Cartan Matrix

In mathematics, the term Cartan matrix has three meanings. They are all named for the French mathematician Ely Cartan . In fact, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing , while the Killing form belongs to Cartan.

Content

1 Lie Algebras
- 1.1 Classification
  - 1.1.1 Determinants of Cartan matrices of simple Lie algebras
2 Representations of finite-dimensional algebras
3 Cartan Matrices in M-Theory
4 See also
5 notes
6 Literature
7 References

Lie Algebras

The generalized Cartan matrix is a square matrix $A=(a_{ij})$ ${\ displaystyle A = (a_ {ij})}$ with whole elements such that

Diagonal elements a _ii = 2.
Off-diagonal elements $a_{ij}\leq 0$ ${\ displaystyle a_ {ij} \ leq 0}$ .
$a_{ij}=0$ ${\ displaystyle a_ {ij} = 0}$ if and only if $a_{ji}=0$ ${\ displaystyle a_ {ji} = 0}$ .
A can be written as DS , where D is the diagonal matrix and S is symmetric .

For example, the Cartan matrix for G ₂ can be decomposed as follows:

\left[{\begin{smallmatrix}\;\,\,2&-3\\-1&\;\,\,2\end{smallmatrix}}\right]=\left[{\begin{smallmatrix}3&0\\0&1\end{smallmatrix}}\right]\left[{\begin{smallmatrix}2/3&-1\\-1&\;2\end{smallmatrix}}\right].

{\ displaystyle \ left [{\ begin {smallmatrix} \; \, \, 2 & -3 \\ - 1 & \; \, \, 2 \ end {smallmatrix}} \ right] = \ left [{\ begin {smallmatrix } 3 & 0 \\ 0 & 1 \ end {smallmatrix}} \ right] \ left [{\ begin {smallmatrix} 2/3 & -1 \\ - 1 & \; 2 \ end {smallmatrix}} \ right].}

The third condition is not independent and is a consequence of the first and fourth conditions.

We can always choose D with positive diagonal elements. In this case, if S in the expansion is positive definite , then they say that A is the Cartan matrix .

The Cartan matrix of a simple Lie algebra is a matrix whose elements are scalar products

a_{ij}=2{(r_{i},r_{j}) \over (r_{i},r_{i})}

{\ displaystyle a_ {ij} = 2 {(r_ {i}, r_ {j}) \ over (r_ {i}, r_ {i})}}

(sometimes called Cartan integers ), where r _i is the root system of the algebra. Elements are integer due to one of the properties of the root system . The first condition follows from the definition, the second from the fact that for $i\neq j,r_{j}-{2(r_{i},r_{j}) \over (r_{i},r_{i})}r_{i}$ ${\ displaystyle i \ neq j, r_ {j} - {2 (r_ {i}, r_ {j}) \ over (r_ {i}, r_ {i})} r_ {i}}$ is a root, which is a linear combination of simple roots r _i and r _j with a positive coefficient for r _j , and then the coefficient of r _i must be non-negative. The third condition is true in view of the symmetry of the orthogonality relation. And finally, let $D_{ij}={\delta _{ij} \over (r_{i},r_{i})}$ ${\ displaystyle D_ {ij} = {\ delta _ {ij} \ over (r_ {i}, r_ {i})}}$ and $S_{ij}=2(r_{i},r_{j})$ ${\ displaystyle S_ {ij} = 2 (r_ {i}, r_ {j})}$ . Since simple roots are linearly independent, S is their Gram matrix (with coefficient 2), and therefore it is positive definite.

And vice versa, if a generalized Cartan matrix is given, one can find the corresponding Lie algebra (see the article in the article details).

Classification

Matrix A size $n\times n$ ${\ displaystyle n \ times n}$ is decomposable if there is a nonempty subset $I\subset \{1,\dots ,n\}$ ${\ displaystyle I \ subset \ {1, \ dots, n \}}$ such that $a_{ij}=0$ ${\ displaystyle a_ {ij} = 0}$ for all $i\in I$ ${\ displaystyle i \ in I}$ and $j\notin I$ ${\ displaystyle j \ notin I}$ . A is indecomposable if this condition is not satisfied.

Let A be an indecomposable generalized Cartan matrix. We say that A is of finite type if all of its major minors are positive, that A is of affine type , if all of its own major minors are positive and the determinant of A is 0, and that A is of indefinite type in other cases.

Indecomposable matrices of finite type classify simple Lie groups of finite dimension (of type $A_{n},B_{n},C_{n},D_{n},E_{6},E_{7},E_{8},F_{4},G_{2}$ ${\ displaystyle A_ {n}, B_ {n}, C_ {n}, D_ {n}, E_ {6}, E_ {7}, E_ {8}, F_ {4}, G_ {2}}$ ), while indecomposable matrices of affine type classify (over some algebraically closed fields with characteristic 0).

Determinants of Cartan matrices of simple Lie algebras

The determinants of the Cartan matrices of simple Lie algebras are given in the table.

$A_{n}$ ${\ displaystyle A_ {n}}$	$B_{n}$ ${\ displaystyle B_ {n}}$ , $n\geq 2$ ${\ displaystyle n \ geq 2}$	$C_{n}$ ${\ displaystyle C_ {n}}$ , $n\geq 2$ ${\ displaystyle n \ geq 2}$	$D_{n}$ ${\ displaystyle D_ {n}}$ , $n\geq 4$ ${\ displaystyle n \ geq 4}$	$E_{n}$ ${\ displaystyle E_ {n}}$ , $n=6,7,8$ ${\ displaystyle n = 6,7,8}$	$F_{4}$ ${\ displaystyle F_ {4}}$	$G_{2}$ ${\ displaystyle G_ {2}}$
n +1	2	2	four	9- n	one	one

Another property of this determinant is that it is equal to the index of the associated root system, that is, it is equal to $|P/Q|$ ${\ displaystyle | P / Q |}$ where $P, Q$ ${\ displaystyle P, Q}$ denote the and the root grating, respectively.

Representations of finite-dimensional algebras

In the and in the more general theory of representations of finite-dimensional associative algebras that are not , the Cartan matrix is determined by considering the (finite) set of and writing for them in terms simple modules , getting a matrix of integers containing the number of occurrences of a simple module.

Cartan matrices in M-theory

In M-theory, one can represent geometry as the limit of two cycles that intersect each other in a finite number of points as the area of the two cycles tends to zero. In the limit, a group of local symmetry arises. Hypothetically, the matrix of intersection indices of the basis of two cycles is the Cartan matrix of the Lie algebra of this group of local symmetry ^[1] .

This can be explained as follows: in M-theory there are solitons , which are two-dimensional surfaces, called membranes or 2-branes . 2-branes have tension and therefore tend to decrease, but they can be wrapped around bicycles that prevent membranes from collapsing to zero.

One can one dimension in which all two-cycles and their intersection points are located, and take the limit at which the dimension collapses to zero, thereby obtaining a decrease in this dimension. Then we get the string theory of type IIA as the limit of M-theory with 2-branes wrapping bicycles, now presented as open strings stretched between D-branes . There is a group of local symmetry U (1) for each D-brane, similar to the degrees of freedom of motion without changing orientation. The limit where two-cycles have zero area is the limit where these D-branes are on top of each other.

An open string stretched between two D-branes represents a Lie algebra generator, and the commutator of two such generators is the third generator represented by an open string, which can be obtained by gluing the edges of two open strings. Further connections between different open strings depends on the way that 2-branes can intersect in the original M-theory, that is, in the number of intersections of two cycles. Thus, the Lie algebra depends entirely on these intersection numbers. A connection with the Cartan matrix is assumed because it describes commutators of simple roots that are connected with two cycles in the chosen basis.

Note that the generators in the represented by open strings that are stretched between the D-brane and the same brane.

Notes

↑ Ashoke Sen. A Note on Enhanced Gauge Symmetries in M- and String Theory // Journal of High Energy Physics. - IOP Publishing, 1997. - T. 1997 , no. 9 . - DOI : 10.1088 / 1126-6708 / 1997/09/001 .

Literature

William Fulton, Joe Harris. Representation theory: A first course. - Springer-Verlag, 1991. - T. 129. - S. 334. - ( ). - ISBN 0-387-97495-4 .
James E. Humphreys. Introduction to Lie algebras and representation theory. - Springer-Verlag, 1972. - T. 9. - S. 55-56. - ( ). - ISBN 0-387-90052-7 .
Victor G. Kac. Infinite Dimensional Lie Algebras. - 3rd. - 1990. - ISBN 978-0-521-46693-6 .
Michiel Hazewinkel. Encyclopedia of Mathematics. - Springer, 2001. - ISBN 978-1-55608-010-4 .

Links

Weisstein, Eric W. Cartan matrix on Wolfram MathWorld .

[1] Ashoke Sen. A Note on Enhanced Gauge Symmetries in M- and String Theory // Journal of High Energy Physics. - IOP Publishing, 1997. - T. 1997 , no. 9 . - DOI : 10.1088 / 1126-6708 / 1997/09/001 .