Cartan matrix

A Cartan matrix , named after Élie Cartan , is a matrix that is used in the mathematical theory of Lie algebras to classify these algebras.

Cartan matrix of a Lie algebra

To define the Cartan matrices, some terms and facts from the theory of Lie algebras are required, which are briefly summarized here. Let it be a finite-dimensional, semi-simple Lie algebra over the field of complex numbers . be a contained Cartan subalgebra . For be ${\displaystyle L}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle H}$ ${\displaystyle x\in L}$

${\displaystyle \mathrm {ad} \,x:L\rightarrow L,\quad y\mapsto [x,y]}$

the so-called adjunction with . The theory of Lie algebras shows that a symmetrical, non-degenerate bilinear form is defined, the so-called killing form . Their limitation to is also not degenerate, that is, every element of the dual space is of the form ${\displaystyle x}$ ${\displaystyle \langle x,y\rangle :=\mathrm {Spur} ((\mathrm {ad} \,x)(\mathrm {ad} \,y))}$ ${\displaystyle H}$ ${\displaystyle \alpha \in H^{*}}$

${\displaystyle \alpha _{x}:H\rightarrow \mathbb {C} ,\,y\mapsto \langle x,y\rangle }$

for a definite one . Using the vector space isomorphism , the killing form is transferred to a non-degenerate bilinear form , that is, one sets . ${\displaystyle x\in H}$ ${\displaystyle \iota :H\rightarrow H^{*},x\mapsto \alpha _{x}}$ ${\displaystyle H^{*}}$ ${\displaystyle \langle \alpha ,\beta \rangle :=\langle \iota ^{-1}(\alpha ),\iota ^{-1}(\beta )\rangle }$

We also show that there are a finite set of linear functionals such that ${\displaystyle \Phi \subset H^{*}\setminus \{0\}}$ ${\displaystyle \alpha :H\rightarrow \mathbb {C} }$

${\displaystyle L=H\oplus \sum _{\alpha \in \Phi }L_{\alpha }}$

in which

${\displaystyle L_{\alpha }:=\{x\in L|\,\forall h\in H:\,\exists n\in \mathbb {N} :\,(\mathrm {ad} \,h-\alpha (h)\mathrm {id} _{H})^{n}x=0\}}$

and is not the null space . A subset can be selected from this set of so-called roots , so that each unique linear combination of the elements is off, with the coefficients either all positive or all negative. is called a set of fundamental roots, it is a vector space basis of the Cartan subalgebra . ${\displaystyle L_{\alpha }}$ ${\displaystyle \Phi }$ ${\displaystyle \Phi _{0}\subset \Phi }$ ${\displaystyle \alpha \in \Phi }$ ${\displaystyle \Phi _{0}}$ ${\displaystyle \Phi _{0}=\{\alpha _{1},\ldots ,\alpha _{l}\}}$ ${\displaystyle H}$

The Cartan matrix of Lie algebra is defined as the matrix with coefficients . ${\displaystyle A_{i,j}:=2{\frac {\langle \alpha _{i},\alpha _{j}\rangle }{\langle \alpha _{i},\alpha _{i}\rangle }},\quad i,j=1,\ldots l}$

Two Cartan matrices are called equivalent if they emerge from one another by changing the arrangement of the base. Since the basis vectors can be permuted as desired, a Cartan matrix can of course only be uniquely determined up to equivalence. It can be shown that the equivalence class of the Cartan matrix does not depend on the other options in the above construction, that is, on the choice of the Cartan subalgebra and also not on the choice of the fundamental roots . ${\displaystyle \alpha _{1},\ldots ,\alpha _{l}}$ ${\displaystyle \Phi _{0}\subset \Phi }$

Examples

• ${\displaystyle (2)}$is the only matrix that is a Cartan matrix. ${\displaystyle 1\times 1}$
• ${\displaystyle {\begin{pmatrix}2&-1\\-1&2\end{pmatrix}}}$is Cartan matrix of the three-dimensional, special linear Lie algebra .

Since we give a complete classification of all Cartan matrices below, further examples are not necessary at this point.

properties

Be a Cartan matrix. Then: ${\displaystyle A=(A_{i,j})_{i,j}}$

• ${\displaystyle A_{i,i}=2}$   for everyone   . ${\displaystyle i}$
• ${\displaystyle A_{i,j}\in \{0,-1,-2,-3\}}$   for all   ${\displaystyle i\not =j}$
• If   so   ${\displaystyle A_{i,j}\in \{-2,-3\}}$ ${\displaystyle A_{j,i}=-1}$
• ${\displaystyle A_{i,j}=0}$   exactly when   ${\displaystyle A_{j,i}=0}$
• ${\displaystyle A}$is regular , the inverse matrix has only non-negative rational coefficients.
• There is a diagonal matrix and a symmetrical matrix with . ${\displaystyle D}$ ${\displaystyle B}$ ${\displaystyle A=DB}$

Dismantling of the Cartan matrices

Is the Cartan matrix of a Lie algebra equivalent to a matrix of the form ${\displaystyle L}$

${\displaystyle {\begin{pmatrix}A_{1}&0\\0&A_{2}\end{pmatrix}}}$

with sub-matrices and , that's the name of the Cartan matrix that can be dismantled. One can show that and in turn are Cartan matrices again. A direct total decomposition corresponds to this decomposition ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$

${\displaystyle L=L_{1}\oplus L_{2}}$

in ideals and , here is Cartan matrix of . It is therefore sufficient to know all indecomposable Cartan matrices, these then belong to simple Lie algebras . ${\displaystyle L_{1}}$ ${\displaystyle L_{2}}$ ${\displaystyle A_{i}}$ ${\displaystyle L_{i}}$

meaning

The assignment

Isomorphism classes of finite-dimensional simple Lie algebras     Equivalence classes of Cartan matrices ${\displaystyle \mapsto }$

is a complete isomorphism invariant, i.e. H.

• Isomorphic finite-dimensional simple Lie algebras have equivalent Cartan matrices.
• Finite-dimensional simple Lie algebras with equivalent Cartan matrices are isomorphic.

Classification of the indecomposable Cartan matrices

One can specify all indecomposable Cartan matrices (except for equivalence). The designation in the following list corresponds to the usual classification of finite-dimensional simple Lie algebras.

${\displaystyle A_{n}={\begin{pmatrix}2&-1&&&&&&&\\-1&2&-1&&&&&&\\&-1&2&-1&&&&&\\&&-1&\cdot &\cdot &&&&\\&&&\cdot &\cdot &\cdot &&&\\&&&&\cdot &\cdot &-1&&\\&&&&&-1&2&-1&\\&&&&&&-1&2&-1\\&&&&&&&-1&2\end{pmatrix}}\quad \quad n\geq 1}$

${\displaystyle B_{n}={\begin{pmatrix}2&-1&&&&&&&\\-1&2&-1&&&&&&\\&-1&2&-1&&&&&\\&&-1&\cdot &\cdot &&&&\\&&&\cdot &\cdot &\cdot &&&\\&&&&\cdot &\cdot &-1&&\\&&&&&-1&2&-1&\\&&&&&&-1&2&-1\\&&&&&&&-2&2\end{pmatrix}}\quad \quad n\geq 2}$

${\displaystyle C_{n}={\begin{pmatrix}2&-1&&&&&&&\\-1&2&-1&&&&&&\\&-1&2&-1&&&&&\\&&-1&\cdot &\cdot &&&&\\&&&\cdot &\cdot &\cdot &&&\\&&&&\cdot &\cdot &-1&&\\&&&&&-1&2&-1&\\&&&&&&-1&2&-2\\&&&&&&&-1&2\end{pmatrix}}\quad \quad n\geq 3}$

${\displaystyle D_{n}={\begin{pmatrix}2&-1&&&&&&&&\\-1&2&-1&&&&&&&\\&-1&2&-1&&&&&&\\&&-1&\cdot &\cdot &&&&&\\&&&\cdot &\cdot &\cdot &&&&\\&&&&\cdot &\cdot &-1&&&\\&&&&&-1&2&-1&&\\&&&&&&-1&2&-1&-1\\&&&&&&&-1&2&\\&&&&&&&-1&&2\end{pmatrix}}\quad \quad n\geq 4}$

${\displaystyle E_{6}={\begin{pmatrix}2&-1&&&&\\-1&2&-1&&&\\&-1&2&-1&-1&\\&&-1&2&&\\&&-1&&2&-1\\&&&&-1&2\end{pmatrix}}}$

${\displaystyle E_{7}={\begin{pmatrix}2&-1&&&&&\\-1&2&-1&&&&\\&-1&2&-1&&&\\&&-1&2&-1&-1&\\&&&-1&2&&\\&&&-1&&2&-1\\&&&&&-1&2\end{pmatrix}}}$

${\displaystyle E_{8}={\begin{pmatrix}2&-1&&&&&&\\-1&2&-1&&&&&\\&-1&2&-1&&&&\\&&-1&2&-1&&&\\&&&-1&2&-1&-1&\\&&&&-1&2&&\\&&&&-1&&2&-1\\&&&&&&-1&2\end{pmatrix}}}$

${\displaystyle F_{4}={\begin{pmatrix}2&-1&&\\-1&2&-1&\\&-2&2&-1\\&&-1&2\end{pmatrix}}}$

${\displaystyle G_{2}={\begin{pmatrix}2&-3\\-1&2\end{pmatrix}}}$

Existence proposition

In the theory of Lie algebras one shows with some effort that every finite-dimensional simple Lie algebra must have a Cartan matrix from the list above. Much more difficult is the proof that there is actually a suitable finite-dimensional simple Lie algebra for each of these Cartan matrices . The so-called existence principle says that this is indeed the case. Of course, one could give a finite-dimensional simple Lie algebra for each of the given matrices and check that its Cartan matrix is ​​the given matrix. In a general construction one considers the Lie algebra freely generated by generators with relations ${\displaystyle A=(A_{i,j})_{i,j}}$ ${\displaystyle \{e_{1},\ldots ,e_{n},h_{1},\ldots ,h_{n},f_{1},\ldots ,f_{n}\}}$

${\displaystyle [h_{i},h_{j}]}$

${\displaystyle [h_{i},e_{j}]-A_{i,j}e_{j}}$

${\displaystyle [h_{i},f_{j}]+A_{i,j}f_{j}}$

${\displaystyle [e_{i},f_{i}]-h_{i}}$

${\displaystyle [e_{i},f_{j}]}$   For   ${\displaystyle i\not =j}$

${\displaystyle [e_{i},[e_{i}[\ldots [e_{i},e_{j}]]]]}$   with and occurrences of ${\displaystyle i\not =j}$ ${\displaystyle 1-A_{i,j}}$ ${\displaystyle e_{i}}$

${\displaystyle [f_{i},[f_{i}[\ldots [f_{i},f_{j}]]]]}$   with and occurrences of . ${\displaystyle i\not =j}$ ${\displaystyle 1-A_{i,j}}$ ${\displaystyle f_{i}}$

From this Lie algebra one can show that it is a finite-dimensional simple Lie algebra with a suitable Cartan matrix. A particular difficulty lies in proving the finite dimension. This is the proof of the existence theorem going back to Jean-Pierre Serre . Note that this general construction only depends on the data of the Cartan matrix presented. This shows once again that knowledge of the Cartan matrix determines the finite-dimensional simple Lie algebra.

Relationship to Dynkin Diagrams

The related Dynkin diagrams

The Cartan matrices are closely and reciprocally related to the Dynkin diagrams . For each Cartan matrix with the lower index one constructs a graph, which one then calls the associated Dynkin diagram, with nodes and connects two different nodes and through edges. If and are connected by more than one edge, then an angle> is made through these edges, the pointed end pointing to exactly when . The Cartan matrix can be recovered from the Dynkin diagram. The indivisibility on the side of the Cartan matrices corresponds exactly to the context of the Dynkin diagrams. In the adjacent drawing, all Dynkin diagrams for the non-decomposable Cartan matrices are given. ${\displaystyle A=(A_{i,j})_{i,j}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \{x_{1},\ldots ,x_{n}\}}$ ${\displaystyle x_{i}}$ ${\displaystyle x_{j}}$ ${\displaystyle A_{i,j}A_{j,i}}$ ${\displaystyle x_{i}}$ ${\displaystyle x_{j}}$ ${\displaystyle x_{j}}$ ${\displaystyle |A_{j,i}|>|A_{i,j}|}$ ${\displaystyle A_{n},B_{n},C_{n},D_{n},E_{6},E_{7},E_{8},F_{4},G_{2}}$

Individual evidence

1. ^ Roger Carter : Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , Chapter 6.1: The Cartan matrix
2. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory , Springer-Verlag (1972), ISBN 0-387-90052-7 , Chapter 11.1: Cartan matrix of ${\displaystyle \Phi }$
3. ^ Roger Carter: Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , sentence 10.18
4. ^ Roger Carter: Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , Chapter 6.4: Classification of Cartan matrices
5. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory , Springer-Verlag (1972), ISBN 0-387-90052-7 , Chapter 11.4: Classification Theorem
6. ^ Roger Carter: Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , Chapter 7.5: The existence theorem
7. ^ Roger Carter: Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , Chapter 6.4: Classification Cartan matrices