LOG#090. Group theory(X).

grouptheoryX

The converse of the first Lie theorem is also generally true.

Theorem. Second Lie Theorem. Given a set of N hermitian matrices or operators L_j, closed under commutation with the group multiplication, then these operators L_j define and specify a Lie group and they are their generators.

Remark(I): We use hermitian generators in our definition of “group generators”. Mathematicians use to define “antihermitian” generators in order to “erase” some “i” factors from the equations of Lie algebras, especially in the group exponentiation.

Remark (II): Mathematicians use to speak about 3 different main Lie theorems, more or less related to the 2 given theorems in my thread. I am a physicist, so somehow I do prefer this somewhat informal distinction but the essence and contents of the Lie theory is the same.

Definition(39). Lie algebra. The set of N matrices N\times N L_j and their linear combinations, closed under commutation with the group multiplication, is said to be a Lie algebra. Lie algebra are characterized by structure constants.

Example: In the 3D worl of space, we can define the group of 3D rotations

SO(3)=\left\{O\in M_{3\times 3}(\mathbb{R}),OO^+=I, \mbox{det}O=1\right\}

and the generators of this group J_1, J_2,J_3 satisfy a Lie algebra called so(3) with commutation rules

\left[J_i,J_j\right]=i\varepsilon_{ijk}J_k

There \varepsilon_{ijk} is the completely antisymmetric symbol in the three indices. We can form linear combinations of generators:

J_{\pm}=J_1\pm J_2

and then the commutation rules are changed into the following relations

\left[J_3,J_\pm\right]=\pm J_\pm

\left[J_+,J_-\right]=2J_3

The structure constants are different in the two cases above but they are related through linear expressions involving the coefficients of the combinations of the generators. In summary:

1) It is possible to get by direct differentiation that every Lie algebra forms a Lie group. The Lie algebra consists of those matrices or operators X for which the exponential \exp(tX)\in G \forall real number t. The Lie bracket/commutator of the Lie algebra is given by the commutator of the two matrices.

2) Different groups COULD have the same Lie algebra, e.g., the groups SO(3) and SU(2) share the same Lie algebra. Given a Lie algebra, we can build by exponentiation, at least, one single Lie group. In fact, usually we can build different Lie groups with the same algebra.

Definition (40). Group rank. By definition, the rank is the largest number of generators commuting with each other. The rank of a Lie group is the same that the rank of the corresponding Lie algebra.

Definition (41). Casimir operator.  A Casimir operator is a matrix/operator which commutes with all the generators of the group, and therefore it also commutes with all the elements of the algebra and of the group.

Theorem. Racah’s theorem. The number of Casimir operators of a Lie group is equal to its rank.

Before giving a list of additional results in the theory of Lie algebras, let me provide some extra definitions about (Lie) groups.

Definition (42). Left, right and adjoint group actions.  Let L_g, R_g, \mbox{Ad}_g be group function isomorphisms:

L_g: G\longrightarrow G/ g\Rightarrow L_g=L_g(h)=gh

R_g:G\longrightarrow G/ g\Rightarrow R_g(h)=hg^{-1}

\mbox{Ad}_g:G\longrightarrow G/ g\Rightarrow \mbox{Ad}_g(h)=ghg^{-1}

then they are called respectively left group action, right group action and adjoint group action. They are all group isomorphisms.

In fact, we can be more precise about what a Lie algebra IS. A Lie algebra is some “vector space” with an external operation, the commutator definining the Lie algebra structure constants, with some beautiful properties.

Definition (43). Lie algebra. Leb (A,+,\circ) be a real (or complex) vector space and define the binary “Lie-bracket” operation

\left[,\right]:\mathcal{A}\times\mathcal{A}\longrightarrow \mathcal{A}

This Lie bracket is a bilinear and antisymmetric operation in the (Lie) algebra \mathcal{A} such as it satisfies the so-called Jacobi identity:

\left[\left[A,B\right],C\right]+\left[\left[B,C\right],A\right]+\left[\left[C,A\right],B\right]=0

In fact, if the Jacobi identity holds for some algebra \mathcal{A}, then it is a real(or complex) Lie algebra.

Remark(I): A bilinear antisymmetric operation satisfies (in our case we use the bracket notation):

Bilinearity: \left[A,B+\lambda C\right]=\left[A,B\right]+\lambda\left[A,C\right]

Antisymmetry: \left[A,B\right]=-\left[B,A\right]

Remark(II): The Jacobi identity is equivalent to the expressions:

i) \left[\left[A,B\right],C\right]+\left[B,\left[A,C\right]\right]=\left[A,\left[B,C\right]\right]

ii) Let us define a “derivation” operation with the formal equation D_A(X)\equiv \left[A,X\right]. Then, it satisfies the Leibniz rule

D_A(\left[B,C\right])=\left[D_A(B),C\right]+\left[B,D_A(C)\right]

Remark(III): From the antisymmetry, in the framework of a Lie algebra we have that \left[A,A\right]=0

The commutators of matrices/operators are examples of bilinear antisymmetric operations, but even more genral operations can be guesses. Ask a mathemation or a wise theoretical physicist for more details! 🙂

In the realm of Lie algebras, there are some “basic results” we should know. Of course, the basical result is of course:

\left[A_i,A_j\right]=c_{ijk}A_k

Since the Lie algebra is a vector space, we can form linear combinations or superpositions

B_i=\sum_j a_{ij}A_j

with some a_{ij} non singular matrix. Thus, the new elements will satisfy the commutation relation swith some new “structure constants”:

\left[B_i,B_j\right]=c'_{ijk}B_k

such as

c'_{ijk}=a_{ik}a_{jl}(A^{-1})_{mn}c_{klm}

In particular, if A=a_{ij} is a unitary transformation (or basis change) with A^{-1}=A^+, then

c'_{ijk}=a_{ik}a_{jl}a^*_{nm}c_{klm}

Definition (44). Simple Lie algebra.  A simple Lie algebra is a Lie algebra that has no non-trivial ideal and that is not abelian. An ideal is certain subalgebra generated by a subset of generators A_i such that \left[B_j,A_i\right]=\sum a_{ijk}A_k \forall B_j.

Definition (45). Semisimple Lie algebra. A Lie algebra is called semisimple if it does not contain any non-zero abelian ideals (subalgebra finitely generated).

In particular, every simple Lie algebra is semisimple. Reciprocally, any semisimple Lie algebra is the direct sum of simple Lie algebras. Semisimplicity is related with the complete reducibility of the group representations. Semisimple Lie algebras have been completely by Cartan and other mathematicians using some more advanced gadgets called “root systems”. I am not going to discuss root systems in this thread (that would be too advanced for my current purposes), but I can provide the list of semisimple Lie algebras in the next table. This table contains the summary of t:

Algebra Rank Dimension Group
A_{n-1}(n\geq 2) n-1 n^2-1 SL(n),SU(n)
B_{n}(n\geq 1) n n(2n+1) SO(2n+1)
C_{n}(n\geq 3) n n(2n+1) Sp(2n)
D_{n}(n\geq 4) n n(2n-1) SO(2n)
\mathfrak{G}_2 2 12 G_2
\mathfrak{F}_4 4 48 F_4
\mathfrak{e}_6 6 72 E_6
\mathfrak{e}_7 7 126 E_7
\mathfrak{e}_8 8 240 E_8

We observe that there are 4 “infinite series” of “classical groups” and 5 exceptional semisimple Lie groups. The allowed dimensions for a given rank can be worked out, and the next table provides a list up to rank 9 of the dimension of Lie algebras for the above Cartan group classification:

Rank Dimension
1 3(SU(2)
2 8(SU(3)),10,12
3 15(SU(4)),21,21
4 24,36,36,28,48
5 35,55,55,45
6 48,78,78,66,72
7 63,105,105,91,72
8 80,126,136,136,120
9 99,240,171,171,153

Final remark: in general, Lie algebras deriving or providing some Lie group definition can be represented with same letters of the group but in gothic letters(fraktur style) or in lower case.

See you in the next blog post!

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