Group generators

Not to be confused with a group generator .

A group generator ( infinitesimal operator ) is a concept used in the theory of Lie groups . Generators of the group G are the elements that form the basis of its Lie algebra , or, in the general case, the basis of the Lie algebra of the image of G.

This is an operator that is a derivative of the operator (or matrix) representation of an element of a group with respect to a certain presentation parameter for all parameters at zero (it is assumed, without loss of generality, that at zero values of the parameters, the operator representing this element is equal to one and corresponds to a single element of the group). The representation of an arbitrary element of a group that is close enough to a single element is expressed linearly through the generators of the group (generators are first-order terms in the expansion of the representation operator in a power series in parameters). Moreover, under certain weak assumptions, any element of the group (its representation) can be expressed through generators, since members of the second and higher order are expressed through generators. For a certain class of connected Lie groups with pairwise commuting generators $A_{1},A_{2},\dots A_{n}$ ${\ displaystyle A_ {1}, A_ {2}, \ dots A_ {n}}$ $A_ {1}, A_ {2}, \ dots A_ {n}$ group elements can be represented using exponential mapping in the form $\exp(A_{1}\alpha _{1}+\cdots +A_{n}\alpha _{n})$ ${\ displaystyle \ exp (A_ {1} \ alpha _ {1} + \ cdots + A_ {n} \ alpha _ {n})}$ $\ exp (A_ {1} \ alpha _ {1} + \ cdots + A_ {n} \ alpha _ {n})$ . The properties of the group obviously follow from the identity $\exp(A+B)=\exp(A)\exp(B)$ ${\ displaystyle \ exp (A + B) = \ exp (A) \ exp (B)}$ $\ exp (A + B) = \ exp (A) \ exp (B)$ for commuting matrices. If the generators do not commute, then not all elements of the group can be represented in this form, even if the group is connected.

Content

Definition

Let an arbitrary element of the group $G$ ${\ displaystyle G}$ It has $k$ ${\ displaystyle k}$ -parametric representation (operator function $k$ ${\ displaystyle k}$ parameters, the operators act on a certain vector space) $g(\alpha _{1},\dots ,\alpha _{k})$ ${\ displaystyle g (\ alpha _ {1}, \ dots, \ alpha _ {k})}$ , and the unit element corresponds to the value of the operator function at zero values of the parameters. Then the group generators are the quantities:

A_{k}=\left.{\frac {\partial g(\alpha _{1},\dots ,\alpha _{n})}{\partial \alpha _{k}}}\right\vert _{\alpha =\,0}

{\ displaystyle A_ {k} = \ left. {\ frac {\ partial g (\ alpha _ {1}, \ dots, \ alpha _ {n})} {\ partial \ alpha _ {k}}} \ right \ vert _ {\ alpha = \, 0}}

Then an arbitrary element $g(\alpha _{1},\dots ,\alpha _{n})$ ${\ displaystyle g (\ alpha _ {1}, \ dots, \ alpha _ {n})}$ from the considered neighborhood (where the parameters $\alpha _{k}$ ${\ displaystyle \ alpha _ {k}}$ naturally small) can be decomposed near the unit transformation up to terms of the second order of smallness:

g(\alpha _{1},\dots ,\alpha _{n})=1+\sum _{k=1}^{n}A_{k}\alpha _{k}+O(\sum _{k,\,j=1}^{n}\alpha _{k}\alpha _{j}),

{\ displaystyle g (\ alpha _ {1}, \ dots, \ alpha _ {n}) = 1+ \ sum _ {k = 1} ^ {n} A_ {k} \ alpha _ {k} + O ( \ sum _ {k, \, j = 1} ^ {n} \ alpha _ {k} \ alpha _ {j}),}

Algebra Lee. Exponential Mapping

Let a group be a connected Lie group - a group of transformations $T(\alpha )$ ${\ displaystyle T (\ alpha)}$ depending on a finite set of parameters so that any element of the group can be connected to a single element in a way that lies entirely within this group. We denote $t_{a}$ ${\ displaystyle t_ {a}}$ - group generators. Then it can be shown that they generate a Lie algebra with the commutation relation

[t_{b},t_{c}]=iC_{bc}^{a}t_{a}

{\ displaystyle [t_ {b}, t_ {c}] = iC_ {bc} ^ {a} t_ {a}}

,

Where $C_{bc}^{a}$ ${\ displaystyle C_ {bc} ^ {a}}$ - so-called Lie algebra (they also say, “structural constants of a group”).

Evidence

The group law of multiplication has the form

T(\alpha )T(\beta )=T(f(\alpha ,\beta ))

{\ displaystyle T (\ alpha) T (\ beta) = T (f (\ alpha, \ beta))}

,

Where $f$ ${\ displaystyle f}$ - some function. Since the zero parameter vector is taken as the "coordinates" of the unit element, this function must have the properties $f^{a}(\alpha ,0)=f^{a}(0,\alpha )=\alpha ^{a}$ ${\ displaystyle f ^ {a} (\ alpha, 0) = f ^ {a} (0, \ alpha) = \ alpha ^ {a}}$ . In addition, this function can be expanded in a power series

f^{a}(\alpha ,\beta )=\alpha ^{a}+\beta ^{a}+f_{bc}^{a}\alpha ^{b}\beta _{c}+...

{\ displaystyle f ^ {a} (\ alpha, \ beta) = \ alpha ^ {a} + \ beta ^ {a} + f_ {bc} ^ {a} \ alpha ^ {b} \ beta _ {c} + ...}

,

Moreover, the terms proportional to the squares of the parameters would violate the above property of this function, therefore they are absent in the expansion.

Let a group representation be given $U(T(\alpha ))$ ${\ displaystyle U (T (\ alpha))}$ . It can be expanded in a neighborhood of zero in terms of parameters in the form of the following series (we add the imaginary unit for the approach used in physics).

U(T(\alpha ))=1+i\alpha ^{a}t_{a}+1/2\alpha ^{b}\alpha _{c}t_{bc}+...

{\ displaystyle U (T (\ alpha)) = 1 + i \ alpha ^ {a} t_ {a} +1/2 \ alpha ^ {b} \ alpha _ {c} t_ {bc} + ...}

,

Where $t_{a},t_{bc}$ ${\ displaystyle t_ {a}, t_ {bc}}$ - operators, independent of parameters $\alpha$ ${\ displaystyle \ alpha}$ .

In case of unitarity of representation $U$ ${\ displaystyle U}$ operators $t_{a}$ ${\ displaystyle t_ {a}}$ (group generators) are Hermitian. It is assumed that the presentation is non-projective, that is, ordinary and therefore can be written

U(T(\alpha ))U(T(\beta ))=U(T(f(\alpha ,\beta ))

{\ displaystyle U (T (\ alpha)) U (T (\ beta)) = U (T (f (\ alpha, \ beta))}

.

The left side of this relation is

(1+i\alpha ^{a}t_{a}+1/2\alpha ^{b}\alpha _{c}t_{bc}+...)*(1+i\beta ^{a}t_{a}+1/2\beta ^{b}\beta _{c}t_{bc}+...)=1+i(\alpha ^{a}+\beta ^{a})t_{a}-\alpha ^{b}\beta _{c}t_{b}t_{c}+....

{\ displaystyle (1 + i \ alpha ^ {a} t_ {a} +1/2 \ alpha ^ {b} \ alpha _ {c} t_ {bc} + ...) * (1 + i \ beta ^ {a} t_ {a} +1/2 \ beta ^ {b} \ beta _ {c} t_ {bc} + ...) = 1 + i (\ alpha ^ {a} + \ beta ^ {a} ) t_ {a} - \ alpha ^ {b} \ beta _ {c} t_ {b} t_ {c} + ....}

.

The right-hand side can be represented as follows (using the expansion of the representation and the expansion of the function f)

1+i(\alpha ^{a}+\beta ^{a}+f_{bc}^{a}\alpha ^{b}\beta _{c}+...)t_{a}+1/2(\alpha ^{b}+\beta ^{b}+...)(\alpha ^{c}+\beta ^{c}+...)t_{bc}+...=1+i(\alpha ^{a}+\beta ^{a})t_{a}+f_{bc}^{a}\alpha ^{b}\beta _{c}t_{a}+\alpha ^{b}\beta _{c}t_{bc}+...

{\ displaystyle 1 + i (\ alpha ^ {a} + \ beta ^ {a} + f_ {bc} ^ {a} \ alpha ^ {b} \ beta _ {c} + ...) t_ {a} +1/2 (\ alpha ^ {b} + \ beta ^ {b} + ...) (\ alpha ^ {c} + \ beta ^ {c} + ...) t_ {bc} + ... = 1 + i (\ alpha ^ {a} + \ beta ^ {a}) t_ {a} + f_ {bc} ^ {a} \ alpha ^ {b} \ beta _ {c} t_ {a} + \ alpha ^ {b} \ beta _ {c} t_ {bc} + ...}

,

where unmixed second-order terms are omitted due to their obvious coincidence with the left-hand side. Obviously, the first order terms also coincide. Relations for mixed second-order terms are nontrivial. Namely, for the left and right sides of the group condition to be equal, for the representation U it is necessary to fulfill the relation

t_{bc}=-t_{b}t_{c}-if_{bc}^{a}t_{a}

{\ displaystyle t_ {bc} = - t_ {b} t_ {c} -if_ {bc} ^ {a} t_ {a}}

.

Thus, the second-order operator for expanding the representation of the group turned out to be expressed through first-order operators - through group generators. However, for complete consistency, symmetry of the operator is required $t_{bc}$ ${\ displaystyle t_ {bc}}$ by indices. Using an expression through generators, the requirement of symmetry means

0=t_{cb}-t_{bc}=(t_{b}t_{c}-t_{c}t_{b})-i(f_{cb}^{a}-f_{bc}^{a})t_{a}

{\ displaystyle 0 = t_ {cb} -t_ {bc} = (t_ {b} t_ {c} -t_ {c} t_ {b}) - i (f_ {cb} ^ {a} -f_ {bc} ^ {a}) t_ {a}}

.

From here we get the expression for the group generator switch

[t_{b},t_{c}]=iC_{bc}^{a}t_{a}

{\ displaystyle [t_ {b}, t_ {c}] = iC_ {bc} ^ {a} t_ {a}}

,

Where $C_{bc}^{a}=f_{cb}^{a}-f_{bc}^{a}$ ${\ displaystyle C_ {bc} ^ {a} = f_ {cb} ^ {a} -f_ {bc} ^ {a}}$ - so-called structural constants of the group.

Such a set of commutation relations is a Lie algebra. Thus, the generators of the group generate a Lie algebra.

These commutation relations are the only condition guaranteeing a recurrent expression of the operators appearing in the expansion of the group representation in terms of the second and higher order. Thus, all terms of the expansion can be expressed through generators. This means that the group representation operators in at least some neighborhood of the unit element can be uniquely expressed through group generators.

In one particular case, when $C_{bc}^{a}=0$ ${\ displaystyle C_ {bc} ^ {a} = 0}$ switching relations show that generators commute in pairs $[t_{b},t_{c}]=0$ ${\ displaystyle [t_ {b}, t_ {c}] = 0}$ . Such a group is abelian. For such a group, it is possible to express the group representation operators through generators

U(T(\alpha ))=e^{i\alpha ^{a}t_{a}}

{\ displaystyle U (T (\ alpha)) = e ^ {i \ alpha ^ {a} t_ {a}}}

.

Such a map of a Lie algebra to a Lie group is called an exponential map.

Evidence

In such a group $f(\alpha ,\beta )=\alpha +\beta$ ${\ displaystyle f (\ alpha, \ beta) = \ alpha + \ beta}$ , Consequently $f(\alpha /n,\alpha /n,...,\alpha /n)=\sum _{n}\alpha /n=\alpha$ ${\ displaystyle f (\ alpha / n, \ alpha /n,...,\alpha / n) = \ sum _ {n} \ alpha / n = \ alpha}$ . Therefore, we can write the following group relation

U(T(\alpha ))=U(T(\alpha /n))^{n}

{\ displaystyle U (T (\ alpha)) = U (T (\ alpha / n)) ^ {n}}

at a sufficiently large $n$ ${\ displaystyle n}$ you can use the infinitesimal representation due to smallness $\alpha /n$ ${\ displaystyle \ alpha / n}$ . We get

U(T(\alpha ))=(1+i/n\alpha ^{a}t_{a})^{n}

{\ displaystyle U (T (\ alpha)) = (1 + i / n \ alpha ^ {a} t_ {a}) ^ {n}}

.

Passing to the limit by $n$ ${\ displaystyle n}$ we obtain the desired expression for the representation of the group for arbitrary parameters through the exponential

U(T(\alpha ))=e^{i\alpha ^{a}t_{a}}

{\ displaystyle U (T (\ alpha)) = e ^ {i \ alpha ^ {a} t_ {a}}}

.

Generator Examples

The imaginary unit is the generator of the group U (1) .
The Pauli matrices are generators of the special unitary group SU (2).
Gell-Mann matrices are generators of a special unitary group SU (3).

Links

V. S. Zamiralov “Basic concepts of group theory and their representations and some applications to particle physics” on the website of the Research Institute of Nuclear Physics, Moscow State University .