots are odd roots?

Kindly explain to me with some examples. Thank you.

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    $\begingroup$ Hi GA316. May i ask for some reference for your definitions (i.e. rel. 1-6 and 1-9) ? $\endgroup$ – Konstantinos Kanakoglou Dec 25 '19 at 21:44
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    $\begingroup$ I mean, the way the questions are posed, it would be helpful to know wthat texts you are stydying. $\endgroup$ – Konstantinos Kanakoglou Dec 26 '19 at 0:05
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    $\begingroup$ 1) Automorphic Forms and Lie Superalgebras Authors: Ray, Urmie. Chapter 2. 2) infinite-dimensional Lie Algebras By Minoru Wakimoto. Thanks for your interest. $\endgroup$ – GA316 Dec 26 '19 at 0:50
  • $\begingroup$ @KonstantinosKanakoglou can you help me with this, please. $\endgroup$ – GA316 Dec 27 '19 at 18:36
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Although i am not an expert in the topic, i did some studying on the references you provided together with Kac's monograph on Infinite dimensional Lie algebras.

I do not have very clear answers to your questions, but here is what i can understand: The relations you are providing (the def of the generalized Cartan matrix $A$ and the Lie superalgebra $\mathfrak g(A)$ associated to it) are not generally valid for any Lie superalgebra but specifically for the Kac-Moody Lie (super)algebras and the BKM Lie (super) algebras. However, for the general case of the basic, classical, simple, complex, Lie superalgebras (BCSCLS for short; these were classified by Kac) these relations are not enough (and should be altered including higher order relations that is the Chevalley-Serre relations; however this is another story).

In my understanding, relations (5) (and (8)) have the following consequences on the structure of the root spaces:

First, these relations guarantee that the Kac-Moody Lie (super)algebras and the BKM superalgebras $\mathfrak g(A)$, associated to the generalized Cartan matrix $A$, are generated by copies of $sl_2$, $sl(0|1)$ and the three dim Heisenberg Lie (super)algebra and decompose -as vector spaces- into finite dimensional $sl_2$ and $sl(0|1)$ modules under the adjoint action.
I think this is the content of Proposition 2.1.13, p. 19, from Ray's book.

Second, the most important consequence of these relations (as i get it) is the uniqueness of the Cartan matrices and the corresponding Dynkin diagrams for the description of these algebras:
For the general case of the f.d., BCSCLS, the Cartan matrices are not uniquely determined (generally, there are more than one non-equivalent Cartan matrices corresponding to the same BCSCLS) and so are the Dynkin diagrams (non-equivalent Dynkin diag may represent the same BCSCLS). Furthermore, the notion of Cartan matrix for the BCSCLS is different than the generalized Cartan matrices (they generally satisfy somewhat different relations).
However, for the Kac-Moody Lie (super)algebras and the BKM Lie superalgebras the generalized Cartan matrices (as defined above) are uniquely associated to the algebras and so are the corresponding Dynkin diagrams constructed via these matrices. In this sense, the structure and the mechanism of the classification is now much closer to the case of the ordinary, fin dim, simple Lie algebras.
As i understand it, the fact that rel (5) and ((7),(8)) imply the uniqueness of the generalized Cartan matrices and of the Dynckin diagrams for BKM Lie superalgebras, is a result of Theorem 2.1.17, p.20, of Ray's book.

P.S.: I hope the above help a bit towards your questions. However, since i have mostly been based upon your references, maybe you are already aware of these remarks and you are looking for something deeper. In that case, i am sorry i cannot help more.

References:

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  • $\begingroup$ Thank you. I was not aware that, these relations play a vital role in the uniqueness of Cartan matrices. Also, do you have some idea why the even integers are appearing in the definition of the generalized Cartan matrix if we assume some odd $𝑎_{𝑖𝑖}=2$? $\endgroup$ – GA316 Dec 30 '19 at 6:25
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If you unpack the standard definition of Lie superalgebras you will see that the even part $\mathfrak{g}_0$ is ordinary Lie algebra and the odd part $\mathfrak{g}_1$ is a representation of $\mathfrak{g}_0$. So odd roots are in fact weights of this representation and the half integrality appears because sometimes $\mathfrak{g}_1$ can be spinorial representation of $\mathfrak{g}_0.$

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  • $\begingroup$ I checked the definition of Spin representations but I couldn't understand much. can you please explain it to me a little further about the half integers? Thank you. $\endgroup$ – GA316 Dec 30 '19 at 6:11

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