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| In [[mathematics]], the term '''Cartan matrix''' has three meanings. All of these are named after the [[France|French]] [[mathematician]] [[Élie Cartan]]. In fact, Cartan matrices in the context of [[Lie algebra]]s were first investigated by [[Wilhelm Killing]], whereas the [[Killing form]] is due to Cartan.
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| == Lie algebras ==
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| {{Lie groups}}
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| A '''generalized Cartan matrix''' is a [[square matrix]] <math>A = (a_{ij})</math> with [[integer|integral]] entries such that
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| # For diagonal entries, ''a<sub>ii</sub>'' = 2.
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| # For non-diagonal entries, <math>a_{ij} \leq 0 </math>.
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| # <math>a_{ij} = 0</math> if and only if <math>a_{ji} = 0</math>
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| # ''A'' can be written as ''DS'', where ''D'' is a [[diagonal matrix]], and ''S'' is a [[symmetric matrix]].
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| For example, the Cartan matrix for [[G2_(mathematics)#Dynkin_diagram_and_Cartan_matrix|''G''<sub>2</sub>]] can be decomposed as such:
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| :<math>
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| \left [
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| \begin{smallmatrix}
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| \;\,\, 2&-3\\
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| -1&\;\,\, 2
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| \end{smallmatrix}\right ]
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| = \left [
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| \begin{smallmatrix}
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| 3&0\\
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| 0&1
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| \end{smallmatrix}\right ]
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| \left [
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| \begin{smallmatrix}
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| 2/3&-1\\
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| -1&\;2
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| \end{smallmatrix}\right ].
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| </math>
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| The third condition is not independent but is really a consequence of the first and fourth conditions.
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| 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'''.
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| The Cartan matrix of a simple [[Lie algebra]] is the matrix whose elements are the [[scalar product]]s
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| :<math>a_{ij}=2 {(r_i,r_j)\over (r_i,r_i)}</math>
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| (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.
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| Conversely, given a generalized Cartan matrix, one can recover its corresponding Lie algebra. (See [[Kac-Moody algebra]] for more details).
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| === Classification ===
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| A <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.
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| 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.
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| 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).
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| ==== Determinants of the Cartan matrices of the simple Lie algebras ====
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| The determinants of the Cartan matrices of the simple Lie algebras given in the following table.
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| {| class="wikitable" border="1"
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| |-
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| | <math>A_n</math>
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| | <math>B_n</math>, <math> n\geq 2 </math>
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| | <math>C_n</math>, <math> n\geq 2 </math>
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| | <math>D_n</math>, <math> n\geq 4 </math>
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| | <math>E_n</math>, <math> n\geq 5 </math>
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| | <math>F_4</math>
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| | <math>G_2</math>
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| |- align=center
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| | ''n''+1
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| | 2
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| | 2
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| | 4
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| | 9-''n''
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| | 1
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| | 1
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| |}
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| == Representations of finite-dimensional algebras ==
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| 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.
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| == Cartan matrices in M-theory ==
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| In [[M-theory]], one may consider a geometry with [[Cycle (mathematics)|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 [[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 [http://arxiv.org/abs/hep-th/9707123].
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| 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.
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| 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.
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| 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.
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| 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.
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| Note that generators in the [[Cartan subalgebra]] are represented by open strings which are stretched between a D-brane and itself.
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| ==References==
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| * {{cite book | author=William Fulton | authorlink=William Fulton (mathematician) | coauthors=[[Joe Harris (mathematician)|Joe Harris]] | 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 }}
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| * {{cite book | author=James E. 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 }}
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| * {{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}}.
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| ==See also==
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| * [[Dynkin diagram]]
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| * [[Exceptional Jordan algebra]]
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| * [[Fundamental representation]]
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| * [[Killing form]]
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| * [[Simple Lie group]]
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| == External links ==
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| * {{springer|title=Cartan matrix|id=p/c020530}}
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| * {{mathworld | urlname = CartanMatrix | title = Cartan matrix }}
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| {{DEFAULTSORT:Cartan Matrix}}
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| [[Category:Matrices]]
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| [[Category:Lie algebras]]
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| [[Category:Representation theory]]
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