Editing Cartan matrix

{{Short description|Matrices named after Élie Cartan}}In [[mathematics]], the term '''Cartan matrix''' has three meanings.  All of these are named after the French [[mathematician]] [[Élie Cartan]]. Amusingly, the Cartan matrices in the context of [[Lie algebra]]s were first investigated by [[Wilhelm Killing]], whereas the [[Killing form]] is due to Cartan.{{fact|date=December 2017}}

== Lie algebras ==
{{Lie groups}}

A (symmetrizable) '''generalized Cartan matrix''' is a [[square matrix]] <math>A = (a_{ij})</math> with [[integer]] entries such that

# For diagonal entries, <math>a_{ii} = 2 </math>.
# For non-diagonal entries, <math>a_{ij} \leq 0 </math>.
# <math>a_{ij} = 0</math> if and only if <math>a_{ji} = 0</math>
# <math>A</math> can be written as <math>DS</math>, where <math>D</math> is a [[diagonal matrix]], and <math>S</math> is a [[symmetric matrix]].

For example, the Cartan matrix for [[G2 (mathematics)#Dynkin diagram and Cartan matrix|''G''<sub>2</sub>]] can be decomposed as such:
:<math>
  \begin{bmatrix}
     2 & -3 \\
    -1 &  2
  \end{bmatrix} =
  \begin{bmatrix}
    3&0\\
    0&1
  \end{bmatrix}\begin{bmatrix}
     \frac{2}{3} & -1 \\
    -1           &  2
  \end{bmatrix}.
</math>

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 matrix|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 the [[scalar product]]s

:<math>a_{ji}=2 {(r_i,r_j)\over (r_j,r_j)}</math><ref>{{cite book |last1=Georgi |first1=Howard |title=Lie Algebras in Particle Physics |publisher=Westview Press |isbn=0-7382-0233-9 |page=115 |edition=2|date=1999-10-22 }}</ref>

(sometimes called the '''Cartan integers''') where ''r<sub>i</sub>'' are the [[root system|simple roots]] of the algebra. The entries are integral from one of the properties of [[root system|root]]s. The first condition follows from the definition, the second from the fact that for <math>i\neq j, r_j-{2(r_i,r_j)\over (r_i,r_i)}r_i</math> is a root which is a [[linear combination]] of the simple roots ''r<sub>i</sub>'' and ''r<sub>j</sub>'' with a positive coefficient for ''r<sub>j</sub>'' and so, the coefficient for ''r<sub>i</sub>'' has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let <math>D_{ij}={\delta_{ij}\over (r_i,r_i)}</math> and <math>S_{ij}=2(r_i,r_j)</math>. Because the simple roots span a [[Euclidean space]], S is positive definite.

Conversely, given a generalized Cartan matrix, one can recover its corresponding Lie algebra. (See [[Kac–Moody algebra]] for more details).

=== Classification ===
An <math>n \times n</math> matrix ''A'' is '''decomposable''' if there exists a nonempty proper subset <math>I \subset \{1,\dots,n\}</math> such that <math>a_{ij} = 0</math> whenever <math>i \in I</math> and <math>j \notin I</math>. ''A'' is '''indecomposable''' if it is not decomposable.

Let ''A'' be an indecomposable generalized Cartan matrix. We say that ''A'' is of '''finite type''' if all of its [[principal minor]]s are positive, that ''A'' is of '''affine type''' if its proper principal minors are positive and ''A'' has [[determinant]] 0, and that ''A'' is of '''indefinite type''' otherwise.

Finite type indecomposable matrices classify the finite dimensional [[simple Lie algebra]]s (of types <math>A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2 </math>), while affine type indecomposable matrices classify the [[affine Lie algebra]]s (say over some algebraically closed field of characteristic 0).

==== Determinants of the Cartan matrices of the simple Lie algebras ====
The determinants of the Cartan matrices of the simple Lie algebras are given in the following table (along with A<sub>1</sub>=B<sub>1</sub>=C<sub>1</sub>, B<sub>2</sub>=C<sub>2</sub>, D<sub>3</sub>=A<sub>3</sub>, D<sub>2</sub>=A<sub>1</sub>A<sub>1</sub>, E<sub>5</sub>=D<sub>5</sub>, E<sub>4</sub>=A<sub>4</sub>, and E<sub>3</sub>=A<sub>2</sub>A<sub>1</sub>).<ref>[https://deepblue.lib.umich.edu/bitstream/handle/2027.42/70011/JMAPAQ-23-11-2019-1.pdf Cartan-Gram determinants for the simple Lie Groups] Alfred C. T. Wu, J. Math. Phys. Vol. 23, No. 11, November 1982</ref>
{| class="wikitable" border="1"
|- style="vertical-align:top"
! A<sub>''n''</sub>
! B<sub>''n''</sub>
! C<sub>''n''</sub>
! D<sub>''n''</sub><br/>''n'' ≥ 3
! E<sub>''n''</sub><br/>3 &le; ''n'' &le; 8
! F<sub>4</sub>
! G<sub>2</sub>
|- align=center
| ''n'' + 1 || 2 || 2 || 4 || 9 − ''n'' || 1 || 1
|}
Another property of this determinant is that it is equal to the index of the associated root system, i.e. it is equal to <math>|P/Q| </math> where {{mvar|P, Q}} denote the weight lattice and root lattice, respectively.

== Representations of finite-dimensional algebras ==
In [[modular representation theory]], and more generally in the theory of representations of finite-dimensional [[associative algebra]]s ''A'' that are ''not'' [[Semisimple algebra|semisimple]], a '''Cartan matrix''' is defined by considering a (finite) set of [[principal indecomposable module]]s and writing [[composition series]] for them in terms of [[irreducible module]]s, yielding a matrix of integers counting the number of occurrences of an irreducible module.

== Cartan matrices in M-theory ==
In [[M-theory]], one may consider a geometry with [[Cycle graph|two-cycles]] which intersects with each other at a finite number of points, in the limit where the area of the two-cycles goes to zero. At this limit, there appears a [[gauge group|local symmetry group]]. The matrix of [[intersection number]]s of a basis of the two-cycles is conjectured to be the Cartan matrix of the [[Lie algebra]] of this local symmetry group.<ref>{{cite journal|last=Sen|first=Ashoke|title=A Note on Enhanced Gauge Symmetries in M- and String Theory|journal=Journal of High Energy Physics|volume=1997|issue=9|pages=001|year=1997|doi=10.1088/1126-6708/1997/09/001|arxiv=hep-th/9707123|s2cid=15444381}}</ref>

This can be explained as follows. In M-theory one has [[soliton]]s which are two-dimensional surfaces called ''membranes'' or ''2-branes''. A 2-brane has a [[tension (physics)|tension]] and thus tends to shrink, but it may wrap around a two-cycles which prevents it from shrinking to zero.

One may [[Compactification (physics)|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 IIA [[string theory]] as a limit of M-theory, with 2-branes wrapping a two-cycles now described by an open string stretched between [[D-brane]]s. There is a [[U(1)]] local symmetry group for each D-brane, resembling the [[Degrees of freedom (physics and chemistry)|degree of freedom]] of moving it without changing its orientation. The limit where the two-cycles have zero area is the limit where these D-branes are on top of each other, so that one gets an enhanced local symmetry group.

Now, an open string stretched between two D-branes represents a Lie algebra generator, and the [[commutator]] 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 dependent on the way 2-branes may intersect in the original M-theory, i.e. in the intersection numbers of two-cycles. Thus the Lie algebra depends entirely on these intersection numbers. The precise relation to the Cartan matrix is because the latter describes the commutators of the [[Simple root (root system)|simple root]]s, which are related to the two-cycles in the basis that is chosen.

Generators in the [[Cartan subalgebra]] are represented by open strings which are stretched between a D-brane and itself.

==See also==
* [[Dynkin diagram]]
* [[Exceptional Jordan algebra]]
* [[Fundamental representation]]
* [[Killing form]]
* [[Simple Lie group]]

==Notes==
{{reflist}}

==References==
* {{cite book | first1=William | last1=Fulton | authorlink=William Fulton (mathematician) | first2=Joe | last2=Harris |authorlink2=Joe Harris (mathematician)  | title=Representation theory: A first course | series=[[Graduate Texts in Mathematics]] | volume=129 | publisher=Springer-Verlag | year=1991 | isbn=0-387-97495-4 | page=334 }}
* {{cite book | first=James E. | last=Humphreys | title=Introduction to Lie algebras and representation theory | series=[[Graduate Texts in Mathematics]] | volume=9 | publisher=Springer-Verlag | year=1972 | isbn=0-387-90052-7 | pages=55–56 |doi=10.1007/978-1-4612-6398-2}}
* {{Cite book|last=Kac|first= Victor G.|title=Infinite Dimensional Lie Algebras|edition=3rd|publisher=Cambridge University Press|year= 1990|isbn=978-0-521-46693-6}}.

== External links ==
* {{springer|title=Cartan matrix|id=p/c020530}}
* {{mathworld | urlname = CartanMatrix | title = Cartan matrix }}

{{Matrix classes}}

[[Category:Matrices (mathematics)]]
[[Category:Lie algebras]]
[[Category:Representation theory]]