Physics 195Supplementary NotesGroups, Lie algebras, and Lie groups020922 F. PorterThis note defines some mathematical structures which are useful in thediscussion of angular momentum in quantum mechanics (among other things).Def: Apair(G, ◦), where G is a non-empty set, and ◦ is a binary operationdefined on G, is called a group if:1. Closure: If a, b ∈ G,thena ◦ b ∈ G .2. Associativity: If a, b, c ∈ G,thena ◦ (b ◦ c)=(a ◦ b) ◦ c.3. Existence of right identity: There exists an element e ∈ G suchthat a ◦ e = a for all a ∈ G.4. Existence of right inverse: For some right identity e,andforanya ∈ G, there exists an element a−1∈ G such that a ◦ a−1= e.The ◦ operation is typically referred to as “multiplication”.The above may be termed a “minimal” definition of a group. It is amusing(and useful) to prove that:1. The right identity element is unique.2. The right inverse element of any element is unique.3. The right identity is also a left identity.4. The right inverse is also a left inverse.5. The solution for x ∈ G to the equation a ◦ x = b exists and is unique,for any a, b ∈ G.We will usually drop the explicit ◦ symbol, and merely use juxtaposition todenote group multiplication. Note that both G (the set) and ◦ (the “mul-tiplication table”) must be specified in order to specify a group. Where theoperation is clear, we will usually just refer to “G” as a group.Def: An abelian (or commutative) group is one for which the multiplica-tion is commutative:ab = ba ∀ a, b ∈ G. (1)1Def: The order of a group is the number of elements in the set G.Ifthisnumber is infinite, we say it is an “infinite group”.In the discussion of infinite groups of relevance to physics (in particular,Lie groups), it is useful to work in the context of a richer structure called analgebra. For background, we start by giving some mathematical definitionsof the underlying structures:Def: A ring is a triplet R, +, ◦ consisting of a non-empty set of elements(R) with two binary operations (+ and ◦) such that:1. R, + is an abelian group.2. (◦) is associative.3. Distributivity holds: for any a, b, c ∈ Ra ◦ (b + c)=a ◦ b + a ◦ c (2)and(b + c) ◦ a = b ◦ a + c ◦ a (3)Conventions:We use 0 (“zero”) to denote the identity of R, + . We speak of (+) as ad-dition and of (◦) as multiplication, typically omitting the (◦)symbolentirely(i.e., ab ≡ a ◦ b).Def: A ring is called a field if the non-zero elements of R form an abeliangroup under (◦).Def: An abelian group V, ⊕ is called a vector space over a field F, +, ◦by a scalar multiplication (∗) if for all a, b ∈ F and v, w ∈ V :1. a ∗ (v ⊕ w)=(a ∗ v) ⊕ (a ∗ w) distributivity2. (a + b) ∗ v =(a ∗ v) ⊕ (b ∗ v) distributivity3. (a ◦ b) ∗ v = a ∗ (b ∗ v) associativity4. 1 ∗ v = v unit element (1 ∈ F )2Conventions:We typically refer to elements of V as “vectors” and elements of F as“scalars.” We typically use the symbol + for addition both of vectors andscalars. We also generally omit the ∗ and ◦ multiplication symbols. Notethat this definition is an abstraction of the definition of vector space givenin the note on Hilbert spaces, page 1.Def: An algebra is a vector space V over a field F on which a multiplication(◦) between vectors has been defined (yielding a vector in V ) such thatfor all u, v, w ∈ V and a ∈ F :1. (au) ◦ v = a(u ◦ v)=u ◦ (av)2. (u+v) ◦ w =(u ◦ w)+(v ◦ w)andw ◦ (u + v)=(w ◦ u)+(w ◦ v)(Once again, we often omit the multiplication sign, and hope that it isclear from context which quantities are scalars and which are vectors.)We are interested in the following types of algebras:Def: An algebra is called associative if the multiplication of vectors is as-sociative.We note that an associative algebra is, in fact, a ring. Note also thatthe multiplication of vectors is not necessarily commutative. An importantnon-associative algebra is:Def: A Lie algebra is an algebra in which the multiplication of vectorsobeys the further properties (letting u, v, w be any vectors in V ):1. Anticommutivity: u ◦ v = −v ◦ u.2. Jacobi Identity: u ◦ (v ◦ w)+w ◦ (u ◦ v)+v ◦ (w ◦ u)=0.We may construct the idea of a “group algebra”: Let G be a group,and V be a vector space over a field F , of dimension equal to the order ofG (possibily ∞). Denote a basis for V by the group elements. We can nowdefine the multiplication of two vectors in V by using the group multiplicationtable as “structure constants”: Thus, if the elements of G are denoted by gi,a vector u ∈ V may be written:u =aigi3We require that, at most, a finite number of coefficients aiare non-zero. Themultiplication of two vectors is then given by:aigi bjgj =gigj=gkaibjgk[Since only a finite number of the aibjcan be non-zero, the sumgigj=gkaibjpresents no problem, and furthermore, we will have closure under multipli-cation.]Since group multiplication is associative, our group algebra, as we haveconstructed it, is an associative