On Classical and Exceptional Semisimple Lie Algebras

Medhat Ahmed Rakha

Abstract

Numerical procedures for obtaining the root systems of both classical and exceptional Lie algebras are slow and tedious when done by hand. This is a contribution in the field that uses a modern computer language Mathematica. This gives the benefits of high portability, efficiency and makes it easier to use either classical or exceptional Lie algebras in user notebook. Root systems, Positive roots and their sum, Simple roots, Cartan's matrices, highest and Fundamental Weights are all calculated in the notebook.

Introduction

Lie algebras are algebraic structures used for the study of continuous Lie groups, they were introduced by and named after Sophus Lie (1842-99). Besides Sophus Lie, the important contributors to this theory were W. Killing, E. Cartan and H. Weyl.
The theory of Lie algebras is an outgrowth of the Lie theory of continuous groups. Three fundamental theorems of Sophus Lie establishes a one - to - one correspondence between local Lie groups and Lie algebras, one associates with every Lie group a Lie algebra over the reals or complexes and one establishes a correspondence between the analytic subgroups of Lie group and the subgroups of its Lie algebra.
The number of applications of the theory of both simple and semi-simple Lie algebras to mathematics and physics continuous to grow with no end in sight. Although in recent years many of the new developments have been in the direction of infinite dimensional representations, the finite dimensional representations continue to be the basic tool in the majority of applied situations.
The study of the properties of the root systems is one of the most important application of the different types of Lie algebras.

Basic Concepts

Lie algebras arise ''in nature'' as vector spaces of linear transformations endowed with a new operation which is in general neither commutative nor associative : [x,y]=x y - y x (where the operations on the right side are the usual ones).

What is the Definition of a Lie algebra?

A vector space L over a field F, with an operation L×L -> L, denoted (x,y)->[xy] and called the bracket or commutator of x and y, is called a Lie algebra over F if the following axioms are satisfied :
1- The bracket operation is bilinear.
2- [x x] = 0, for all x in L.
3- [x[yz]] + [y[zx]] + [z[xy]]=0, (x,y,z ϵ L).
Axiom 3 is called the Jacobi identity.

For further examples,which are central to the main work in this article, they fall into four families (Subscript[A, l]) (l≥1), (Subscript[B, l]) (l≥2), (Subscript[C, l]) (l≥3) and (Subscript[D, l]) (l≥4) and are called the Classical Lie algebras (because they correspond to certain of the classical linear Lie groups).

What is the Definition of a Root System?

A  subset ϕ of the Euclidean space E is called a root system in E if the following axioms are satisfied :
1- ϕ is finite, spans E, and does not contain 0.
2- if α∈ϕ, the only multiples of α in ϕ are ±α.
3- if α∈ϕ, the reflections of σα leaves ϕ invarient.
4- if α,β∈ϕ, then <β,α>∈Z.

We call l=dimE the rank of the root system ϕ.
A part from these four classical simple Lie algebras,there are only five other simple Lie algebras. These are the Cartan's exceptional Lie algebras (Subscript[G, 2]), (Subscript[F, 4]),(Subscript[E, i])(i=6,7,8). 
The following table, describes the root systems of both classical and exceptional Lie algebras.

Classical Lie Algebras

Here we will present a different example about the Classical Lie algebras, we will use Mathematica to compute all the items of its  roots system.

Our first question is, How could we calculate the roots system of a special type of the Classical Lie Algebras.

The Answer is, using Mathematica and all the information we were used before in the tedious hand calculations, one can easily use some very interested Mathematica built in functions that can combined together in a very simple algorithm that will be able to compute all the related items of a roots system : 
1- The System Δ of all Roots contained in the specific type of  Lie algebra.
2-  \(TraditionalForm\`\(Δ\^+\)\) , the set of all Positive Roots.
3- The sum of  \(TraditionalForm\`\(Δ\^+\)\).
4- ∏, the system of Simple Roots.
5- The Highest Weights.
6- The Fundamental Weights.
7- Cartan's Matrix.

Here is an example for the Classical Lie algebra (Subscript[A, n]):

Input["Enter the value of n, n>=1"];
e := Array[v, {n + 1, n + 1}]
Rt := Array[R, {n (n + 1), n + 1}]
Pos := Array[P, {(1/2) n (n + 1), n + 1}]
Simple := Array[S, {n, n + 1}]
Cartan := Array[Car, {n, n}]
Do[If[i == j, v[i, j] = 1, v[i, j] = 0], {i, n + 1}, {j, 
        n + 1}]
q := 1
Do[If[i ≠ j, 
        Do[R[q, k] = v[i, k] - v[j, k]; 
          If[k == n + 1, q = q + 1, bb], {k, n + 1}], xx], {i, n + 1}, {j, 
        n + 1}]
q := 1
Do[Do[P[q, k] = v[i, k] - v[j, k]; 
        If[k == n + 1, q = q + 1, bb], {k, n + 1}], {i, n}, {j, i + 1, 
        n + 1}]
q := 1
Do[Do[S[q, k] = v[i, k] - v[j, k]; 
        If[k == n + 1, q = q + 1, bb], {k, n + 1}], {i, n}, {j, i + 1, 
        i + 1}]
H := e[[1]] - 
        e[[n + 1]]
Do[Car[i, 
          j] = ((2 Simple[[i]] . Simple[[j]])/Simple)[[j]] . Simple[[j]], {i, 
        n}, {j, n}]
SUMP := (∑_(i = 1)^((1/2) n (n + 1)))Pos[[i]]
Do[w[
          i] = (∑_(j = 1)^i) e[[
              j]] - ((i (∑_(k = 1)^(n + 1))e[[k]])/(n + 1)), {i, n}]
W := Table[w[i], {i, n}]
Print["The Roots System of A", n "are"]
TableForm[Rt]
Print["The Positive Roots of A", n "are"]
TableForm[Pos]
Print["The Sum of the Positive Roots is :", SUMP]
Print["The Simple Roots of A", 
      n "are  " TableForm[Simple]]
Print["The Highest Weight of A", n  "is  ", H]
Print["Cartan's Matrix of A", n  "is  ", 
      MatrixForm[Cartan]]
Print["The Fundamental Weights of A", n  "are ", 
      TableForm[W]]

Exceptional Lie Algebras

Here we will present different examples about the Exceptional Lie algebras, we will use Mathematica to compute all the items of its  roots system of the exceptional Lie algebras (Subscript[G, 2])and (Subscript[F, 4]).

Here is and example of the Exceptional Lie algebra (Subscript[G, 2])

e:=Array[v,{3,3}]
Cartan:=Array[Car,{2,2}] 
Do[If[i==j,v[i,j]=1,v[i,j]=0],{i,3},{j,3}] 
a1:=e[[1]] - e[[2]]
a2:=e[[1]] - e[[3]]
a3:=e[[2]] - e[[3]]
a4:=2e[[1]] - e[[2]] - e[[3]]
a5:=2e[[2]] - e[[1]] - e[[3]]
a6:=2e[[3]] - e[[1]] - e[[2]]
s1:=e[[1]] - e[[2]]
s2:=-2e[[1]] + e[[2]] + e[[3]]
R:=List[a1,a2,a3,-a1,-a2,-a3,a4,a5,a6,-a4,-a5,-a6]
Simple:=List[s1,s2]
Pos:=List[s1,s2,s1+s2,2s1+s2,3s1+s2,3s1+2s2]
SUMP:= 2*(5*s1 + 3*s2)
Do[Car[i,j]=2*(Simple[[i]].Simple[[j]])/(Simple[[j]].Simple[[j]]),{i,2},{j,2}]
w1:= 2*s1 + s2
w2:= 3*s1 + 2s2
W:=List[w1,w2]
H:=-e[[1]] - e[[2]] + 2e[[3]]
Print["The Roots System of G2 is :"]
TableForm[R]
Print["The Positive Roots of G2 are :"]
TableForm[Pos]
Print["The Sum of the Positive Roots of G2 is :" [SUMP]]
Print["The Simple Roots of G2 are :"] 
TableForm[Simple]
Print["The Adjoint of G2 is :" [H]]
Print["The Fundamental Weights of G2 are :"]
ColumnForm[W]
Print["The Cartan Matrix of G2 is :"]
MatrixForm[Cartan]

and this is for (Subscript[F, 4])

e:=Array[v,{4,4}] 
Rt:=Array[R,{48,4}]
Pos:=Array[P,{24,4}]
Cartan:=Array[Car,{4,4}]
Do[If[i==j,v[i,j]=1,v[i,j]=0],{i,4},{j,4}] 
z1:=Permutations[{1,0,0,0}]
z2:=Permutations[{1,1,0,0}]
z3:=Permutations[{-1,1,0,0}]
z4:=Permutations[{-1,-1,0,0}]
z5:=Permutations[{1,1,1,1}]
z6:=Permutations[{-1,1,1,1}]
z7:=Permutations[{-1,-1,1,1}]
z8:=Permutations[{-1,-1,-1,1}]
z9:=Permutations[{-1,-1,-1,-1}]
z10:=Permutations[{-1,0,0,0}]
q:=1
Do[R[q,j]=z1[[r,j]];If[j==4,q=q+1,vv],{r,4},{j,4}]
Do[R[q,j]=z2[[r,j]];If[j==4,q=q+1,vv],{r,6},{j,4}]
Do[R[q,j]=z3[[r,j]];If[j==4,q=q+1,vv],{r,12},{j,4}]
Do[R[q,j]=z4[[r,j]];If[j==4,q=q+1,vv],{r,6},{j,4}]
Do[R[q,j]=z5[[r,j]];If[j==4,q=q+1,vv],{r,1},{j,4}]
Do[R[q,j]=z6[[r,j]];If[j==4,q=q+1,vv],{r,4},{j,4}]
Do[R[q,j]=z7[[r,j]];If[j==4,q=q+1,vv],{r,6},{j,4}]
Do[R[q,j]=z8[[r,j]];If[j==4,q=q+1,vv],{r,4},{j,4}]
Do[R[q,j]=z9[[r,j]];If[j==4,q=q+1,vv],{r,1},{j,4}]
Do[R[q,j]=z10[[r,j]];If[j==4,q=q+1,vv],{r,4},{j,4}]
Do[R[i,j]=1/2*R[i,j],{i,29,44},{j,4}] 
q:=11
Do[P[i,j]=R[i,j],{i,10},{j,4}]
Do[If[R[r,j]==1,Do[If[R[r,k]!=-1,xx,Do[P[q,j]=R[r,j];If[j==4,q=q+1,vv],{j,4}]],{k,j,4}],yy],{r,11,22},{j,4}]
Do[If[R[r,1]==1/2,Do[P[q,j]=R[r,j];If[j==4,q=q+1,vv],{j,4}],yy],{r,29,48}]
SUMP:=11*e[[1]] + 5*e[[2]] + 3*e[[3]] + e[[4]]
s1:= e[[2]] - e[[3]]
s2:= e[[3]] - e[[4]]
s3:= e[[4]]
s4:= 1/2*(e[[1]] - e[[2]] - e[[3]] - e[[4]])
Simple:= List[s1,s2,s3,s4]
Do[Car[i,j]=2*(Simple[[i]].Simple[[j]])/(Simple[[j]].Simple[[j]]),{i,4},{j,4}]
H:= e[[1]] + e[[2]] 
w1:= e[[1]] + e[[2]]
w2:= 2*e[[1]] + e[[2]] + e[[3]]
w3:= 1/2*(3*e[[1]] + e[[2]] + e[[3]] + e[[4]])
w4:= e[[1]]
W:=List[w1,w2,w3,w4]
Print["The Root System of F4 are :"]
TableForm[Rt]
Print["The Postive Roots of F4 are:"]
TableForm[Pos]
Print["The Sum of the Positive Roots is :"];SUMP  
Print["The Highest root of F4 is :"];H 
Print["The Fundamental Weights of F4 are :"]
TableForm[W] 
Print["The Simple Roots of F4 are :"]
TableForm[Simple]
Print["The Cartan Matrix of F4 is :"]
MatrixForm[Cartan]

References

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2-  B. G. Wybouine, Classical groups in physicists, John Wiley & Sons Inc, NY, 1974.
3. J. E. Humphrey; Introduction to Lie algebras and representations theory; Springer - Verlag 1970.
4. M. R. Bremner, R. V. Moody, J. Patera; Tables of dominant weight multiplicities for representations of simple Lie algebras, Marcel Dekker Inc., 1985.
5. N. Jacobson, Lie algebras; Dover Publications Inc., NY, 1962.
6. N. Bourbaki, Groups et algebra de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1969.
7. S. Helgason, Differential geometry and symmetric groups, Academic press 1962.
8. S. Wolfram, The Mathematica Book, Third Edition, (Wolfram Media, Inc. and Cambridge University Press 1996).
9. W. Miller, Lie theory and special functions, Academic press, 1968.
10. Y. Chow, General theory of Lie algebras, I, II, Gordon \& Breach Science Pub., 1978.
11. Zhe-Xian Wan, Lie algebras, translated by Che-Younf Lee, Pergamon Press, 1975.
12. E. B. Dynkin, The structure of semisimple algebras, Uspehi Mathematiceskih Nauk (N.S.) 2, No. 4 (20), 59-127, 1974.
13. M. Lorente, B. Gruber, Classification of Semisimple Subalgebras od Simple Lie Algebras, J. Math. Phys., Vol. 13, No. 10, 1639-1663, 1972.

Remark:
1- This talk was published under the title : On the Root Systems of the Classical & Exceptional Lie Algebras. Mathematica Programs.;  Advances in Modeling & Analysis, 1999 - Vol. 35, No. 1,2; pp 1-12.
2- For other notebooks concerning the remaining Classical and Exceptional Lie algebras, you can refer to the article in the previous remark or by contacting the author at marakha@hotmail.com