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Group Theory Appendix A Group Theory This appendix is a survey of only those topics in group theory that are needed to understand the composition of symmetry transformations and its consequences for fundamental physics. It is intended to be self-contained and covers those topics that are needed to follow the main text. Although in the end this appendix became quite long, a thorough understanding of group theory is possible only by consulting the appropriate literature in addition to this appendix. In order that this book not become too lengthy, proofs of theorems were largely omitted; again I refer to other monographs. From its very title, the book by H. Georgi [211] is the most appropriate if particle physics is the primary focus of interest. The book by G. Costa and G. Fogli [102] is written in the same spirit. Both books also cover the necessary group theory for grand unification ideas. A very comprehensive but also rather dense treatment is given by [428]. Still a classic is [254]; it contains more about the treatment of dynamical symmetries in quantum mechanics. A.1 Basics A.1.1 Definitions: Algebraic Structures From the structureless notion of a set, one can successively generate more and more algebraic structures. Those that play a prominent role in physics are defined in the following. Group A group G is a set with elements gi and an operation ◦ (called group multiplication) with the properties that (i) the operation is closed: gi ◦ g j ∈ G, (ii) a neutral element g0 ∈ G exists such that gi ◦ g0 = g0 ◦ gi = gi , (iii) for every gi exists an −1 ∈ ◦ −1 = = −1 ◦ inverse element gi G such that gi gi g0 gi gi , (iv) the operation is associative: gi ◦ (g j ◦ gk) = (gi ◦ g j ) ◦ gk. The closure property can be expressed K. Sundermeyer, Symmetries in Fundamental Physics, 509 Fundamental Theories of Physics 176, DOI: 10.1007/978-3-319-06581-6, © Springer International Publishing Switzerland 2014 510 Appendix A: Group Theory Table A.1 The group D4 1 i −1 −i C Ci −C −Ci 1 1 i −1 −i C Ci −C −Ci i i −1 −i 1 −Ci C Ci −C −1 −1 −i 1 i −C −Ci C Ci −i −i 1 i −1 Ci −C −Ci C C C Ci −C −Ci 1 i −1 −i Ci Ci −C −Ci C −i 1 i −1 −C −C −Ci C Ci −1 −i 1 i −Ci −Ci C Ci −C i −1 −i 1 by considering the operation as a mapping ◦:G × G → G. There is indeed only one neutral element and the inverse of a group element is unique, as easily can be shown. A group is called finite if the number of its elements (also called the order of the group) is finite. An example of a finite group is the rotations of a square by ◦ 90 (in mathematics and crystallography called C4; a group of order 4). Examples of infinite groups are the integers with respect to addition (countably infinite) and the symmetry group of the circle, characterized by a parameter ϕ [0 ≤ ϕ < 2π] (continuous group). The center of a group is defined as the set of those elements of G which commute with all other elements of the group: Cent(G) ={c ∈ G | cg = gc for all g ∈ G}. Example: From the multiplication table of D4 (see Table A.1; more about this group later), one can see that its center is given by the set {1, −1}. 1 A group is called Abelian if the operation is commutative: gi ◦ g j = g j ◦ gi . Obviously the center of an Abelian group is the group itself. The symmetry group C4 of the square is Abelian; the analogous group of the cube is an example of a non–Abelian group. In physics we often deal with groups (or subgroups of) GL(n, F). This is the set of non–singular n×n matrices with elements from the field F which are in the majority of cases real numbers R or complex numbers C. GL(n, F) is in general a non–Abelian group with matrix multiplication as the group operation. The restriction to non–singular matrices is necessary because otherwise the inverse would not be defined. A subset of G, which itself is a group, is called a subgroup of G. By this definition, every group has two trivial subgroups, namely itself (G ⊂ G) and the set consisting of the neutral element ({g0}⊂G). 1 This term derives from the name of the Norwegian mathematician Niels Henrik Abel (1802– 1829). He is treated with some disrespect by the fact that contemporary theoreticians use his name in a negated form, by talking for instance of “non–Abelian” gauge theories. Appendix A: Group Theory 511 (b) (a) i 1 i–i–1 (c) –1 1 C C Ci–C –Ci –i Fig. A.1 Symmetries of the square Although in the context of symmetries in fundamental physics, continuous group are the most important ones, many notions from group theory will subsequently be explained on a finite group. This is the group D4, defined by its multiplication table above. The notation of its elements is taken from [410]. D4 is a group of order 8, its neutral element is (1), and it is non–Abelian (for instance (Ci)C =−i, but C(Ci) = i). This abstract group is realized by 90◦-rotations of the square and reflections along its diagonals; see Fig. A.1. Here the vertices of the square are represented by the points 1, i, –1, –i. The group D4 has various nontrivial subgroups; these are ◦ • order 2: rotations by 180 realized by C2 ={1, −1}, and four reflections realized by {1, C}, {1, Ci}, {1, −C}, {1, −Ci}. ◦ • order 4: rotations by 90 realized by C4 ={1, i, −1, −i}, reflections along diag- onals, that is {1, −1, C, −C}, and reflections along the two parallels to the edges of the square: {1, −1, Ci, −Ci}. Field A field is a set F with elements fi together with two operations +: F × F → F ∗:F × F → F with the properties that (i) F is an Abelian group under the operation + with f0 as the identity, (ii) F \{f0} is a group under ∗, and (iii) there is a distributive law fi ∗ ( f j + fk) = fi ∗ f j + fi ∗ fk,(fi + f j ) ∗ fk = fi ∗ fk + f j ∗ fk. Prominent examples of fields are the real and the complex numbers under the usual addition and multiplication operations. Further fields are known, as for instance quaternions (building a 4–dimensional number system and having some link to Dirac matrices), octonions, sedenions,…the latter bearing a relation to the exceptional Lie groups. 512 Appendix A: Group Theory Vector Space A (linear) vector space consists of a set V with elements vi and a field F together with two operations +: V × V → V : F × V → V with the properties that (i) (V, +) is an Abelian group, (ii) the operation fulfills fi (f j vk) = ( fi ∗ f j )vk 1 vi = vi = vi 1 fi (v j + vk) = fi v j + fi vk,(fi + f j )vk = fi vk + f j vk. Further definitions: • A set of vectors vk is called linearly dependent iff f j ∈ F exist such that fk vk ≡ 0. k • The dimension D of a vector space is defined as the maximal number of linearly independent vectors. • In every vector space of dimension D one can find a set of basis vectors ek which are linearly independent and which make it possible to expand any vector D as v = f j e j . Prominent examples of vector spaces are the Euclidean space, the solution space of linear differential and integral operators, the sets of N × M–matrices, and Hilbert spaces. Algebra A (linear) algebra A consists of a set V with elements vi , a field F together with three operations +,,. The algebra is a vector space with respect to (V, F, +,). The is a mapping : V × V → V with (vi + v j )vk = vi vk + v j vk vi (v j + vk) = vi v j + vi vk. Prominent examples are Lie algebras, which aside from the algebra properties addi- tionally fulfill vi (av j + bvk) = avi v j + bvi v j , a, b ∈ F vi v j =−v j vi vi (v j vk) = (vi v j )vk + v j (vi vk), Appendix A: Group Theory 513 where the latter relation is the Jacobi identity, as it is known in physics for instance for the Poisson brackets or the generators of bosonic/Grassmann-even2 symmetry transformations. A.1.2 Mapping of Groups Group Homomorphism Let f be a group mapping f : G → Gˆ . If the mapping respects the group structure, that is if ( ) ◦ ( ) = ( ◦ ), f g1 Gˆ f g2 f g1 G g2 (where ◦F denotes the operation in the respective group) it is called a (group) homo- morphism. The definition of a homomorphism can be generalized to fields, vector spaces, and algebras as structure–preserving maps between two of these algebraic structures. The kernel of the homomorphism f is defined as the subset of G for which each element is mapped to the neutral element in Gˆ : ˆ Ker f := {g ∈ G | f (g) =ˆg0 ∈ G}. The definition of homomorphism does not exclude that f (g1) = f (g2) for g1 = g2 (a so-called “many-to-one” relation). For instance, mapping all elements of an arbi- trary group to the real number 1 is a homomorphism between the group and the group of real numbers. A “one-to-one” mapping is called an isomorphism.
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