Kac–Moody algebra

2013-12-22T00:28:41Z

74.83.182.22:

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.

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

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

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

For example, the Cartan matrix for [[G2_(mathematics)#Dynkin_diagram_and_Cartan_matrix|''G''2]] can be decomposed as such:
:<math>
\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 ].
</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_{ij}=2 {(r_i,r_j)\over (r_i,r_i)}</math>

(sometimes called the '''Cartan integers''') where ''ri'' 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 ''ri'' and ''rj'' with a positive coefficient for ''rj'' and so, the coefficient for ''ri'' 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 ===
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.

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 given in the following table.
{| class="wikitable" border="1"
|-
| <math>A_n</math>
| <math>B_n</math>, <math> n\geq 2 </math>
| <math>C_n</math>, <math> n\geq 2 </math>
| <math>D_n</math>, <math> n\geq 4 </math>
| <math>E_n</math>, <math> n\geq 5 </math>
| <math>F_4</math>
| <math>G_2</math>
|- align=center
| ''n''+1
| 2
| 2
| 4
| 9-''n''
| 1
| 1
|}

== 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 (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].

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.

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

==References==
* {{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 }}
* {{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 }}
* {{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}}.

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

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

{{DEFAULTSORT:Cartan Matrix}}
[[Category:Matrices]]
[[Category:Lie algebras]]
[[Category:Representation theory]]

Theoretical gravity

2013-10-07T22:23:17Z

74.83.182.22:

{{Use dmy dates|date=August 2013}}
'''[[Feedback]]-controlled electromigration''' (FCE) is an experimental technique to investigate the phenomenon known as [[electromigration]]. By controlling the voltage applied as the [[Electrical conductance|conductance]] varies it is possible to keep the [[voltage]] at a critical level for [[electromigration]].

==Theory==
FCE has been shown to be reversible, demonstrating the fact that the electrons are moving rather than [[thermomigration]] or [[sublimation (phase transition)|sublimation]]. The migration occurs due to the [[Ion wind|electronic wind force]] experienced by the metallic [[adatom]].<ref name=Ishida1>{{cite journal | author = H.Ishida | title = Driving force for adatom electromigration within mixed Cu/Al overlayers on Al(111) | journal = Phys. Rev. | volume = 89 |date=September 2000 | doi = 10.1063/1.1325385 | url = http://link.aip.org/link/doi/10.1063/1.1325385 }}</ref> The electromigration occurs at a critical power dissipation <math>P=I/G</math> in the neck of the bridge.<ref name=Strachan1>{{cite journal | author = D.R. Strachan et al. | title =Clean Electromigrated Nanogaps Imaged by Transmission Electron Microscopy | journal = Nano Lett | volume = 6 | year = 2006 | url = http://pubs.acs.org/wls/journals/query/citationFindResults.html?op=findCitation&cit_qjrn=nalefd&vol=6&spn=441&mscid=&submit.x=24&submit.y=9&submit=FIND }}</ref> This leads to [[Electromigrated Nanogaps]].

==Uses==
FCE is often used in forming nanogaps in metallic bridges.

==Problems==
[[Thermal runaway]] can occur when the neck is narrower than about 20 nm.<ref name=Mahadevan1>{{cite journal | author = M.Mahadevan and R.m. Bradley | title =Simulations and theory of electromigration-induced slit formation in unpassivated single-crystal metal lines | journal = Phys. Rev. B | volume = 59 | year = 1999 | url = http://link.aps.org/doi/10.1103/PhysRevB.59.11037 |doi = 10.1103/PhysRevB.59.11037 }}</ref>

==References and external links==
{{reflist|2}}

[[Category:Electronic design automation]]
[[Category:Semiconductor device defects]]
[[Category:Nanoelectronics]]
[[Category:Emerging technologies]]

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Kac–Moody algebra

Theoretical gravity