Group Theory Glossary

Group: A set with a binary operation \(*\) with the following properties:
closure, associativity, unit/neutral element, inverse element.
Closure: The fact that a group is closed under the group action.
Associativity: The binary operation \(*\) is associative if \((a\ast b)\ast c=a \ast (b\ast c)\).
Unit/Neutral Element: The unique element \(e\) of a group \(G\) with the property \(e \ast a= a\ast e=a\) for \(a\in G\).
Inverse Element: Given an element \(a\) of a group \(G\), the unique element \(a^{-1}\) such that \(a^{-1} \ast a= a\ast a^{-1}=e\) for \(a\in G\).
Finite Group: A group with a finite number of elements, i.e. with a finite order.
Rearrangement Lemma: We have \(aG=G\) with \(aG:=\{ab | b\in G\}\) for an arbitrary element \(a\in G\). Arranging all elements of \(G\) in a sequence, multiplication by \(a\) corresponds to a rearrangement.
Homomorphism: Given two groups \((G,\ast)\) and \((H,\cdot)\), a group homomorphism from \((G,\ast)\) to \((H,\cdot)\) is a function \(h:G\to H\) such that for all \(a\) and \(b\) in \(G\) we have \(h(a\ast b)=h(a)\cdot h(b)\).
Isomorphism: Two groups are said to be \(\it{isomorphic}\) if there exists a one-to-one correspondence (an isomorphism) between their elements which preserves the group operation.
Order of a Group: The number of elements of a group.
Multiplication Table: The table displaying the results for all possible group multiplications.
Cyclic Group: The group generated by the repeated action of a group element of order \(n\).
Order of a Group Element: The minimal number \(n\) for which \(a^n=e\) with \(a\in G\) and \(e\) the unit/neutral element of the group.
Abelian/Non-Abelian Group: A group for which any two of its elements commute with each other is called Abelian. Groups without this property are called non-Abelian.
Dihedral Group: The symmetry group \( D_n\) of a regular polygon with \(n\) vertices. The word dihedral is composed of the greek words "di" (two) and "hédra" (face of a geometrical solid).
Subgroup: A subset of a group \(G\) is a subgroup, if its elements form a group with respect to the group operation of the original group \(G\).
Direct/Kronecker Product Group: For two groups \(G\) and \(K\) of order \(n_G\) and \(n_K\), respectively, with elements \(g_a\) and \(k_i\), the direct product group \(G\times K\) has elements \((g_a,k_i)\equiv g_ak_i\) and order \(n_G n_K\).
Elements of the direct product group are multiplied as \((g_a,k_i)(g_b,k_j)=(g_a\ast g_b,k_i\ast k_j)\).
Coset: Given a group \(G\) with subgroup \(H\), the left cosets of \(H\) in \(G\) are the sets obtained by multiplying each element of \(H\) by a fixed element \(g\) of \(G\) that is not an element of \(H\), i.e. \(gH=\{gh: h\in H\}\). Similarly for the right cosets.
Coset Decomposition: Decomposition of a group \(G\) in terms of its cosets with respect to a subgroup \(H\), e.g. its left coset decomposition \(G= H+g_1\ast H+\dots+g_k\ast H\). Here the +-sign denotes the addition of sets, i.e. the sum of two sets \(A\) and \(B\) is the set of objects which are contained in at least one of the sets \(A\) or \(B\).
Lagrange's Theorem: If a group \(G\) of order \(N\) has a subgroup \(H\) of order \(n\), then \(N\) is necessarily an integer multiple of \(n\).
Index of a Subgroup: If a group \(G\) of order \(N\) has a subgroup \(H\) of order \(n\), the integer ratio \(k=N/n\) is called the index of \(H\) in \(G\).
Order of a Group Element: When \(a^n=e\) for an element \(a\) of a group with \(e\) being the unit element, we say that \(a\) has order \(n\).
Generators of a Group: The elements of a subset of the group set which allow to express every group element as their combination (under the group operation).
Rank of a Group: The cardinality of the smallest subset of group elements which generates the whole group.
Presentation of a Group: A list of generators together with their relations that fully specify the group.
Symmetric Group \(S_n\): The group of all permutations of \(n\) objects.
Dihedral Group \(D_n\): Invariance group of an \(n\)-sided polygon consisting of rotations by \(2\pi/n\) and mirror reflections about
\(n\) odd: axis from vertex-to-midpoint-of-an-edge
\(n\) even: vertex-to-vertex and midpoint-to-midpoint-of-an-edge
A presentation is given by \(D_n: \langle a,b|a^n=b^2=e;bab^{-1}=a^{-1}\rangle\)
Quaternion Group \(Q\): The group formed out of elements of the quaternions with presentation \(\langle a,b|a^4=e,a^2=b^2,bab^{-1}=a^{-1}\rangle\). It is the first member of the family of dicyclic groups \(Q_{2n}\), i.e. \(Q=Q_4\) (sometimes also denoted by \(Q_8\)).
Dicyclic Groups \(Q_{2n}\): Infinite family of finite groups with presentation \(Q_{2n}: \langle a,b|a^{2n}=e,b^2=a^n,bab^{-1}=a^{-1}\rangle\). They include the quaternions \(Q=Q_{4}\) and the group \(\Gamma=Q_6\).
Tetrahedral Group \(T\): The non-Abelian order-12 symmetry group of the regular tetrahedron. It forms part of the infinite family of anternating groups \(A_n\), i.e. \(T=A_4\).
Group \(\Gamma\): Non-Abelian group of order 12 with presentation \(\langle aa,b|a^6=e,b^2=a^3,bab^{-1}=a^{-1}\rangle\).
Degree of a Permutation: Number of objects the permutation acts on.
Parity of a Permutation: A permutation is called even/odd, if it can be written as an even/odd number of transpositions.
Alternating Group \(A_n\): The group of even permutations of order \(n!/2\).
Cayley's Theorem: Every group of finite order \(n\) is isomorphic to a subgroup of the symmetric group \(S_n\).
Regular Representation of a Degree-\(n\) Permutation: The representation in terms of \((n\times n)\)-matrices acting on \(n\) objects arranged as column vectors.
Conjugate Group Element: Given a group \(G\) with elements \(g_a\), the conjugate of \(g_a\) with respect to any other group element \(g\in G\) is defined as \(\tilde g_a=g g_a g^{-1}\).
Self-Conjugate Group Element: If two group elements \(g\) and \(h\) commute, then \(g\) is self-conjugate with respect to \(h\) and vice versa.
Group Homomorphism: Given two groups \((G,\ast)\) and \((H,\cdot)\) with their respective group operations, a \(\textit{group homomorphism}\) from \(G\) to \(H\) is a a map which preserves the group operation, i.e. \(f: G\to H\) such that for all \(a,b\in G\) we have \(f(a\ast b)=f(a)\cdot f(b)\).
Group Automorphsim: A group automorphism is an isomorphism from the group to itself.
Inner Group Automorphism: An inner automorphism of a group \(G\) is an automorphism defined by conjugation, i.e. of the form \(\varphi(g)=h g h^{-1}\) with \(h\) being a fixed element of \(G\).
Conjugacy Class: Elements of a group which are conjugate to each other form a (conjugacy) class. Each element of a group belongs to a single class.
Normal Subgroup: A \(\textit{normal subgroup} \) \(H\) is a subgroup of a group \(G\) that is invariant under conjugation by all elements of \(G\). This is denoted by \(H\lhd G\).
Simple Group: A group that has no nontrivial normal subgroup, i.e. its only normal subgroups are the trivial group and the group itself.
Proper Subgroup: A proper subgroup \(H\) of a group \(G\) is a subgroup that is formed by a proper subset of \(G\), i.e. the sets \(H\) and \(G\) do not agree.
Quotient/Factor Group: Given a group \(G\) with a normal subgroup \(H\). The quotient/factor group \(G/H\) is formed by the cosets of \(H\) considered as elements with the product of cosets as group operation.
Simple Group: A group with no normal subgroups.
Sporadic Groups: The 26 finite simple groups that do not belong to one of the infinite families in which the other simple groups are classified.
Composition Series: The series of inclusions one obtains by starting from a group \(G\), identifying its largest normal subgroup \(H_1\) then looking for its largest normal subgroup \(H_2\) and iterating this procedure until one finds a simple group \(H_k\) whose only normal subgroup is the trivial group \(\{e\}\).
Composition Length: The number of steps in a composition theory of a group, i.e. the number of groups in the composition series (involving the full and the trivial group) minus one. This number is the same for any composition series of a fixed group.
Composition Indices: For a given composition series one can compute the quotient groups of all consecutive groups in the series. The set of orders of the quotient groups are \(\textit{composition indices} \). The number of composition indices is given by the composition length and they are the same for any composition series of a fixed group.
Jordan-Hölder Theorem: For a given group, all of its composition series have the same composition length. Furthermore the set one obtains by computing all quotient groups of consecutive groups in a composition series is the same up to permutation and isomorphism for all composition series of a fixed group. In particular the composition indices are the same for different composition series of the same group.
Solvable/Soluble Group: A group is \(\textit{solvable/soluble}\) if any quotient group formed out of two consecutive groups in a composition series of the group, is a cyclic group of prime order.
Commutator: For two group elements \(a,b\) their \(\textit{commutator}\) is defined as \( [a,b]:=a^{-1}b^{-1}ab \).
Derived (Commutator) Subgroup: The normal subgroup of a group \(G\) generated by all possible commutators of elements of \(G\). It is typically denoted by \(G'\).
Perfect Group: A group \(G\) which is equal to its derived subgroup, i.e. which satisfies \(G=G'\).
\(p\)-Group: A group where the order of each element is a power of the prime \(p\).
\(p\)-Subgroup: For a given prime \(p\) and group \(G\) a \(p\textit{-subgroup}\) is a subgroup of \(G\), which is also a \(p\)-group.
Sylow \(p\)-Subgroup: For a given prime \(p\) and group \(G\) a \(\textit{Sylow }p\textit{-subgroup}\) is a \(p\)-subgroup of \(G\), which is not a proper subgroup of any other \(p\)-subgroup of \(G\).
Sylow Theorems: Let \(G\) be a group of order \(n=p^m r\), where \(p\) is a prime, \(m\) is a positive integer and \(r\) is an integer which is not a multiple of \(p\). Then \(G\) contains \(n_p\) Sylow \(p\)-subgroups of order \(p^m\) which are related to each other by conjugation and hence isomorphic. Furthermore, \(n_p\) is a divisor of \(r\) and satisfies \(n_p=1 \text{ mod } p\).
Action of a Group: A group \(G\) \(\textit{acts on}\) another group \(K\) if, for any element \(g\) in \(G\) there is a group homorphism \(K\rightarrow K\), typically written as \(k\mapsto k^g\).
Semi-Direct Product Group: For two groups \(G\) and \(K\) with \(G\) acting on \(K\). Then the set \(\{(k,g)|\,k\in K,g\in G\} \) together with the composition \( (k_1,g_1)\circ(k_2,g_2)=((k_1)^{g_2}\, k_2,g_1 g_2 ) \) forms a group, the \(\textit{semi-direct product} \) of \(G\) with \(K\), denoted by \(K\rtimes G\).
Young Diagram: A \(\textit{Young diagram}\) is a graphical representation for a partition, i.e. an ordered set of \(n\) positive integers \( \{\lambda_1\geq \lambda_2 \geq \dots \geq \lambda_n \} \). Given a partition one obtains the Young diagram by drawing \(\lambda_1 \) boxes horizontally side to side, then in the same way \(\lambda_2\) boxes just below and so on down to the lowest value.
Young Tableau (Plural: Tableaux): A Young Tableau is a Young diagram with boxes filled with numbers \(1\) to \(n\). A Young tableau has \(\textit{standard filling}\) when the numbers increase, both along the rows and along the columns from left to right and top to bottom, respectively.
Hook-length Formula: The hook-length formula states that the number of standard Young tableaux of shape \(\lambda\) is \(n_\lambda=\frac{n!}{\prod_b h_b}\), where \(b\) runs over all boxes of \(\lambda\) and \(h_b\) is the number of boxes of the `hook', given by the boxes directly to the right or below the box \(b\).
Representation of a Group: A \(\textit{representation}\) of a group \(G\) on a vector space \(V\) is a group homomorphism \(G\rightarrow \mathrm{GL}(V)\), where \(\mathrm{GL}(V)\) is the group of non-singular (i.e. invertible) \(\mathrm{dim}V\times\mathrm{dim}V\) matrices acting as linear transformations on \(V\).
Dimension of a Group Representation: The \(\textit{dimension}\) of a representation of a group \(G\) on a vector space \(V\), is defined as the dimension of the vector space \(\mathrm{dim}V\).
Regular Representation: The \(\textit{regular representation}\) of a finite group \(G\) of order \(n\) is a representation of dimension \(n\) obtained by interpreting the group elements as the basis vectors of an \(n\)-dimensional vector space. Explicitly if we label the elements of \(G\) as \(\{g_1,\dots,g_n\}\), then we can write for any product \( g_ig_j=g_m M_{ij}^m\) for \(M_{ij}^m=1\) for \(m=k\) and \(M_{ij}^m=1\) for \(m\neq k\). The representation matrix for element \(g_j\) is then \((M_j)_{ki}=M_{ij}^k \).
Invariant Subspace: Let \(G\) be a group and \(M\) a representation of \(G\) on a vector space \(V\). An \(\textit{invariant subspace}\) of \(V\) with respect to \(M\) is a vector subspace \(V_1\) of \(V\) such that \(M(g)|w\rangle\in V_1\) for all \(g\in G,\, |w\rangle\in V_1\).
Reducible Representation: A representation \(M\) of a group \(G\) on a vector space \(V\) is \(\textit{reducible}\) if \(V\) has a non-trivial (i.e. not \(\{0\}\) or \(V\)) invariant subspace with respect to \(M\).
Irreducible Representation: A representation which is not reducible.
Fully Reducible Representation: A represention \(M\)of a group \(G\) on a vector space \(V\) is \(\textit{fully reducible}\) if it is reducible and there is a basis of \(V\) such that all representation matrices \(M(g)\) are block-diagonal.
Schur's First Lemma: Let \(A,B\) be two irreducible representations of a group \(G\) of dimensions \(d_A,d_B\) with \(d_A\neq d_B\). If there is a \(d_B\times d_A\) matrix \(S\) satisfying \( SA(g)=B(g)S\) for all \(g\in G\), then \(S=0\).
Similarity Transformation: For an \(n\times n\) matrix \(M\) a \(\textit{similarity transformation}\) is a transformation of the form \(M\mapsto PMP^{-1}\), where \(P\) is an invertible \(n\times n\) matrix.
Equivalent Representations: Two representations \(A,B\) of a group \(G\) on the same vector space \(V\) are \(\textit{equivalent}\), if they are related by a similarity transformation, i.e. if there is an invertible matrix \(P\in\mathrm{GL}(V)\) such that \(A(g)=PB(g)P^{-1}\) for all \(g\in G\).
Schur's Second Lemma: Let \(M\) be an irreducible \(n\)-dimensional representation of a group \(G\). If there is an \(n\times n\) matrix \(S\) satisfying \(M(g)S=SM(g)\) for all \(g\in G\), then \(S\) is a multiple of the identity.
Completeness of Characters: Characters satisfy the following \(\textit{completeness}\) relation: \(\sum_{\alpha=1}^{n_R}\chi_i^{[\alpha]}\bar{\chi}_j^{[\alpha]}=\frac{n}{n_i}\delta_{ij}\), where \(n_R\) is the number of irreducible representations, \(n\) is the number of elements in the group and \(n_i\) is the number of elements that transform in the irreducible representation \(R_i\).
Character Table: The \(\textit{character table}\) lists the values of the characters for the different representations.
Conjugate Representations: Two irreducible representations are referred to as \(\textit{conjugate representations}\) when their representation matrices are complex conjugate to each other. The conjugate of a fundamental representation is also called \(\textit{anti-fundamental representation}\).
Kronecker Product: Consider two irreducible representations of a group \(G\) acting on two vector spaces spanned by \(\bra{i}_{\alpha}\,:\, i,j=1,...,d_{\alpha}\) and \(\bra{s}_{\beta}\,:\, s,t=1,...,d_{\beta}\) with \(\ket{i}_{\alpha}\rightarrow\ket{i(g)}_{\alpha}=M_{ij}^{[\alpha]}(g)\ket{j}_{\alpha}\) and \(\ket{s}_{\alpha}\rightarrow\ket{s(g)}_{\beta}=M_{st}^{[\beta]}(g)\ket{t}_{\beta}\). The \(\textit{Kronecker product}\) representation acts on the vector spaced spanned by \(\ket{A}:=\ket{i}_{\alpha}\ket{s}_{\beta}\) as \(\ket{A}\rightarrow\ket{A(g)}=M_{AB}^{[\alpha\times\beta]}(g)\ket{B}=M_{ij}^{[\alpha]}(g)M_{st}^{[\beta]}(g)\ket{j}_{\alpha}\ket{t}_{\beta}\).
Clebsch-Gordon Coefficients: The expansion of the Kronecker product into irreducible representations can be written as \(R_{\alpha}\times R_{\beta}=\sum_{\gamma}d(\alpha,\beta|\gamma)R_{\gamma}\), where \(d(\alpha,\beta|\gamma)\) are called \(\textit{Clebsch-Gordon coefficients}\).
Maschke's Theorem': A theorem which essentially states that under reasonable assumptions a representation \(R\) of a finite group \(G\) can be decomposed into irreducible representations.
Alternating Representation: The \(\textit{alternating}\) (also called \(\textit{sign}\)) representation associates +1(-1) to group elements with an even (odd) number of transpositions.
Standard Representation: For every \(n\) the \(n\)-dimensional permutation representation of \(S_n\) can be reduced into a 1-dimensional and an \((n − 1)\)-dimensional representation, with the latter being called the \(\textit{standard}\) representation.
Real/Pseudoreal Representation: If a representation is related to its conjugate by \(M=S\bar M S^{-1}\), it is called \(\textit{real}\) if \(S\) is symmetric and \(\textit{pseudoreal}\) if \(S\) is anti-symmetric.
Frobenius-Schur Indicator: An irreducible representation is real/pseudoreal/complex, if the expression \(\frac{1}{n} \sum_g \chi(g^2)\) evaluates to \(+1/-1/0\).
Embedding Coefficients: The coefficients \(f^\alpha{}_a\) in the expression \(R^{[\alpha]}=\sum_a f^\alpha{}_a T^{[a]}\), which relates an irreducible representation \(R^{[\alpha]}\) of a group to irreducible representations \(T^{[a]}\) of its subgroup.
Inner/Outer Automorphism of a Group: A \(\textit{group automorphism}\) is an isomorphism of a group to itself. If it is generated by an element of the group it is called \(\textit{inner}\), otherwise it is called \(\textit{outer automorphism}\).
Induced Representation: Given a group \(G\) and a subgroup \(H\) with a given representation \(\mathbf{r}\) of \(H\) with representation matrices \(M^{[\mathbf{r}]}\) that on a vector space \(V_H\) spanned by \(d\) vectors \(\ket{i}\). Consider the \(N\) cosets \(H,g_1H,\dots,g_{N-1}H\) with \(N\) being the index of the subgroup \(H\). The induced representation of \(G\) acts on the space of elements of the form \((g_k,\ket{i})\) as \(g:(g_k,\ket{i})\to (g_mh_b,\ket{i})=(g_m,M(h_b)_{ij}^{[\mathbf{r}]}\ket{j})\), where \(gg_k=g_mh_b\)
Covering Group: Let \(G\) be a finite group generated by a set of elements \(a_i\), called letters, with a presentation defined by a set of relations \(w_\alpha(a_1,a_2,\dots)=e\), called words. Now define a group \(G'\) where the words are not equal to the identity anymore but commute with the generators of \(G\). The group \(G'\) is then called the \(\textit{covering group}\) of \(G\) by the central subgroup \(C\) generated by the words with \(G=G'/C\).
Double Cover: Consider a finite group \(G \) with covering group \(G'\), such that \(G=G'/C\), where \(C\) is the central subgroup generated by the words. If the group \(C\cap C'\), where \(C'\) is the derived subgroup of \(C\), is of order two, the covering group is called a \(\textit{double cover}\).
Central Subgroup: A subgroup of a group whose elements commute with all elements of the group.
Dirac Group: The \(\textit{Dirac group} \) of order \(2^{2n+1}\) is the set of objects \(\{\pm\gamma_1^{l_1}\gamma_2^{l_2}\dots \gamma_{2n}^{l_{2n}}\, |\, l_k=0,1\} \) with multiplication as group operation. Here \(\gamma_j=b^j+b_j^\dagger ,\, \gamma_{n+j}=i(b^j-b_j^{\dagger})\) for \(j=1=1,\dots ,n \), in terms of some Fermionic oscillators.
Clifford Algebra: The relation \(\{\gamma_a,\gamma_b\}=\delta_{ab}\) is typically referred to as \(\textit{Clifford algebra} \).
Dirac Matrices: The representation matrices of the unique \(2^n\)-dimensional irreducible representations of the Dirac group. They are unique up to equivalence and obey the Clifford algebra.
Unitary Operator: An operator on a Hilbert space that preserves the inner product. This means that it satisfies \(UU^\dagger=1\).
Heisenberg Algebra: The commutation relation \([X,P]=i \) is typically referred to as \(\textit{Heisenberg algebra}\).
Hermite Polynomials: An orthogonal set of polynomials defined by \(H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2} \).
Levi-Civita Symbol: The \(\textit{Levi-Civita symbol}\) \(\epsilon^{ABC} \) with \(A,B,C\in\{1,2,3\}\) is determined by its value \(\epsilon^{123}=1\) and the fact that it is antisymmetric under the exchange of any pair of indices. In particular it vanishes if two or more of its indices are equal.
Jacobi Identity: The relation \([A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0\) for three objects \(A,B,C\) is referred to as the \( \textit{Jacobi identity}\).
Lie Bracket: A bilinear, antisymmetric map \([\cdot,\cdot] \) satisfying the Jacobi identity.
Lie Algebra: A vector space \(g\) equipped with a Lie bracket \([\cdot,\cdot]:g\times g\rightarrow g \).
Casimir Operator: A polynomial function of the generators of a Lie algebra which commutes with all of the generators.
Lie Algebra Representation: A \(\textit{representation}\) of a Lie algebra \(g\) on a vector space \(V\) is a Lie algebra homomorphism, i.e. a map \(\rho: g\rightarrow \mathrm{gl}(V)\) satisfying \(\rho([X,Y])=\rho(X)\rho(Y)-\rho(Y)\rho(X)\) for all \(X,Y\in g\).
Highest Weight: The state in an irreducible \(\mathfrak{su}(2)\) representation which is annihilated by \(T^+\).
Dynkin Label: The \(\textit{Dynkin label}\) of the \(2j+1\)-dimensional irreducible representation of \(\mathfrak{su}(2)\) is \(2j\).
Trivial Representation of \(\mathfrak{su}(2): \underline{1}\): The Lie algebra of \(\mathfrak{su}(2)\) has infinite number of irreps labeled by integer value of 2\(j\): the \(\textit{trivial representation}\) has \(j=0\) and the generators act as \(T^A=0\).
Fundamental Representation of \(\mathfrak{su}(2): \underline{2}\): The \(\textit{fundamental representation}\) has \(j=\frac{1}{2}\) and the generators can be expressed in terms of the Pauli matrices: \(T^A=\frac{\sigma^A}{2}\).
Adjoint Representation of \(\mathfrak{su}(2): \underline{3}\): The \(\textit{adjoint representation}\) has \(j=1\) and \((T^A)_{BC}=-i\epsilon^{ABC}\).
Rank of a Lie Algebra: The \(\textit{rank}\) of a Lie algebra is the number of Casimir operators.
Clebsch-Gordon Coefficients: In the physics literature, the \(\textit{Clebsch-Gordon coefficients}\) are the coefficients of the decomposition of the Kronecker product of representations (of e.g. \(\mathfrak{su}(2)\)) into irreps.
Group Manifold: When any group element can be viewed as a point on a manifold, that manifold is called \(\textit{group manifold}\). Each Lie group can be understood as a manifold.
Simply Connectedness: A manifold is \(\textit{simply connected}\) if every closed curve can be contracted to a point.
Universal Covering Group: A Lie group \(G\) is called the \(\textit{universal covering group}\) of another Lie group \(H\), when they are generated by the same Lie algebra but the group manifold of \(G\) is simply connected as opposed to the group manifold of \(H\).
Lie Group: A \(\textit{Lie group}\) \(G\) is a smooth differentiable manifold which is also a group, and where the group multiplication \(\ast\) has the following properties: \(i)\) \(\ast : G\times G\rightarrow G\) with \((g_1,g_2)\rightarrow g_1\ast g_2\) is a smooth map; \(ii)\) The inverse map \(G\rightarrow G\) with \(g\rightarrow g^{-1}\) is a smooth map.
Matrix Lie Groups: \(\textit{Matrix Lie groups}\) are Lie groups that are subgroups of GL\((n,\mathbb{C})\).
Connected Group: A Lie group is \(\textit{connected}\), if any two points are connected by a smooth curve.
Compact and Non-compact Lie Algebra: A Lie algebra is called \(\textit{compact}\) when it is associated to a compact Lie group. It is called \(\textit{non-compact}\) if the associated Lie group is non-compact.
Singleton Representation: A \(\textit{singleton representations}\) is a representation in which the generators are constructed from a single oscillator. The singleton representation of \(\mathfrak{so}(2,1)\) was discussed in the lecture.
Defining/Fundamental Representation of \(\mathfrak{su}(3)\) and Gell-Mann Matrices: The \(\textit{defining representation}\) of the Lie algebra \(\mathfrak{su}(3)\) is given by \(3\times 3\) traceless anti-hermitian matrices. The generators can be written as \(T^A=\lambda^A/2\), where \(\lambda^A\) denotes the \(\textit{Gell-Mann matrices}\) and \(A=1,\dots,8\). In physics this is also called the \(\textit{fundamental representation}\), while in mathematics the term fundamental representation is used differently.
Roots of the Lie Algebra \(\mathfrak{su}(3)\): In the case of the Lie algebra \(\mathfrak{su}(3)\) we associate two-component root vectors \(\beta^{(a)}\) to the generators according to \([H^i,E_{\beta^{(a)}}]=\beta^{(a)} E_{\beta^{(a)i}}\), with \(i=1,2\) and \(a=1,2,3\). Here \(H^i\) denotes the Cartan generators of and \(E_{\beta^{(a)}}\) the raising and lowering generators. The root vectors obtained in this way are in the \(H\)-basis. Depending on the application, other bases may be more convenient. This construction generalizes to other Lie algebras.
Root Diagram of the Lie Algebra \(\mathfrak{su}(3)\): The two-dimensional plot of the roots of \(\mathfrak{su}(3)\) in the \(H^1\)-\(H^2\) plane.
Positive Roots/\(\alpha\)-Basis of the Lie Algebra \(\mathfrak{su}(3)\): \(\textit{Positive roots}\) of \(\mathfrak{su}(3)\) are those with positive \(H^2\)-component, or positive \(H^1\)-component if the \(H^2\)-component is zero.
Dependent/Simple Roots of the Lie Algebra \(\mathfrak{su}(3)\): We call \(\beta^{(2)}\) a \(\textit{dependent root}\) since it can be expressed in terms of the \(\textit{simple roots}\) \(\alpha_1\equiv\beta^{(1)}\) and \(\alpha_2\equiv\beta^{(3)}\) with positive coefficients: \(\beta^{(2)}=\beta^{(1)}+\beta^{(3)}\). The basis of root space spanned by the positive roots is called the \(\alpha\)-basis.
Cartan Subalgebra/Cartan Generators of \(\mathfrak{su}(3)\): The subalgebra of \(\mathfrak{su}(3)\) spanned by the \(\textit{Cartan generators}\) \(H^1,H^2\).
Cartan Matrix: The matrix defined as \(A_{ji}=2\frac{(\alpha_i,\alpha_j)}{(\alpha_j,\alpha_j)}\) with positive roots \(\alpha_i\) and scalar product \((\cdot,\cdot)\).
Dynkin Diagram: A \(\textit{Dynkin diagram}\) is a graphical representation of a semisimple Lie algebra. To construct it draw a point for each simple root. Then connect these points by lines depending on the angles between the two respective roots. If the angle is \(\frac{\pi}{2}\) the roots are not connected, if the angle is \(\frac{2}{3}\pi\) they are connected by a single line, if the angle is \(\frac{3}{4}\pi\) they are connected by a double line and if the angle is \(\frac{5}{6}\pi\) they are connected by a triple line.
Dynkin Diagram of \(\mathfrak{su}(3)\): The Dynkin diagram of \(\mathfrak{su}(3)\) is given by two dots connected by a single line.
Weyl Reflection: The \(\textit{Weyl reflection}\) with respect to a root is the reflection about the direction of this root.
Weyl Chamber of the \(\mathfrak{su}(3) \) Root System: The \(\omega\)-basis of the \(\mathfrak{su}(3) \) root system is defined by the vectors \(\omega_1=\left(\frac{1}{2} ,\frac{1}{2\sqrt{3}} \right)\) and \(\omega_2=\left(0 ,\frac{1}{\sqrt{3}} \right)\). The \(\textit{Weyl chamber}\) is defined as the area between (and including) these two vectors.
Weights: The \(\textit{weights}\) of a state (basis element) of a finite-dimensional representation of a semisimple Lie algebra are defined as the eigenvalues of this state corresponding to the Cartan generators. The roots are the weights of the adjoint representation.
Weight Diagram: The \(\textit{weight diagram}\) of a representation is constructed by plotting the weights of all the basis elements of the representation space.
Highest Weight State: An element of the representation space of a semisimple Lie algebra is called \(\textit{highest weight state}\) if it is an eigenstate of the Cartan generators and annihilated by all generators associated with positive roots.
Conjugation: In the context of Lie algebras, for example \(\mathfrak{su}(2) \) and \(\mathfrak{su}(3) \), \(\textit{conjugation}\) is a natural operation because \(\mathfrak{su}(2) \) and \(\mathfrak{su}(3) \) act on complex vectors.
Cartan-Weyl Basis: The \(\textit{Cartan-Weyl}\) basis of a semisimple Lie algebra is a basis in terms of the Cartan generators \(H_i\) and the generators \(E_\alpha\) indexed by roots and their negatives with commutation relations. Its defining commutators are \([H_i,H_j]=0\), \([H_i,E_\alpha]=\alpha_iE_\alpha\), \([E_\alpha,E_{-\alpha}]=\alpha_iH^i\), \([E_\alpha,E_\beta]=N_{\alpha\beta}E_{\alpha+\beta}\) and \(N_{\alpha,\beta}=0\) if \(\alpha+\beta\) is not a root.
Chevalley Basis: The \(\textit{Chevalley basis}\) of a simple Lie algebra is distinguished by the fact that all structure functions are integer. It is a refinement of the \(\textit{Cartan-Weyl}\) basis.
Defining Representation: For a matrix Lie group of \(n\times n\) matrices, the \(\textit{defining representation}\) acts on an \(n\)-dimensional vector. In physics is also known as fundamental representation.
Fundamental Representation: In mathematics, a \(\textit{fundamental representation}\) is a finite dimensional irreducible representation of a semisimple Lie group or algebra whose highest weight is a fundamental weight.
Complexification: The \(\textit{complexification}\) of a Lie algebra is the same Lie algebra but allowing for complex linear combinations of its elements.
Schur-Weyl Duality: The \(\textit{Schur-Weyl duality}\) relates finite dimensional irreducible representations of the symmetric group \(S_n\) and the general linear group, acting on tensor product spaces. Here the symmetric group acts by permuting the factors of the product space.
Cartan Subalgebra: A \(\textit{Cartan Subalgebra}\) is a maximal commuting subalgera of a Lie algebra.
Rank: The \(\textit{rank}\) of a Lie algebra is the dimension of any of its Cartan subalgebras.
Racah's Theorem: \(\textit{Racah's theorem}\) states that the rank of a complex semisimple Lie algebra is equal to the number of Casimirs.
Simple Lie Algebra: A Lie algebra is \(\textit{simple}\) if it is non-Abelian and has no nonzero proper ideals.
Adjoint Representation: The \(\textit{adjoint representation}\) of an \(N\)-dimensional Lie algebra is (in our conventions) defined via \(i\,(T^A_{\mathrm{adj}})^B_C=f^{AB}_C\) and acts on an \(N\)-dimensional vector space.
Killing Form: The Killing form \(\kappa\) is defined via the adjoint representation as \(\kappa(X,Y)=\mathrm{tr}(\mathrm{ad}(X)\cdot\mathrm{ad}(Y))\), where \(X,Y\) are Lie algebra elements. In terms of Lie algebra generators \(T^A\) we have \(\kappa^{AB}=-f^{AC}{}_D\,f^{BD}{}_C=\mathrm{Tr}(T^A_{\mathrm{adj}}T^B_{\mathrm{adj}})\).
Semisimple Lie Algebra: A Lie algebra is \(\textit{semisimple}\) if it is a direct sum of simple Lie algebras. Equivalently, its Killing form is non-degenerate.
Quark Model: The \(\textit{quark model}\) is a model for fermionic matter. In particular, some particles were explained in 1964 using quarks (anti-quarks) with three flavors \((u,d,s)\) that transform in the \(\underline{3}\) (\(\bar{\underline{3}}\)) representation of \(\mathfrak{su}(3) \).
Mesons and Baryons: Bound states of a quark and an anti-quark are called \(\textit{mesons}\). Bound states made of three quarks are called \(\textit{baryons}\).
Color SU(3): Apart from the \(\textit{flavor}\) \(SU(3)\), which is not an exact symmetry due to different quark masses, quarks transform under a \(\textit{color \(\mathrm{SU(3)}\)}\), which is the gauge group of quantum chromodynamics (QCD).
Structure Constants: Lie algebra elements \(X^A\), \(A=1,..,N\) satisfy \([X^A,X^B]=\,i f^{AB}_C\,X^C\), where the \(f^{AB}{}_C\) are called \(\textit{structure constants}\). Note the conventional factor of \(i\).
Root Vector (or simply Root): A Lie algebra can be defined by \(r\) (the rank) mutually commuting (Cartan) generators and with the remaining elements \(X^a\) characterized by a \(\textit{root vector}\) \((\beta^1(a),...,\beta^r(a))\) with \([H^i,X^a]=\beta^i(a)X^a\). The roots are the weights of the states in the adjoint representation.
Positive Root: A root is called \(\textit{positive}\) if its first non-zero component of eigenvalues under \(H^1,\dots,H^r\) is positive. This is a basis dependent statement. In the context of \(\mathfrak{su}(3)\) we swapped the order of \(H^1,H^2\) following the convetion of Ramond.
Simple Root: A positive root \(\alpha_i\) is called simple if it cannot be expressed as a linear combination of other positive roots with positive coefficients.
Cartan Matrix: The \(\textit{Cartan Matrix}\;A\) is characteristic of a Lie algebra with its components defined in terms of simple roots \(\alpha_j\) as \(A_{ij}=2\frac{(\alpha_j,\alpha_i)}{(\alpha_j,\alpha_j)}\).
Dynkin Diagram: A \(\textit{Dynkin diagram}\) is a graph containing the simple roots of a Lie algebra. Short roots are depicted by filled dots and long roots are depicted by unfilled circles. Depending on the relative angles between the roots in root space, roots are connected by 0,1,2 or 3 lines.
Classical Lie Algebras: In the classification of compact simple Lie algebras one finds four infinite families of so-called \(\textit{classical Lie algebras} \). They are called \(A_n,B_n,C_n\) for\(n\geq 1\) and \(D_n \) for \(n\geq 2\). They correspond to the algebras \(\mathfrak{su}(n+1,\mathbb{C}),\mathfrak{so}(2n+1,\mathbb{C}),\mathfrak{sp}(2n,\mathbb{C}),\mathfrak{so}(2n,\mathbb{C}) \), respectively.
Exceptional Lie Algebras: Besides the infinite families of classical Lie algebras, the classification of compact simple Lie algebras also contains five \(\textit{exceptional} \) Lie algebras, which are called \(G_2,F_4,E_6,E_7 \) and \( E_8 \).
Simply Laced Lie Algebras: A compact semi-simple Lie algebra is called \(\textit{simply laced} \) if its Dynkin diagram does not contain double or triple lines. Explicitly these are the classical families \(A_n,D_n \), as well as the exceptional Lie algebras \(E_6,E_7,E_8\).
Fundamental Weights: For simple roots \(\alpha_i\), the \(\textit{fundamental weights}, \mu_i \) are defined by \(\frac{2(\mu_i,\alpha_j)}{(\alpha_j,\alpha_j)}=\delta_{ij}\) for \(i,j=1,..,r\), where \(r\) is the rank.
Dynkin Labels: A state of a representation with weight \(w=\sum_{i=1}^r a_i\mu_i\) can be labeled by the \(\textit{Dynkin labels}\) \(\lambda=(a_1,...,a_r)\).
Weyl Chamber: When all the Dynkin labels \(a_i\) are positive, the state is said to be in the \(\textit{Weyl chamber}\) of the algebras lattice, or on its boundary if any of its entries are zero.
Height or Level of a Representation: The \(\textit{height}\) or \(\textit{level of a representation}\) is given by the dot product of the vector of Dynkin labels of the highest weight state with the level vector \(R=(R_1,...,R_r)\) with \(R_i=\sum_j(A^{-1})_{ij}\), where \(A\) is the Cartan matrix.
Fundamental Representation: The \(\textit{fundamental representations}\) are those with Dynkin labels \((0^{p-1}\,1\,0^{r-p})\) for \(p=1,...,r\) (by \(0^i\) we mean 0 repeated \(i\) times).
Coxeter Number: The \(\textit{Coxeter number}\) \(h\) of the algebra is given by \(h=1/2 l_{\mathrm{adj}}+1\), where \(l_{\mathrm{adj}}\) is the height of the adjoint representation. It is related to the dimension \(D\) and the rank \(r\) of the Lie algebra via \(D=r(h+1)\).
Weyl Group: Repeated actions of the Weyl reflections generate the finite \(\textit{Weyl group}\).
Clifford Algebra: The generators of the \(\textit{Clifford Algebra}\), \(\gamma_a\), satisfy \(\{\gamma_a,\gamma_b\}=2\delta_{ab}\).
Lie Super Algebra: A Lie super algebra is an extension of a Lie algebra that contains even and odd elements. The even elements of the Lie algebra obey commutation relations and the odd elements anti-commutation relations.
Graded Commutator: The \(\textit{graded commutator}\) of a Lie superalgebra \(\mathfrak{g}\) reads \(\{X,Y]= XY- (-1)^{xy} YX\) with \(X,Y\in \mathfrak{g}\) and \(x=0\) for \(X\) even and \(x=1\) for \(X\).
Graded Jacobi Identity: The \(\textit{graded Jacobi identity}\) of a Lie super algebra reads \(\{X,\{Y,Z]]+\{Y,\{Z,X]]+\{Z,\{X,Y]]=0\).
Lorentz and Poincare Group: The group of Lorentz transformations is identified with \(O(3,1)\). The \(\textit{proper Lorentz group}\) is identified with \(\mathrm{SO}(3,1)\). The Poincare group is the semidirect product of the Lorentz group with translations in Minkowski space.
Conformal Group: The \(\textit{conformal group}\) is identified with \(SO(4,2)\).
Pauli-Lubanski Vector: The \(\textit{Pauli-Lubanski vector}\) is defined as \( W_{\mu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}P^{\nu}M^{\rho\sigma}\). Its square yields a Casimir operator for the Poincaré algebra.
Dilatations: Transformations of the form \( x^{\mu}\rightarrow \lambda x^{\mu}\) with \( \lambda\neq 0 \) are called \(\textit{dilatations}\).
Special Conformal Transformations: Transformations of the form \( x^{\mu}\rightarrow \frac{x^{\mu}-c^{\mu}x^2}{1-2c^{\nu}x_{\nu}+c^2x^2}\) with \( c^{\mu}\) some four-vector, are called \(\textit{special conformal transformations}\). Note that they are non-linear.
The Coleman-Mandula Theorem and its Assumptions: The \(\textit{Coleman-Mandula Theorem}\) states that the most general symmetries of the scattering matrix are a direct product of the Poincaré group with internal symmetries, assuming that these symmetries are captured by a Lie algebra, we are considering scattering of massive particles and that the spacetime dimension is greater than two. Dropping these assumptions leads to exceptions of the theorem, namely conformal symmetry, supersymmetry and integrability, respectively.