Cartan matrix

2020 Mathematics Subject Classification: Primary: 17B [MSN][ZBL]

The Cartan matrix of a finite-dimensional semi-simple Lie algebra $\def\fg{\mathfrak{g}}\fg$ over an algebraically closed field $k$ of characteristic $0$ is a matrix

$$\def\a{\alpha} A= \Bigg( 2\frac{(\a_i,\a_j)}{(\a_j,\a_j)}\Bigg)_{i,j = 1,\dots,r}$$ where $\a_1,\dots,\a_r$ is some system of simple roots of $\fg$ with respect to a fixed Cartan subalgebra $\def\ft{\mathfrak{t}}$ and $(\;,\;)$ is the scalar product on the dual space of $\ft$ defined by the Killing form on $\fg$. (For the Cartan matrix of an arbitrary system of roots see Root system.) Up to a transformation induced by a permutation of the indices $1,\dots,r$, a Cartan matrix is an invariant of $\fg$, that is, it does not depend on the choice of $\ft$ or the system of simple roots. This invariant completely determines $\fg$: Two semi-simple Lie algebras are isomorphic if and only if their Cartan matrices are the same, up to a transformation induced by permutation of the indices. A semi-simple Lie algebra is simple if and only if its Cartan matrix is indecomposable, that is, not expressible as a block-diagonal matrix after some permutation of indices.

Let $\fg = \fg_1+\cdots + \fg_m$ be a decomposition of $\fg$ into a direct sum of simple subalgebras and let $A_j$ be the Cartan matrix of the simple Lie algebra $\fg_j$. Then the block-diagonal matrix

$$\begin{pmatrix} A_1 & \cdots & 0\\ \vdots & \ddots & \vdots \\ 0 &\cdots&A_m\end{pmatrix}$$ is the Cartan matrix of $\fg$. (For the explicit form of the Cartan matrix of simple Lie algebras, see Lie algebra, semi-simple.)

The entries $a_{ij} = 2 (\a_i,\a_j)/(\a_j,\a_j)$ of a Cartan matrix have the following properties:

$$\begin{equation} \left.\begin{aligned} a_{ii}=2;\quad a_{ij}\le 0 \ &\textrm{and}\ a_{ij} \in \Z\textrm{ for }i\ne j,\\ a_{ij}=0\ &\;\;\Rightarrow\ a_{ji}=0. \end{aligned} \right\}\qquad\tag{1} \end{equation}$$ The Cartan matrix is closely related to the description of $\fg$ in terms of generators and relations. Namely, there exist in $\fg$ linearly independent generators $e_i, f_i, h_i,\;i=1,\dots,r$, (the so-called canonical generators), connected by the following relations:

$$\def\d{\delta} [e_i,f_j]=\d_{ij}h_i;\quad [h_i,e_j]=a_{ij}e_j;\quad [h_i,f_j] = -a_{ij}f_j;\quad [h_i,h_j] = 0.\tag{2}$$

Any two systems of canonical generators can be transformed one into the other by automorphism of $\fg$. In addition to (2), the canonical generators satisfy the relations

$$\def\ad{\textrm{ad}\;} (\ad e_i)^{-\a_{ij}+1} e_j = 0,\quad (\ad f_i)^{-\a_{ij}+1} f_j = 0,\quad i\ne j,\tag{3}$$ where, by definition, $(\ad x)y = [x,y].$. For the chosen system of generators $e_i$, $f_i$, $h_i$, $i=1,\dots,r$, relations (2) and (3) are defining for $\fg$ (see [Se]).

For any matrix $A$ satisfying (1), the Lie algebra $\fg(A)$ over the field $k$ with generators $e_i$, $f_i$, $h_i$, $i=1,\dots,r$, and defining relations (2) and (3) is finite-dimensional if and only if $A$ is the Cartan matrix of a semi-simple Lie algebra [Ka].

Comments

A matrix $A$ satisfying (1) defines a finite-dimensional Lie algebra if and only if it is positive definite; in other cases, notably the semi-positive definite case, other interesting algebras arise, cf. Kac–Moody algebra, [Ka2].

Let $L$ be a semi-simple Lie algebra over an algebraically closed field of characteristic zero. Then a set of generators $e_i$, $f_i$, $h_i$ such that (2) holds is also called a set of Chevalley generators or a Chevalley basis. That such a set of generators exists is Chevalley's theorem. The result that the relations (2) and (3) together define the Lie algebra is often called Serre's theorem.

The Cartan matrix of a finite-dimensional associative algebra $A$ with a unit over a field $k$ is the matrix $c_{ij}$, $i,j=1,\dots,s$, defined by a complete set $N_1,\dots,N_s$ of finite-dimensional irreducible left $A$-modules. Specifically, $c_{ij}$ is the multiplicity of occurrence of $N_j$ in a composition series of an indecomposable projective left $A$-module $P_i$ for which $\textrm{Hom}(P_1,\dots,N_i)\ne 0$. Such a module $P_i$ exists for each $N_i$ and is uniquely defined up to an isomorphism.

In certain cases the Cartan matrix $C$ turns out to be symmetric, positive definite, and even $C=D^TD$, where $D$ is an integral, not necessarily square, matrix (and ${}^T$ denotes taking the transpose matrix). Such is the case for the Cartan matrix of the group algebra $A=k[G]$ of a finite group $G$ over a field $k$ of characteristic $p>0$ (see [CuRe]); in this case the $P_1,\dots,P_s$ form a complete set of non-isomorphic principal indecomposable left $A$-modules, that is, indecomposable $A$-modules into which the left $AAA$-modules $A$ is decomposed as a direct sum. Another example when such an equality holds for a Cartan matrix: $A$ is the restricted universal enveloping algebra of a Lie algebra $\fg$ over an algebraically closed field of characteristic $p>0$, obtained from a semi-simple complex Lie algebra $\fg$ by reduction to characteristic $p$ (see [Hu2]).

References

[CuRe]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras", Interscience (1962) MR0144979 Zbl 0131.25601
[Hu]	J.E. Humphreys, "Introduction to Lie algebras and representation theory", Springer (1972) MR0323842 Zbl 0254.17004
[Hu2]	J.E. Humphreys, "Modular representations of classical Lie algebras and semi-simple groups" J. of Algebra, 19 (1971) pp. 51–79 Zbl 0219.17003
[Ja]	N. Jacobson, "Lie algebras", Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[Ka]	V.G. [V.G. Kats] Kac, "Simple irreducible graded Lie algebras of finite growth" Math. USSR Izv., 2 : 6 (1968) pp. 1271–1311 Ivz. Akad. Nauk SSSR Ser. Mat., 32 : 6 (1968) pp. 1323–1367
[Ka2]	V.G. Kac, "Infinite dimensional Lie algebras", Cambridge Univ. Press (1985) MR0823672 Zbl 0574.17010
[Se]	J.-P. Serre, "Algèbres de Lie semi-simples complexes", Benjamin (1966) MR0215886 Zbl 0144.02105