# LOG#091. Group theory(XI).

Today, we are going to talk about the Lie groups $U(n)$ and $SU(n)$, and their respective Lie algebras, generally denoted by $u(n)$ and $su(n)$ by the physics community. In addition to this, we will see some properties of the orthogonal groups in euclidena “signature” and general quadratic metrics due to their importance in special relativity or its higher dimensional analogues.

Let us remember what kind of groups are $U(n)$ and $U(n)$:

1) The unitary group is defined by:

$U(n)\equiv\left\{ U\in M_{n\times n}(\mathbb{C})/UU^+=U^+U=I\right\}$

2) The special unitary group is defined by:

$SU(n)\equiv\left\{ U\in M_{n\times n}(\mathbb{C})/UU^+=U^+U=I,\det (U)=1\right\}$

The group operation is the usual matrix multiplication. The respective algebras are denoted as we said above by $u(n)$ and $su(n)$. Moreover, if you pick an element $U\in U(n)$, there exists an hermitian (antihermitian if you use the mathematician “approach” to Lie algebras/groups instead the convention used mostly by physicists) $n\times n$ matrix $H$ such that:

$U=\exp (iH)$

Some general properties of unitary and special unitary groups are:

1) $U(n)$ and $SU(n)$ are compact Lie groups. As a consequence, they have unitary, finite dimensional and irreducible representations. $U(n)$ and $SU(n)$ are subgroups of $U(m)$ if $m\geq n$.

2) Generators or parameters of unitary and special unitary groups. As we have already seen, the unitary group has $n^2$ parameters (its “dimension”) and it has rank $n-1$ (its number of Casimir operators). The special unitary group has $n^2-1$ free parameters (its dimension) and it has rank $n-1$ (its number of Casimir operators).

3) Lie algebra generators. The unitary group has a Lie algebra generated by the space of $n^2$ dimensional complex matrices. The special unitary group has a Lie algebra generated by the $n^2-1$ dimensional space of hermitian $n\times n$ traceless matrices.

4) Lie algebra structures. Given a basis of generators $L_i$ for the Lie algebra, we define the constants $C_{ijk}$, $f_{ijk}$, $d_{ijk}$ by the following equations:

$\left[L_i,L_m\right]=C_{ijk}L_k=if_{ijk}L_k$

$L_iL_j+L_jL_i=\dfrac{1}{3}\delta_{ij}I+d_{ijk}L_k$

These structure constants $f_{ijk}$ are totally antisymmetric under the exchange of any two indices while the coefficients $d_{ijk}$ are symmetric under those changes. Moreover, we also have:

$d_{ijk}=2\mbox{Tr}(\left\{L_i,L_j\right\}L_k)$

$f_{ijk}=-2i\mbox{Tr}(\left[L_i,L_j\right]L_k)$

Remark(I):   From $U=e^{iH}$, we get $\det U=e^{i\mbox{Tr} (H)}$, and from here we can prove the statement 3) above.

Remark(II): An arbitrary element of $U(n)$ can be expressed as a product of an element of $U(1)$ and an element of $SU(n)$. That is, we can write $U(n)\cong U(1)\cup SU(n)$, where the symbol $\cong$ means “group isomorphism”.

Example 1. The SU(2) group.

In particular, for $n=2$, we get

$SU(2)=\left\{U\in M_{2\times 2})(\mathbb{C})/UU^+=U^+U=I_{2\times 2},\det U=1\right\}$

This is an important group in physics! It appears in many contexts: angular momentum (both classical and quantum), the rotation group, spinors, quantum information theory, spin networks and black holes, the Standard Model, and many other places. So it is important to know it at depth. The number of parameters of SU(2) is equal to 3 and its rank is equal to one (1). As generators of the Lie algebra associated to this Lie group, called su(2), we can choose for free 3 any independent traceless (trace equal to zero) matrices. As a convenient choice, it is usual to select the so called Pauli matrices $\sigma_i$:

$\sigma_1=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}$

$\sigma_2=\begin{pmatrix}0 & -i\\ i & 0\end{pmatrix}$

$\sigma_3=\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}$

In general, these matrices satisfy an important number of mathematical relations. The most important are:

$\left\{\sigma_i,\sigma_j\right\}=2\sigma_i\sigma_j+2i\varepsilon_{ijk}\sigma_k$

and

$\sigma_i\sigma_j=i\varepsilon_{ijk}\sigma_k$

The commutators of Pauli matrices are given by:

$\left[\sigma_i,\sigma_j\right]=2if_{ijk}\sigma_k$

$f_{ijk}=\dfrac{1}{2}\varepsilon_{ijk}$ $d_{ijk}=0$

The Casimir operator/matrix related to the Pauli basis is:

$C(\sigma_i)=\sigma_i^2=\sigma_1^2+\sigma_2^2+\sigma_3^2$

This matrix, by Schur’s lemma, has to be a multiple of the identity matrix (it commutes with each one of the 3 generators of the Pauli algebra, as it can be easily proved). Please, note that using the previous Pauli representation of the Pauli algebra we get:

$\displaystyle{C=\sum_i\sigma_i^2=3I}$

Q.E.D.

A similar relation, with different overall prefactor, must be true for ANY other representation of the Lie group algebra su(2). In fact, it can be proved in Quantum Mechanics that this number is “four times” the $j(j+1)$ quantum number associated to the angular momentum and it characterizes completely the representation. The general theory of the representation of the Lie group SU(2) and its algebra su(2) is known in physics as the general theory of the angular momentum!

Example 2. The SU(3) group.

If n=3, the theory of $SU(3)$ is important for Quantum Chromodynamics (QCD) and the quark theory. It is also useful in Grand Unified Theories (GUTs) and flavor physics.

$SU(3)=\left\{U\in M_{3\times 3})(\mathbb{C})/UU^+=U^+U=I_{3\times 3},\det U=1\right\}$

The number of parameters of SU(3) is 8 (recall that there are 8 different massless gluons in QCD) and the rank of the Lie algebra is equal to two, so there are two Casimir operators.

The analogue generators of SU(3), compared with the Pauli matrices, are the so-called Gell-Mann matrices. They are 8 independent traceless matrices. There are some “different” elections in literature, but a standard choice are the following matrices:

$\lambda_1=\begin{pmatrix}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 &0\end{pmatrix}$

$\lambda_2=\begin{pmatrix}0 & -i & 0\\ i & 0 & 0\\ 0 & 0 &0\end{pmatrix}$

$\lambda_3=\begin{pmatrix}1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 &0\end{pmatrix}$

$\lambda_4=\begin{pmatrix}0 & 0 & 1\\ 0 & 0 & 0\\ 1 & 0 &0\end{pmatrix}$

$\lambda_5=\begin{pmatrix}0 & 0 & -i\\ 0 & 0 & 0\\ i & 0 &0\end{pmatrix}$

$\lambda_6=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 &0\end{pmatrix}$

$\lambda_7=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & -i\\ 0 & i &0\end{pmatrix}$

$\lambda_8=\dfrac{1}{\sqrt{3}}\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 &-2\end{pmatrix}$

Gell-Mann matrices above satisfy a normalization condition:

$\mbox{Tr}(\lambda_i\lambda_j)=2\delta_{ij}$

where $\delta_{ij}$ is the Kronecker delta in two indices.

The two Casimir operators for Gell-Mann matrices are:

1) $\displaystyle{C_1(\lambda_i)=\sum_{i=1}^8\lambda_i^2}$

This operator is the natural generalization of the previously seen SU(2) Casimir operator.

2) $\displaystyle{C_2(\lambda_i)=\sum_{ijk}d_{ijk}\lambda_i\lambda_j\lambda_k}$

Here, the values of the structure constans $f_{ijk}$ and $d_{ijk}$ for the su(3) Lie algebra can be tabulated in rows as follows:

1) For $ijk=123,147,156,246,257,345,367,458,678$ we have $f_{ijk}=1,\dfrac{1}{2},-\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},-\dfrac{1}{2},\dfrac{\sqrt{3}}{2},\dfrac{\sqrt{3}}{2}$.

2) If

$ijk=118,146,157,228,247,256,338,344,355,366,377,448,558,668,778,888$

then have

$d_{ijk}=\dfrac{1}{\sqrt{3}},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{\sqrt{3}},-\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{\sqrt{3}},\dfrac{1}{2},\dfrac{1}{2},-\dfrac{1}{2},-\dfrac{1}{2},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{\sqrt{3}}$

Example 3. Euclidean groups, orthogonal groups and the Lorentz group in 4D and general $D=s+t$ dimensional analogues.

In our third example, let us remind usual galilean relativity. In a 3D world, physics is the same for every inertial observer (observers moving with constant speed). Moreover, the fundamental invariant of “motion” in 3D space is given by the length:

$L^2=x^2+y^2+z^2=\delta_{ij}x^ix^j$ $\forall i,j=1,2,3$

In fact, with tensor notation, the above “euclidean” space can be generalized to any space dimension. For a ND space, the fundamental invariant reads:

$\displaystyle{L_N^2=\sum_{i=1}^NX_i^2=x_1^2+x_2^2+\cdots+x_N^2}$

Mathematically speaking, the group leaving the above metrics invariant are, respectively, SO(3) and SO(N). They are Lie groups with dimensions $3$ and $N(N-1)/2$, respectively and their Lie algebra generators are antisymmetric traceless $3\times 3$ and $N\times N$ matrices. Those metrics are special cases of quadratic forms and it can easily proved that orthogonal transformations with metric $\delta_{ij}$ (the euclidean metric given by a Kronecker delta tensor) are invariant in the following sense:

$A^\mu_{\;\;\; i}\delta_{\mu\nu}A^\nu_{\;\;\; j}=\delta_{ij}$

or equivalently

$A\delta A^T=\delta$

using matric notation. In special relativity, the (proper) Lorentz group $L$ is composed by every real $4\times 4$ matrix $\Lambda^\mu_{\;\;\;\nu}$ connected to the identity through infinitesimal transformations, and the Lorentz group leaves invariant the Minkovski metric(we use natural units with $c=1$):

$s^2=X^2+Y^2+Z^2-T^2$ if you use the “mostly plus” 3+1 metric ($\eta=\mbox{diag}(1,1,1,-1)$) or, equivalentaly

$s^2=T^2-X^2-Y^2-Z^2$ if with a “mostly minus” 1+3 metric ($\eta=\mbox{diag}(1,-1,-1,-1)$).

These equations can be also genearlized to arbitrary signature. Suppose there are s-spacelike dimensions and t-time-like dimensions ($D=s+t$). The associated non-degenerated quadratic form is:

$\displaystyle{s^2_D=\sum_{i=1}^sX_i^2-\sum_{j=1}^tX_j^2}$

In matrix notation, the orthogonal rotations leaving the above quadratic metrics are said to belong to the group $SO(3,1)$ (or $SO(1,3)$ is you use the mostly minus convention) real orthogonal group over the corresponding quadratic form. The signature of the quadratic form is said to be $S=2$ or $(3,1)$ (equivalently $\Sigma=3-1=2$ and $(1,3)$ with the alternative convention). We are not considering “degenerated” quadratic forms for simplicity of this example. The Lorentzian or Minkovskian metric are invariant in the same sense that the euclidean example before:

$L^\mu_{\;\;\;\alpha}\eta_{\alpha\beta}L^\mu_{\;\;\;\beta}=\eta_{\alpha\beta}$

$LGL^T=G$

The group $SO(s,t)$ has signature $(s,t)$ or $s-t$ or $s+t$ in non-degenerated quadratic spaces. Obviously, the rotation group $SO(3)$ is a subgroup of $SO(3,1)$ and more generally $SO(s)$ is a subgroup of $SO(s,t)$. We are going to focus now a little bit on the common special relativity group $SO(3,1)$. This group have 6 parameters or equivalently its group dimension is 6. The rank of this special relativity group is equal to 1. We can choose as parameters for the $SO(3,1)$ group 3 spatial rotation angles $\omega_i$ and three additional parameters, that we do know as rapidities $\xi_i$. These group generators have Lie algebra generators $S_i$ and $K_i$ or more precisely, if we define the Lorentz boosts as

$\xi=\dfrac{\beta}{\parallel\beta\parallel}\tanh^{-1}\parallel \beta\parallel$

In the case of $SO(3,1)$, a possible basis for the Lie algebra generators are the next set of matrices:

$iS_1=\begin{pmatrix}0 & 0 & 0& 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & -1 & 0\end{pmatrix}$

$iS_2=\begin{pmatrix}0 & 0 & 0& 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\end{pmatrix}$

$iS_3=\begin{pmatrix}0 & 0 & 0& 0\\ 0 & 0 & 1 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}$

$iK_1=\begin{pmatrix}0 & 1 & 0& 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}$

$iK_2=\begin{pmatrix}0 & 0 & 1& 0\\ 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}$

$iK_3=\begin{pmatrix}0 & 0 & 0& 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0\end{pmatrix}$

And the commutation rules for these matrices are given by the basic relations:

$\left[S_a,S_b\right]=i\varepsilon_{abc}S_c$

$\left[K_a,K_b\right]=-i\varepsilon_{abc}S_c$

$\left[S_a,K_b\right]=i\varepsilon_{abc}K_c$

Final remark: $SO(s,t)$ are sometimes called isometry groups since they define isometries over quadratic forms, that is, they define transformations leaving invariant the “spacetime length”.

# LOG#090. Group theory(X).

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!

# LOG#089. Group theory(IX).

Definition (36). An infinite group $(G,\circ)$ is a group where the order/number of elements $\vert G\vert$ is not finite. We distinguish two main types of groups (but there are more classes out there…):

1) Discrete groups: their elements are a numerable set. Invariance under a discrete group provides multiplicative conservation laws. Elements are symbolized as $g_i$ $\forall i=1,\ldots,\infty$ for a discrete group.

2) Continuous groups: their elements are not numerable, since they depend “continuously” on a finite number of parameters (real, complex,…):

$g=g(\alpha_1,\alpha_2,\ldots)$

Note that the number or paraters can be either finite or infinite in some cases. The number of parameters define the so-called “dimension” of the group. Please, don’t confuse group order with its dimension. Group order is the number of elements, group dimension is the number of parameters we do need to characterize/generate the group! Invariance under a continuous group has some consequences (due to the Noether’s theorems):

1) Invariance under a finite dimensional r-parametric continuous group provides conservation laws.

2) Invariance under an infinite dimensional continuous group (parametrized by some set of “functions”) provides some relationships between field equations called “dependencies” or “noether identities” in modern language.

Definition (37). Composition rule/law for a group. Let $G$ be a continuous group and two elements $g(\alpha_1),g(\alpha_2)\in G$, then

$g(\alpha_1)\circ g(\alpha_2)=g(\alpha_3)$

and we define the composition law of a continuous  group as the function that gives $\alpha_3=f(\alpha_1,\alpha_2)$ and similarly

$g(\alpha_2)=g^{-1}(\alpha_1)$

so

$\alpha_1=f^{-1}(\alpha_2)$

Theorem (Lie). Every continuous group is a Lie Group. It means that whenever you have a group where the composition rule is given, as the inverse element, then the group elements are differentiable functions (analytic in the complex case) on its arguments.

Some examples of Lie groups (some of them we have already quoted in this thread):

1) The euclidean real space $\mathbb{R}^n$ or the hermitian complex space $\mathbb{C}$ with ordinary vector addition form (in any of that two cases) a n-dimensional noncompact abelian Lie group.

2) The general linear (Lie) group of non-singular matrices over the real number or the complex numbers is a Lie group $GL_n(\mathbb{R})$ or $GL_n(\mathbb{C})$.

3) The special linear group $SL(n,\mathbb{R})$ or the complex analogue $SL(n,\mathbb{C})$ of square matrices with determinant equal to one.

4) The orthogonal group $O(n)$ over the real numbers, $n\times n$ matrices with real entries is a $n(n-1)/2$ dimensional Lie group.

5) The special orthogonal group $SO(n,\mathbb{R})$ is the subgroup of the orthogonal group whose matrices have determinant equal to one.

6) The unitary group $U(n,\mathbb{C})$ of complex $n\times n$ unitary matrices, $UU^+=U^+U=\mathbb{I}_n$. Its dimension is equal to $n^2$ over the complex numbers. SU(n) is the $n^2-1$ dimensional subgroup formed by unitary matrices with determinant equal to one.

7) The symplectic group $\mbox{Sp}(2n,\mathbb{R})$.

8) The group of upper triangular matrices $n\times n$ is a group with dimension $n(n+1)/2$.

9) The Lorentz group and the Poincaré group. The are non-compact Lie groups (Poincaré is non-compact due to the fact that the Lorentz subgroup is non-compact). Their dimensions in 4D spacetime are 6 and 10 dimensions respectively.

10) The Standard Model “gauge” (Lie) group $U(1)\times SU(2)\times SU(3)$ is a group formed with direct group (in the group sense) of three groups and it has dimension $1+3+8=12$. The dimensions of the gauge groups in the Standard Model is in direct correspondence with the numbers of gauge bosons: 1 massless photon, 3 vector bosons for the electroweak interactions, and 8 gluons for the quantum chromodynamics (QCD).

11) The exceptional Lie groups $\mathcal{G}_2,\mathcal{F}_4, E_6, E_7, E_8$, the so called Cartan exceptional groups. Their dimensions are respectively 14, 52, 78, 133 and 248.

The continuous group made of matrices (finite and infinite matrices/operators) play an important role in Physics. Moreover, as Lie groups depend continuously on their arguments AND their dependence is generally differentiable, it makes sense to take derivatives in the group elements. In fact, this fact allow us to define the idea of group generator.

Definition(38).  Group generator. If $g=U(\alpha)$ is a continuous (therefore differentiable; remember that continuity implies differentiability but the converse is not necessarily true), then we define the generators of the group $L_i$ in the following (hermitian) way:

$-iL_j=\dfrac{\partial U(\alpha)}{\partial \alpha_j}\bigg|_{\alpha=0}$

Theorem (Lie). Let us choose some $G=U(\alpha)$ one-parameter continuous group and K its generator. Then, the following facts hold:

i) K fully determines the group $U(\alpha)$.

ii) Group elements are obtained using “exponentiation” of generators. That is,

$U(\alpha)=\exp\left(-iK\alpha\right)$

The “proof” involves a group parametrization and an expansion as a series. We have $U(0)=1$ and $U(x+y)=U(x)U(y)$. Therefore,

$\dfrac{dU(x)}{dx}=\dfrac{d}{dy}\left(U(x+y)\right)\vert_{y=0}=\dfrac{d}{dy}(U(x)U(y))\vert_{y=0}$

$\dfrac{dU(x)}{dx}=U(x)\dfrac{dU(y)}{dy}\bigg|_{y=0}=\dfrac{dU(y)}{dy}\bigg|_{y=0}U(x)=-iKU(x)$

so

$-iKU(x)=\dfrac{dU(x)}{dx}$ and $U(0)=\mathbb{I}$

This differential equation has one and only one solution for every K-value$\forall x$. The general solution of this equation is the exponential:

$U(x)=U(0)\exp\left(-iKx\right)$

Taking into account the initial conditions $U(0)=\mathbb{I}$ (elements nearby of any group element are the identity element) we have the desired result for every $x=\alpha$. Q.E.D.

Theorem (Lie). A multiparametric Lie group (N-dimensional) is a Lie group G with functions $g=U(\alpha_j)$,$\forall j=1,2,\ldots, N$ and generators $L_j$ obtained by exponentiation. That is:

$\boxed{\displaystyle{U(\alpha_j)=\prod_{j=1}^N\exp \left(-iL_j\alpha_j\right)}}$

Check (Easy simplified proof): Using the previous result, we have to fix only all the parameters $\alpha_j\forall j=1,\ldots,N$. Then, a simple “empatic mimicry” of the previous one dimensional provides:

$U(\alpha_N)=U(0,\alpha_2,\alpha_3,\ldots,\alpha_N)\exp\left(-iL_1\alpha_1\right)$

and then

$U(\alpha_N)=U(0,0,\alpha_3,\ldots,\alpha_N)\exp\left(-iL_1\alpha_1\right)\exp\left(-iL_2\alpha_2\right)$

and finally, iterating the process N-times we get

$\displaystyle{U(\alpha_N)=U(0,0,0,\ldots,0)\prod_j^N\exp\left(-iL_j\alpha_j\right)}$

The generators of any Lie group satisfy some algebraic and important relations. In the case of dealing with matrix or operator groups, the generators are matrices or operator theirselves. These mathematical relations can be written in terms of (ordinary) algebraic commutators. There is a very important theorem about this fact:

Theorem. First Lie theorem. Lie group generators form a closed commutator algebra under “matrix/operator” products. That is:

$\boxed{\left[L_i,L_j\right]=C_{ij}^{k}L_k}$ or $\boxed{\left[L_i,L_j\right]=C^{ijk}L_k}$

without distinction of lower and upper “labels”.

There the commutator of two matrices/operators is defined to be $\left[A,B\right]=AB-BA$ and the contants $C_{ijk}$ or $C^k_{ij}$ are the so-called structure constants of the Lie group. The structure constants of a Lie group are:

1) Antisymmetric with respect to the first two indices (or the paired ones, $ij$, with our notation).

2) Characteristic of the group but they do change, in a particular way, if we form linear combinations of the Lie group generators.

There is a nice formula called Baker-Campbell-Hausdorff identity that relates group exponentials and group commutators. It is specially important in the theory of Lie groups and Lie algebras:

The Baker-Campbell-Hausdorff (BCH) formula. For any matrix/operator A,B, under certain very general conditions, we have:

$\exp(A)\exp(B)=\exp\left(A+B+\dfrac{1}{2}\left[A,B\right]+\dfrac{1}{12}\left[\left[A,B\right],B\right]-\dfrac{1}{12}\left[\left[A,B\right],A\right]-\ldots\right)$

In the case that the matrices A and B do commute, then we recover the usual ordinary exponentiation of “elements”:

$\exp(A)\exp(B)=\exp(A+B)$

A beautiful and simple application of the BCH formula is the next feature which allows us to write ANY member of a Lie group as the exponential of a sum of the Lie group generators. Let us write the group elements, firstly, as

$g=U(\alpha_j)\forall j=1,2,\ldots,N$

and let us write the group generators as $L_j$. Then, we have

$\displaystyle{U(\alpha_j)=\prod_{j=1}^N\exp\left(-i\alpha_jL_j\right)=\exp\left(-i\sum_ {j=1}^N\omega_jL_j\right)}$

where the parameters $\omega_j$ are related to the $\alpha_j$ parameters in a simple continuous way

$\omega_j=\omega_j(\alpha_k)$

The specific form of this realtion can be expanded and computed/calculated term by term using the BCH formula, as given before.

See you in the next blog post of this group theory thread!

# LOG#088. Group theory(VIII).

Schur’s lemmas are some elementary but very useful results in group theory/representation theory. They can be also used in the theory of Lie algebras so we are going to review these results in this post (for completion).

FIRST SCHUR’S LEMMA. If $D_1$ and $D_2$ are 2 finite-dimensional irreducible representations of G, and if A is certain linear map (generally a matrix) from $D_1$ to $D_2$ such as it commutes with the action of the group (for any element $g\in G$), i.e.,

$AD_1(g)=D_2(g)A$

then at least one of the following consequences holds:

1) A is invertible, and then the representation are necessarily equivalent.

2) A=0.

SECOND SCHUR’S LEMMA. If $A$ is a complex matrix of order $n$ that commutes with every matrix from an irreducible representation $D(G)$

$AD(g)=D(g)A\forall g\in G$

Then, A must be a scalar matrix multiple of the identity matrix, i.e., $A=\lambda I$.

Schur’s lemmas and their corollaries are used to prove the so-called (Schur’s) orthogonality relations and to develop the basics of the representation theory of finite groups. Schur’s lemmas admits some generalizations to Lie groups and some other kind of structures, such as Lie algebras and other sets of operators or matrices. Therefore, it is important to understand Schur’s lemmas!

Consequences of these lemmas:

1) Irreducible representations of an abelian finite group are one dimensional.

2) Orthonormality of group representations. Let $G$ be a finite group, and $D_\mu, D_\nu$ two irreducible representations of G. Then,

$\displaystyle{\dfrac{N_{(\mu)}}{N_G}\sum_{g\in G}\left[D^{-1}_\mu (g')\right]_{ki}\left[D_\nu (g)\right]_{jl}=\delta_{\nu\mu}\delta_{jl}\delta_{kl}}$

3) Completeness of irreducible representations. Let $G$ be a finite group and $(D_\mu)_{\mu\in A}$ the set of every irreducible representation of G. Then,

$\displaystyle{\sum_{\mu \in A}\dfrac{N_{(\mu)}}{N_G}\mbox{Tr}\left[D_\mu (g)D^{-1}_\mu (g')\right]=\delta_{gg'}}$

4) The number of irreducible representations of a finite group is finite and they can be computed with the aid of the following formula:

$\displaystyle{\sum_{\mu\in A}(\mbox{dim}D_\mu)^2=\vert G\vert}$

We can give some elementary examples of irreducible representations of abelian (finite) groups.

1) For $G_2=\left\{e,a\right\}$, then $2=1^2+1^2$ and so there are two and only two irreducible representations, both one dimensional:

$D_1(e)=1$ $D_1(a)=1$

$D_2(e)=1$ $D_2(a)=-1$

2) For $G_3=\left\{e,a,a^2\right\}$, then $3=1^2+1^2+1^2$ and there are 3 irreducible representations:

$D_1(e)=1$ $D_1(a)=1$ $D_1(a^2)=1$

$D_2(e)=1$ $D_2(a)=\exp\left(\dfrac{2\pi i}{3}\right)$ $D_2(a^2)=\exp\left(\dfrac{4\pi i}{3}\right)$

$D_3(e)=1$ $D_3(a)=\exp\left(-\dfrac{2\pi i}{3}\right)$ $D_3(a^2)=\exp \left(-\dfrac{4\pi i}{3}\right)$

3) For $G_4$ we have two cases: $4=1^2+1^2+1^2+1^2$ and $2^2$. And then we have:

3.1) The case of the cyclic group $C_4=\left\{e,a,a^2,a^3\right\}$ with 4 unidimensional representations

$D_1(e)=1$ $D_1(a)=1$ $D_1(a^2)=1$ $D_1(a^3)=1$

$D_2(e)=1$ $D_2(a)=i$ $D_2(a^2)=-1$ $D_2(a^3)=-i$

$D_3(e)=1$ $D_3(a)=-1$ $D_3(a^2)=1$ $D_3(a^3)=-1$

$D_4(e)=1$ $D_4(a)=-i$ $D_4(a^2)=-1$ $D_4(a^3)=i$

3.2) The case of the Klein group $K_4=\left\{e,a,b,ab\right\}$ where $a^{-1}=a$,$b^{-1}=b$ and $(ab)^{-1}=ab$ with $ab=ba$. The representations of this group, in the unidimensional case, are given by:

$D_1(e)=1$ $D_1(a)=1$ $D_1(a^2)=1$ $D_1(a^3)=1$

$D_2(e)=1$ $D_2(a)=-1$ $D_2(a^2)=1$ $D_2(a^3)=-1$

$D_3(e)=1$ $D_3(a)=-1$ $D_3(a^2)=-1$ $D_3(a^3)=1$

$D_4(e)=1$ $D_4(a)=1$ $D_4(a^2)=-1$ $D_4(a^3)=-1$

Of course, you can build the matrix representation of the above group representations as a nice homework. :).

See you in another blog post!

# LOG#087. Group theory(VII).

Representation theory is the part of Group Theory which is used in the main applications. Matrices acting on the members of a vector space are assigned to every element of a group. The connections between particle physics and representation theory is “natural”. It was noted by Eugene Wigner firstly, and  the properties of elementary particles and the representation theory are due to some special groups,  the so-called Lie groups and Lie algebras. The deep connection between Lie groups and Lie algebras with particle physics is observed in the different quantum states of an elementary particle, as a consequence of the irreducible representations of the Poincaré group. We are going to study this a little bit.

Definition (29). Representation. Group representation is every linear map D in G, $D: G\longrightarrow GL(V)$. That is, a linear representation of a group is a group homomorphism onto the general linear group due to the elements of the group. Informally speaking, it is a rule that assings to every group element a square matrix.

Here V is certain vector space, sometimes called “the space of the representation”, and $GL(V)$ is the “general linear group” of non-singular matrices defined over V.

Definition (30). Equivalent representations. Two representations $D^\mu$ and $D^\nu$ are said to be equivalent representations of G, if there exists some matrix $A\in GL(V)$, an isomorphism, such as

$D^\mu(g)=AD^\nu (g)A^{-1}$ $\forall g\in G$

Definition (31). Character. The character of an element $g\in G$, given the representation

$D:G\longrightarrow G''$

is the number

$\chi^{G'}(g)=\mbox{Tr} D(g)$

Of course, the character is related to the trace of the matrix, so in the case of infinite groups we have to be more careful and precise with the definition of “trace”. But this point is not relevant or important in the present discussion. Clearly, the character could be interpreted as an homomorphism between the group and the group of real (complex) numbers.

Definition (32). Character of a representation. The character of a representation $G$ is the $n-plet$ of numbers

$\chi^{G}=\left(\chi^G(g_i)\right)_{i=1,2,\ldots,\mbox{ord}(G)}$

Property: if $G^\mu$ and $G^\nu$ are two equivalent representations of G, then their characters are the same

$\chi^{G^\mu}=\chi^{G^\nu}$

Remark (I): The space a representation IS the vectorial space V where the matrices (belonging to the general linear group) act.

Remark(II): The dimension of a representation IS the dimension of the vectorial space V.

Remark (III): Do you know some examples of group representations we have seen in fact already? Remember that the trace of a matrix verifies that $\mbox{Tr}(AB)=\mbox{Tr}(BA)$ and $\mbox{Tr}(ABC)=\mbox{Tr}(BCA)=\mbox{Tr}(CAB)$ and so on. This is the cyclic property of the trace.

Given two group representations, $D_1, D_2$ of $G$, we can build other representations from them in at least two simple ways:

Definition (33) . Direct sum representation. The direct sum is defined over the space $V_1\oplus V_2$. The dimension of the direct sum:

$\mbox{dim}(D)=\mbox{dim}(D_1)+\mbox{dim}(D_2)$

so

$D_1\oplus D_2\longrightarrow GL(V_1)\oplus GL(V_2)$

where in block for we have

$(D_1\oplus D_2)(g)=\begin{pmatrix} D_1(g) & 0\\ 0 & D_2(g)\end{pmatrix}$

Definition (34). Tensor product representation. Given two representations $D_1, D_2$ of G, we can build the tensorial product representation

$D_1\otimes D_2$ $\longrightarrow GL(V_1)\otimes GL(V_2)$

so

$(D_1\otimes D_2)(g)=D_1(g)\otimes D_2(g)$

as a tensor product of matrices.

The dimension of the tensor product representation is equal to:

$\mbox{dim}(D)=\mbox{dim}(D_1)\times \mbox{dim}D_2$

Question: Do you know the basic properties of tensorial products in vector spaces? Review their properties and work out some examples yourself.

Definition (35). Invariant subspace of a representation $D$ is a susbspace (or even the whole space itself) such as

$D(g)W\subseteq W$ $\forall g\in G$

We distinguish two types of representations according to the existence or not of non-trivial invariant subspaces with respect to them:

1st. Irreductible representations. These class of representations are those such as V does not contain any invariant subspace, except V itself. That is, irreducible representations are “single” pieces/boxes from which group representations are made of. They are the “atoms” of group representations.

2nd. Reducible representations. In this case, V contains some invariant subspaces different to itself.

3rd. Fully reducible representation is any representation, if exists, such as there is some invariant subspace $W$, such that there exists another $W^\perp$ invariant subspace that satisfies

$V=W\otimes W^\perp$

The study of the group representations is based on the study of the irreducible representations (irreps.), since reducible representations can be contructed from irreducible representations (reducible representations would be “molecules” made of irreducible representations).

Some properties of group representations are interesting:

1) Let G be a finite group. Then, all representation on an inner product space are equivalent to an  unitary representation.

2) Let G be a group and let D be an unitary representation. Then, if the representation is reducible, then it is fully reducible.

3) Every reducible representation of a finite group is fully reducible.

In summary: a finite group has always an unitary representation and if if is reducible, it is fully reducible.

See you in other group theory blog post!

# LOG#085. Group theory(V).

Other important concepts and definitions in group theory!

Definition (22). Normal or invariant group. Let $H$ be a subgroup of other group G. We say that $H$ is a normal or invariant subgroup of G if the following condition holds:

$H=xHx^{-1}$ $\forall x\in G$

Proposition. Let $H$ be a subgroup of G. $H$ is invariant if and only if (iff) $H$ is a union of conjugacy classes of $G$.

Check: H is normal iff $a\in H\Rightarrow xax^{-1}\in H$ $\forall x\in G$. Thus, H is normal iff whenever it contains an element $a$, then it also contains the conjugacy class of $a$, so another way to say this is that H is a union of conjugacy classes.

Definition (23). Simple group. We say that G is a simple group if there is no invariant subgroups except the trivial, the neutral/identity element $\left\{ e\right\}$.

Definition (24). Semisimple group. We say that G is a semisimple group if there is no any “abelian invariant subgroup” except the trivial element $\left\{ e\right\}$.

Simple groups are always semisimple, but the converse is not true. Simple and semisimple finite groups have been completely classified by mathematicians. We will talk about this later.

Definition (25). Coset. Let $H\subseteq G$ be an invariant subgroup. Then we say that

i) The set $xH=\left\{xh/h\in G\right\}$ with $x\in G$ is a coset by the left, and

ii) The set $Hx=\left\{hx/h\in G\right\}$ sith $x \in H$ is a coset by the right.

Theorem. The set $A=\left\{ g_iH/g_i\in G\right\}$

with composition law $g_iH\circ g_jH\equiv (g_i\circ g_j)H$ is a group.

Definition (25). Quotient group. The group defined in the previous theorem is called quotient group and it is generally denoted by $G/H\equiv (A,\circ)$.

Remark: Informally speaking, the elements of the quotient group are “the difference” between the elements of G and those in H.

Now, some additional definitions about morphisms, homomorphisms and isomorphisms in group theory.

Definition (26). (Group) Homomorphism. Let $(G,\circ)$ and $(G',\circ ')$ be two different groups. Any map/application/function/functor

$f: G\longrightarrow G'$

is called an homomorphism if it preserve the operations of the respective groups (their “products” or “multiplications”) in the following sense

$f(x\circ y)=f(x)\circ ' f(y)$ $\forall x,y\in G$

Definition (27). (Group) Isomorphism. Let $f$ be an homomorphism, then $f$ is an isomorphism if $f$ is “bijective”, or one-to-one correspondence between the elements of $G$ and $G'$.

Definition (28). Kernel. Let $(G,\circ)$ and $(G',\circ ')$ be two different groups. and $f:G\longrightarrow G'$ a function between them. Then, we define the kernel of f as the following set

$\mbox{Ker}(f)=\left\{g \in G/f(g)=e'\right\}$

and where $e'$ is the neutral element in the group $G'$.

There are two important theorems for group homomorphisms:

Theorem. First theorem of group homomorphisms. Let $(G,\circ)$ and $(G',\circ ')$ be two different groups. If we hav a function $f:G\longrightarrow G'$ being a group homomorphism, then its kernel is an invariant subgroup of $(G,\circ)$.

Property. It is clear from the above theorem that, if f is bijective, then we have the special case in which $\mbox{Ker}(f)=e$. Then, the theorem is trivially satisfied!

Theorem. Second theorem of group homomorphisms. Let $(G,\circ)$ and $(G',\circ ')$ be two different groups. If $f:G\longrightarrow G'$ is a group homomorphism, then an aplication $\iota$ exists such as

$\iota= G/\mbox{Ker}(f)\longrightarrow \mbox{Im}(f)$

such as $\iota$ is a group isomorphism.

May the group theory be with you until the next blog post!