# Cartan matrix

In

mathematics , the term**Cartan matrix**has two meanings. Both of these are named after the Frenchmathematician Élie Cartan . In an example ofStigler's law of eponymy , Cartan matrices in the context ofLie algebra s were first investigated byWilhelm Killing , whereas theKilling form is due to Cartan.**Lie algebras**A

**generalized Cartan matrix**is asquare matrix $A\; =\; (a\_\{ij\})$ with integer entries such that# For diagonal entries, $a\_\{ii\}\; =\; 2$.

# For non-diagonal entries, $a\_\{ij\}\; leq\; 0$.

# $a\_\{ij\}\; =\; 0$ if and only if $a\_\{ji\}\; =\; 0$

# $A$ can be written as $DS$, where $D$ is adiagonal matrix , and $S$ is asymmetric matrix .The third condition is not independent but is really a consequence of the first and fourth conditions.

We can always choose a D with positive diagonal entries. In that case,if $S$ in the above decomposition is

positive definite , then $A$ is said to be a**Cartan matrix**.The Cartan matrix of a simple

Lie algebra is the matrix whose elements are thescalar product s:$a\_\{ij\}=\{2\; (r\_i,r\_j)over\; (r\_i,r\_i)\}$

(sometimes called the

**Cartan integers**) where $r\_i$ are the simple roots of the algebra. The entries are integral from one of the properties of roots. The first condition follows from the definition, the second from the fact that for $i\; eq\; j$, $r\_j-\{2(r\_i,r\_j)over\; (r\_i,r\_i)\}r\_i$ is a root which is alinear combination of thesimple root s r_{i}and r_{j}with a positive coefficient for r_{i}and so, the coefficient for r_{i}has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let $D\_\{ij\}=\{delta\_\{ij\}over\; (r\_i,r\_i)\}$ and $S\_\{ij\}=2(r\_i,r\_j)$. Because the simple roots span aEuclidean space , S is positive definite.**Representations of finite-dimensional algebras**In

modular representation theory , and more generally in the theory of representations offinite-dimensional algebra s "A" that are "not"semisimple , a**Cartan matrix**is defined by considering a (finite) set ofprincipal indecomposable module s and writingcomposition series for them in terms ofprojective module s, yielding a matrix of integers counting the number of occurrences of a projective module.**Cartan matrices in M-theory**In

M-theory , one may consider a geometry with two-cycles which intersects with each other at a finite number of points, at the limit where the area of the two-cycles go to zero. At this limit, there appears a local symmetry group. The matrix ofintersection number s of a basis of the two-cycles is conjectured to be theCartan matrix of theLie algebra of this local symmetry group [*http://arxiv.org/abs/hep-th/9707123*] .This can be explained as follows. In

M-theory one hassoliton s which are two-dimensional surfaces called "membranes" or "2-branes". A 2-brane has atension and thus tends to shrink, but it may wrap around a two-cycles which prevents it from shrinking to zero.One may compactify one dimension which is shared by all two-cycles and their intersecting points, and then take the limit where this dimension shrinks to zero, thus getting a

dimensional reduction over this dimension. Then one gets type IIAstring theory as a limit ofM-theory , with 2-branes wrapping a two-cycles now described by an open string stretched betweenD-brane s. There is aU(1) local symmetry group for eachD-brane , resembling thedegree of freedom of moving it without changing its orientation. The limit where the two-cycles have zero area is the limit where theseD-brane s are on top of each other, so that one gets an enhanced local symmetry group.Now, an open string stretched between two

D-brane s represents aLie algebra generator, and thecommutator of two such generator is a third one, represented by an open string which one gets by gluing together the edges of two open strings. The latter relation between different open strings is depepnds on the way 2-branes may intersect in the originalM-theory , i.e. in theintersection number s of two-cycles. Thus theLie algebra depends entirely on theseintersection number s. The precise relation to the Cartan matrix is because the latter describes thecommutator s of thesimple root s, which are related to the two-cycles in the basis that is chosen.Note that generators in the

Cartan subalgebra are represented by open strings which are stretched between aD-brane and itself.**References***

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