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The Cartan matrix of a finite-dimensional semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c0205302.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c0205303.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c0205304.png" /> is a matrix
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{{MSC|17B}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c0205305.png" /></td> </tr></table>
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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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c0205306.png" /> is some system of simple roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c0205307.png" /> with respect to a fixed [[Cartan subalgebra|Cartan subalgebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c0205308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c0205309.png" /> is the scalar product on the dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053010.png" /> defined by the [[Killing form|Killing form]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053011.png" />. (For the Cartan matrix of an arbitrary system of roots see [[Root system|Root system]].) Up to a transformation induced by a permutation of the indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053012.png" />, a Cartan matrix is an invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053013.png" />, that is, it does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053014.png" /> or the system of simple roots. This invariant completely determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053015.png" />: 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.
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$$\def\a{\alpha} A= \Bigg( 2\frac{(\a_i,\a_j)}{(\a_j,\a_j)}\Bigg)_{i,j = 1,\dots,r}$$
 +
where $\a_1,\dots,\a_r$ is some system of simple roots of $\fg$ with respect to a fixed
 +
[[Cartan subalgebra|Cartan subalgebra]] $\def\ft{\mathfrak{t}}$ and $(\;,\;)$ is the scalar product on the dual space of $\ft$ defined by the
 +
[[Killing form|Killing form]] on $\fg$. (For the Cartan matrix of an arbitrary system of roots see
 +
[[Root system|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053016.png" /> be a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053017.png" /> into a direct sum of simple subalgebras and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053018.png" /> be the Cartan matrix of the simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053019.png" />. Then the block-diagonal matrix
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053020.png" /></td> </tr></table>
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$$\begin{pmatrix} A_1 & \cdots & 0\\ \vdots & \ddots & \vdots \\ 0 &\cdots&A_m\end{pmatrix}$$
 +
is the Cartan matrix of $\fg$. (For the explicit form of the Cartan matrix of simple Lie algebras, see
 +
[[Lie algebra, semi-simple|Lie algebra, semi-simple]].)
  
is the Cartan matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053021.png" />. (For the explicit form of the Cartan matrix of simple Lie algebras, see [[Lie algebra, semi-simple|Lie algebra, semi-simple]].)
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The entries $a_{ij} = 2 (\a_i,\a_j)/(\a_j,\a_j)$ of a Cartan matrix have the following properties:
  
The entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053022.png" /> of a Cartan matrix have the following properties:
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$$\begin{equation}  \left.\begin{aligned}  a_{ii}=2;\quad a_{ij}\le 0 \ &\textrm{and}\ a_{ij} \in \Z\textrm{ for }i\ne j,\\
 +
        a_{ij}=0\ &\;\;\Rightarrow\  a_{ji}=0.       \end{aligned}
 +
  \right\}\qquad\tag{1}  \end{equation}$$
 +
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$\def\d{\delta} [e_i,f_j]=\d_{ij}h_i;\quad [h_i,e_j]=a_{ij}e_j;\quad [h_i,f_j] = -a_{ij}f_j;\quad [h_i,h_j] = 0.\tag{2}$$
  
The Cartan matrix is closely related to the description of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053024.png" /> in terms of generators and relations. Namely, there exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053025.png" /> linearly independent generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053029.png" /> (the so-called canonical generators), connected by the following relations:
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$\def\ad{\textrm{ad}\;} (\ad e_i)^{-\a_{ij}+1} e_j = 0,\quad (\ad f_i)^{-\a_{ij}+1} f_j = 0,\quad i\ne j,\tag{3}$$
 +
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
 +
{{Cite|Se}}).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053031.png" /></td> </tr></table>
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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
 
+
{{Cite|Ka}}.
Any two systems of canonical generators can be transformed one into the other by automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053032.png" />. In addition to (2), the canonical generators satisfy the relations
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
 
 
 
where, by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053034.png" />. For the chosen system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053038.png" />, relations (2) and (3) are defining for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053039.png" /> (see [[#References|[2]]]).
 
 
 
For any matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053040.png" /> satisfying (1), the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053041.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053042.png" /> with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053046.png" />, and defining relations (2) and (3) is finite-dimensional if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053047.png" /> is the Cartan matrix of a semi-simple Lie algebra [[#References|[3]]].
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "Algèbres de Lie semi-simples complexes" , Benjamin  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>
 
  
 
====Comments====
 
====Comments====
A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053048.png" /> 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|Kac–Moody algebra]], [[#References|[a2]]].
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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|Kac–Moody algebra]],
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053049.png" /> be a semi-simple Lie algebra over an algebraically closed field of characteristic zero. Then a set of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053052.png" /> 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.
+
{{Cite|Ka2}}.
  
====References====
+
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.
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Humphreys,   "Introduction to Lie algebras and representation theory" , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.G. Kac,  "Infinite dimensional Lie algebras" , Cambridge Univ. Press  (1985)</TD></TR></table>
 
  
The Cartan matrix of a finite-dimensional associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053053.png" /> with a unit over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053054.png" /> is the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053056.png" />, defined by a complete set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053057.png" /> of finite-dimensional irreducible left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053058.png" />-modules. Specifically, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053059.png" /> is the multiplicity of occurrence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053060.png" /> in a composition series of an indecomposable projective left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053061.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053062.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053063.png" />. Such a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053064.png" /> exists for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053065.png" /> and is uniquely defined up to an isomorphism.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053066.png" /> turns out to be symmetric, positive definite, and even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053068.png" /> is an integral, not necessarily square, matrix (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053069.png" /> denotes taking the transpose matrix). Such is the case for the Cartan matrix of the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053070.png" /> of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053071.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053072.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053073.png" /> (see [[#References|[1]]]); in this case the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053074.png" /> form a complete set of non-isomorphic principal indecomposable left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053076.png" />-modules, that is, indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053077.png" />-modules into which the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053078.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053079.png" /> is decomposed as a direct sum. Another example when such an equality holds for a Cartan matrix: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053080.png" /> is the restricted universal enveloping algebra of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053081.png" /> over an algebraically closed field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053082.png" />, obtained from a semi-simple complex Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053083.png" /> by reduction to characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020530/c02053084.png" /> (see [[#References|[2]]]).
+
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
 +
{{Cite|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
 +
{{Cite|Hu2}}).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.E. Humphreys,  "Modular representations of classical Lie algebras and semi-simple groups"  ''J. of Algebra'' , '''19'''  (1971)  pp. 51–79</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|CuRe}}||valign="top"| C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras", Interscience  (1962) {{MR|0144979}}  {{ZBL|0131.25601}}
 +
|-
 +
|valign="top"|{{Ref|Hu}}||valign="top"|  J.E. Humphreys,  "Introduction to Lie algebras and representation theory", Springer  (1972)  {{MR|0323842}}  {{ZBL|0254.17004}}
 +
|-
 +
|valign="top"|{{Ref|Hu2}}||valign="top"| J.E. Humphreys,  "Modular representations of classical Lie algebras and semi-simple groups"  ''J. of Algebra'', '''19'''  (1971)  pp. 51–79   {{ZBL|0219.17003}}
 +
|-
 +
|valign="top"|{{Ref|Ja}}||valign="top"|  N. Jacobson,  "Lie algebras", Interscience  (1962)  ((also: Dover, reprint, 1979))  {{MR|0148716}} {{MR|0143793}}  {{ZBL|0121.27504}} {{ZBL|0109.26201}}
 +
|-
 +
|valign="top"|{{Ref|Ka}}||valign="top"|  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 
 +
|-
 +
|valign="top"|{{Ref|Ka2}}||valign="top"|  V.G. Kac,  "Infinite dimensional Lie algebras", Cambridge Univ. Press  (1985)  {{MR|0823672}} {{ZBL|0574.17010}}
 +
|-
 +
|valign="top"|{{Ref|Se}}||valign="top"|  J.-P. Serre,  "Algèbres de Lie semi-simples complexes", Benjamin  (1966)  {{MR|0215886}}  {{ZBL|0144.02105}}
 +
|-
 +
|}

Latest revision as of 08:14, 22 May 2012

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
How to Cite This Entry:
Cartan matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_matrix&oldid=14006
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article