This appendix, a kind of crib sheet,
collects some relevant information on
certain Lie algebras and Lie groups.
Marius Sophus Lie (1842-1899)
It does not cover all Lie algebraic structures.
The main interest is in symplectic, unitary,
and pseudounitary groups of Lorentz signature
(+ + ... + -) and their algebras.
Specifically, not much is said about nilpotent Lie algebras.
The main references consulted for this assemblage are: [Hermann 1965], [Chevalley 1946], [Pontryagin 1966], [Wolf 1967], [Kobayashi 1969], [Helgason 1962], [Bourbaki 1966], [Jacobson 1962]. Specific references in this appendix, within the above, generally indicate where further material can be be found in addition to simply specifying a source for a statement.
For more information on the Internet that may possibly induce less of a headache, or a headache of a different kind try the following remote links.
A Lie algebra L is a linear space spanned by a basis X_k, and possessing an antisymmetry product [.,.] that obeys
[X_i, X_j] = c_i_j^k X_k (B.1)
over some field K, where [., .] is the antisymmetric Lie product,
and real c_i_j^k are the structure constants of the algebra, and
associativity is replaced by (B.2) below.
The Einstein summation convention is assumed, where a lower index and its repetition as an upper index within the same term is assumed to be summed over. The classical algebras are defined as the infinitesimal algebras of matricial groups (the defining representation), and then the Lie product is realized as a matrix commutator. In this appendix, we will only be concerned with the cases when K is R, the real field or C, the complex field, although one could also consider vector spaces L over arbitrary finite fields. Cf. Appendix J Relation B.1 immediately implies that the structure constants are antisymmetric in their two lower indices. While commutators naturally obey the Jacobi identity:
[[X_i, X_j], X_k] + [[X_j, X_k], X_i] + [[X_k, X_i], X_j] = 0 (B.2)
in abstracto, before realizing the Lie product as commutators in a
representation, (B.2) must be assumed axiomatically.
Then, with the Einstein summation convention in force
c_i_j^k c_k_n^p + c_j_n^k c_k_i^p + c_n_i^k c_k_j^p = 0 (B.3)
The algebra L is, of course, non-associative, since, generally
[[X_i, X_j], X_k] =/= [X_i, [X_j, X_k]]
Again, the "associative law" is replaced by (B.2).
The dimensional of a Lie algebra is the cardinality of the set {X_k} of its basis elements. The set of basis change elements is the the general liear group, either gl(n, R) or gl(n, C) depending on whether the algebra is real or complex, and the indicies of Liealgebraic formulas are tensor indicies with respect to this group.
A linear map of L into a set of linear transformations
on a vector space (the carrier space) is called a
representation of L if
the Lie products are mapped to commutators.
This is an embedding
of a nonassociative algebra into an associative algebra.
A Note On Algebra Representations: The mathematical literature represents the compact algebra basis elements as skew-Hermitean matrices (with pure imaginary eigenvalues). These map by exp with a real parameter to compact one-parameter subgroups. The physics literature, on the other hand, represents the compact algebra basis by Hermitean matrices (with real eigenvalues). Lie algebras frequently appear in association with quantum theory, and the latter are more closely related to the physicist's notion of an observable. The choice of the latter reverses the sign in some of the classical mathematical theorems.
It is worth noting that the representations of algebraic structures, themselves, form a ring [Appendix J], that groups extend naturally to group rings, and that group representaions can be embedded in an algebra of representations where the algebra operations are direct product and direct sum, and the structure constants of the algebra are generalizations of the Clebsch-Gordon coefficients familiar to physicists. A similar situation also exists for the structure of abstract groups themselves. That the entire category of groups has such an algebraic structure is not at all unique and is an entrance into the ideas of category theory.
Lie algebras can be classified by the structure of their Cartan metric or Killing form. The Cartan metric is defined by
g_ij := c_i_m^n c_j_n^m (B.4)
The Killing form is defined in terms of adjoint representation
of the algebra:
Associate with any element A of L a linear transformation adj(A) defined by the left action of the algebra on itself. For any Y in L, [A, Y] is also in L. Define the
adj(A)Y = [A, Y] (B.5)
In particular, for fixed k, let A = X_(k), and represent Y on
the algebra basis X_j, so that
Y = y^(j) X_j
then
adj(X_(k)) = y^(j) [X_(k), X_j]
= y^(j) c_(k)_j^i X_i (B.6)
where the y^(j) and the X_i transform
contragrediently to each other under the group of
basis transformations in the algebra.
The adjoint representation of the group is
irreducible for any simple Lie group.
The adjoint action of the algebra on itself also
defines an adjoint action of the associated group on
the algebra. If the group is connected,
any element of the group can be parametrized as a product of one parameter
subgroups. See
[Hermann 1965], p. 10,
and more generally
[Rickart 1960], p. 210.
For g an element of the Lie group
g(theta_1, theta_2, ... theta_n) =
exp( +theta_1 X_1 ) exp( +theta_2 X_2 ) ... exp( +theta_n X_n ) (B.7a)
g^(-1)(theta_1, theta_2, ... theta_n) =
exp( -theta_n X_n ) ... exp( -theta_2 X_2 ) exp( -theta_1 X_1 ) (B.7b)
For k fixed, from the Baker-Campbell-Hausdorff formula
exp( +theta_kX_k ) X_j exp( -theta_kX_k ) =
infinity
SIGMA (theta_k)^m/m! C(X_j: X_k, m) (B.8)
m=0
infinity
= SIGMA (theta_k)^m alpha_kj^i(m) X_i (B.9)
m=0
with C(X_j: X_k, m), the m-fold commutator as defined in
equation
(7.4),
and [SIGMA_m(theta_k)^m alpha_kj^i(m)],
for fixed k, the matrix elements
of the adjoint representation of the group element exp(+theta_k X_k).
The alpha_kj^i(m) are m-th order polynomials in the structure constants.
Then,
g(theta_1, theta_2, ... theta_n) X_j g^(-1(theta_1, theta_2, ... theta_n) =
SIGMA a_k(theta_1, theta_2, ... theta_n) X_k (B.10)
k
where the coefficients
a_k(theta_1, theta_2, ... theta_n) are real
functions of the parameters theta_k and the structure constants.
This partially describes the adjoint action of the group on its algebra.
The form of
(B.8)
can be compared with a similar relation between the
spin and fundamental representations of the orthogonal groups.
In the adjoint representation of the algebra then, the matrices representing the basis elements are formed from the structure constants. The classification of semisimple Lie algebras is essentially the theory of the adjoint representation.
The Killing form on L is defined as a real valued symmetric bilinear form
B( X, Y ) = Tr( adj(X) adj(Y) ) (B.11)
In particular then,
B( X_i, X_j ) = Tr( adj(X_i ) adj(X_j ) )
= g_ij (B.12)
relating the Killing form to the Cartan metric.
The Cartan metric and the structure constants transform under
the linear group of basis transformations as tensors.
Call the lower indices covariant and the upper indicies contravariant.
g_ij, is symmetric in its two covariant indicies,
and can be used to lower indicies. If it is non-singular, an inverse
can be defined as usual with upper indicies:
g_ik g^kj = delta_i^j
with delta_i^j, the invariant Kronecker delta.
The following is a set of elementary definitions:
A Lie algebra L is Abelian iff
[L, L] = 0
A subalgebra in L is a subspace H of L such
that H is closed under the operations of the algebra, i. e.
[H, H] is contained in H
and
H + H is contained in H
An ideal in L is a subspace J of L such that
[L, J] is contained in J
Being an ideal is a stronger requirement than being a subalgebra.
An invariant subalgebra in L is a subalgebra K of L such that
[L, K] is contained in K
Therefore, an invariant subalgebra is an ideal in L.
The derived series for L is defined recursively by
L^(a) := [L^(a-1), L^(a-1)]
with L^(0) := L
Thus if a > b, L^(a) is an invariant subalgebra of L^(b).
If the Lie algebra basis is finite, the derived series must terminate in
L^(a) = L^(a-1)
The upper index (a) obviously labels "equal levels of commutation".
If it also terminates in
L^(a) = 0
L is said to be solvable.
The lower central series for L is defined by
L_(a) := [L, L_(a-1)]
with L_(0) := L
The lower index (a) obviously labels "levels of commutation with L".
The lower central series is also a series of invariant subalgebras.
If it terminates in
L_(a) = 0
L is said to be nilpotent.
Nilpotency immediately implies solvability, but the converse does
not hold.
A group of translations in a Euclidean space is Abelian, so that all commutators of the Lie algebra of generators vanish, being thus an archetypal example of a nilpotent Lie algebra. Another immediately important nilpotent Lie algebra is the Heisenberg Lie algebra expressed by the canonical commutation relations of quantum mechanics.
We are interested here in the Lie algebras classified as simple, semisimple and nilpotent. [It is a theorem of Levi and Malcev that an arbitrary Lie algebra is composed of solvable part and a semisimple part in that it contains a solvable ideal, and further that the algebra of cosets modulo that ideal is semisimple.] The classification of solvable Lie algebras is apparently still rather incomplete. This is an interesting quirk (since we naturally assume commutativity is simpler than non-commutativity), that the solvable algebras most closely related to Abelian algebras are more problematic in classification, than the non-commutative semisimple algebras.
A Lie algebra L is simple if it contains no non-zero invariant subalgebras (ideals). Simplicity implies semisimplicity. The classical simple Lie algebras have been classified into four series, plus five exceptional (they do not fit into the series) algebras:
Cartan name Dimension
A_n su(n+1), n > 1 (n+1)^2 - 1
B_n so(2n+1), n > 2 2n^2 + n
C_n sp(2n), n > 3 2n^2 + n
D_n so(2n), n > 4 2n^2 - n
E_2 g(2) 14
F_4 f(4) 52
E_6 e(6) 78
E_7 e(7) 133
E_8 e(8) 248
[Helgason 1962],
p. 146. SO(4) is not simple, but is semisimple.
The subscript of the Cartan name is the rank of the algebra, also
of the group. [See below.]
A Lie algebra L is semisimple if it contains no non-zero Abelian ideals. It turns out that a semisimple Lie algebra (group) can be represented as a direct sum (direct product) of simple non-Abelian Lie algebras (groups).
Crudely speaking, Lie algebras exponentiate to Lie groups. More precisely, any linear Lie algebra can be integrated to a connected and simply connected Lie group. There are, however, the following caveats: The algebra elements correspond to continuous one parameter subgroups. A finite subgroup (not a Lie subgroup) has no representative structure in the algebra. A standard example is that O(n) which is not connected and has a finite subgroup consisting of two elements (the identity and spatial inversion) which relates the disconnected components, has the same algebra as SO(n). The connected component of O(n) containing the identity is then isomorphic to SO(n). Generally, subalgebras exponentiate to subgroups which may or may not be closed.
Even though here Lie algebras and groups here are specifically semisimple, it is worth noting that the above caveats are not all there is to consider in the case of nilpotent Lie algebras. The abstract Heisenberg algebra:
[p, q] = E
[p, E] = 0
[q, E] = 0
is the foundation of nilpotent Lie algebra theory.
Its representations inevitably involve unbounded operators
[Section I],
and
[Section III],
and not all of the representations of the algebra exponentiate
to a representation of the group.
That is generally true of nilpotent Lie algebras.
Since the Lie algebra can in fact be identified with the set of left-invariant vector fields on the group, [Chevalley 1946], p. 101 it is essentially the story of the local structure of the group manifold. In the case of a semisimple Lie group, the group manifold can be endowed with with an affine connection derived from the structure constants, with which the manifold becomes an Einstein space, that is a Riemannian manifold with a Riemann-Christoffel curvature tensor that satisfies the vacuum n-dimensional Einstein equations. The Cartan metric g_ij extends to the Riemannian metric on the group manifold. [Yano 1955]. , The local story does not tell all, as the global topology of the group manifold is not determined by the local structure. The local group homomorphism between SO(3) and SU(2) is the standard example. As manifolds, SO(2) is doubly connected, while SU(2) is simply connected. All the Special orthogonal groups SO(n), n >= 2 are doubly connected, hence the existence of the groups Spin(n) as their simply connected covering groups [Chevalley 1946], pp. 61-67. Spin(n) acts on a space of spinors. See below.
Informally speaking, a Lie algebra over C is called compact if it integrates to a Lie group, which as a manifold is compact. The parameters of the group then have bounded range. For compact Lie groups, all the irreducible matrix representations are finite dimensional and equivalent to unitary representations [The Peter-Weyl theorem].
Cartan's and other Criteria
1. A Lie algebra L is semisimple if
and only if g_ij is non-singular.
2. If g_ij is singular then L it has a solvable ideal.
3. If L is semisimple, then L is compact iff the Killing form
is negative definite. [Weyl 1925]
(if skew-Hermitean basis, positive if Hermitean)
If a Lie group is not semisimple, it can still
have a compact Lie algebra: the Abelian group of translations
in a Euclidean space, is not semisimple and is noncompact.
Its Lie algebra is also the Lie algebra of the
compact group that is the product of 1-tori.
The Lie algebra is then compact while the translation group is not.
Cosets and Homogeneous Spaces
Let G be a Lie group and H a closed subgroup. Two elements g_1,
g_2 of G are said to be congruent modulo H iff
g_1 H = g_2 H
that is, their left cosets coincide. This is an equivalence relation
between the elements of G. The space of cosets is denoted by G/H.
The natural map of G -> G/H is that which maps g
to its coset gH. G/H inherits a topology from G, and the natural map
is continuous. G operates continuously and transitively on G/H,
and G/H is called a homogeneous space for G.
See also [Appendix D].
The G/H is then the base space of a principle bundle with fibre H, and group G. [Steenrod 1951]
Cartan Canonical Decomposition
The following is an existence theorem of E. Cartan:
Let L be semisimple, then L has a direct sum decomposition
L = K DIRECT-SUM M where K is a subalgebra of L , i.e.,
[K, K] is contained in K
and M is an invariant space of adj(K):
[K, M] is contained in M
[M, M] is contained in K
The Killing form of L restricted to K is negative
definite
(if skew-Hermitean basis, positive definite if Hermitean)
and restricted to M is positive definite.
(if skew-Hermitean basis, negative definite if Hermitean)
K exponentiates to a maximal compact subgroup, and M
exponentiates to a subspace of the group exp( L )
invariant under the adjoint action of the group.
For the adjoint action of the group as spaces,
exp( M ) = exp( K^(-1) ) exp( A ) exp( K )
where A is a Cartan subalgebra. See e.g., [Helgason 1962], p. 334.
For the four series of complex simple Lie algebras:
sl(n+1, C), n > 1
so(2n+1 C), n > 2
sp(n, C), n > 3
so(2n, C), n > 4
n is the corresponding rank. A regular element of a Cartan subalgebra
of u(n) is an Hermitean diagonal matrix with distinct diagonal elements.
The Center of a Lie Group
The center of a group is defined as the set of group elements that
commute with all the elements of the group.
Center
sl(n+1, C), n > 1 Z_n+1
so(2n+1, C), n > 2 Z_2
sp(n, C), n > 3 Z_2
so(2n, C), n > 4 Z_4 for n odd,
Z_4 DIRECT-SUM Z_4 for n even
n is the corresponding rank. Z_(n+1) is the cyclic
group of order n.
Root Vectors in Root Space, Weyl Canonical Form
In the adjoint representation of a semisimple Lie algebra,
the algebra basis is represented by matrices whose elements
are the structure constants. It has the advantage of having
a purely algebraic construction from the abstract algebra itself.
Define the completely covariant structure constants by
c_ijk := g_km c_ij^m (B.13)
then
adj(X_k) = (c_ij)_k (B.14)
The structure constants are only determined up to a basis transformation,
i. e., any non-singular linear transformation on the X_k.
The structure of the Lie algebra is determined by the structure constants,
or equivalently the invariant spectral structure of the adjoint representation,
which is investigated through the eigenvalue equations of the form.
adj(X) Y = w Y (B.15)
where X is given, and w and Y are sought.
The roots, w are the roots of the characteristic equation
Det( adj(X) - w I ) = 0 (B.16)
The fact that the various w can be complex, immediately leads to the
extension of the field of the algebra from C to C,
so that algebraically one is determining the structure of complex Lie
algebras. If L is a real Lie algebra associated with a
Lie group, let L^* denote its complex extension.
Since X commutes with itself, adj(X)X = 0. That is, Y = X with w = 0 always solves the eigenvalue problem. The multiplicity of this zero root depends on the structure of X: the multiplicity of the zero root is equal to the rank; no other roots are degenerate. As X ranges over L^*, the minimal value of the multiplicity is equal to the rank of L^*, (the "rank theorem"). An element of L^* with rank multiplicity of its zero root is a regular element. For a given, X, the regular elements X_0 such that
adj(X)X_0 = [X, X_0] = 0 (B.17)
constitute a Cartan subalgebra. In the Cartan form above, where
For i, j, = 1, 2, ..., r, where r is the rank of
the algebra, and
[H_i, H_j] = 0 (B.18a)
[H_k, E_alpha] = alpha_k E_alpha (B.18b)
alpha_k is considered as the k-th component of
a root vector in an r-dimensional Euclidean
rootspace.
The roots are the weights considered above of the adjoint representation,
where the commuting H_i are associated with degenerate
zero eigenvalue, and the E_alpha are
associated with the nonzero nondegenerate eigenvalues (roots).
The nondegeneracy implies that the vectors alpha
are distinct and can be used to label the E's.
A_(ij) := 2(alpha_i . alpha_j)/(alpha_j . alpha_j) (B.19)
a quotient of "dot products"
The set of root vectors can be proven to have the properties
that:
1) the cosine of the angle subtended by two root vectors,
A_ij is an integer and alpha_i - A_ij a alpha_j
are roots.
2) If alpha not= 0 is a root then k alpha cannot be a root
unless k = (+|-)1. Then also, if alpha is a root,
-alpha is also a root.
3) There is a finite group, W( L ), the Weyl group,
of rotations and reflections in root space which leaves the
configuration of root vectors invariant.
The root
beta - A alpha is a reflection of beta through a hyperplane, called
a Weyl plane, that passes through the origin and is
perpendicular to A alpha.
From property 2), we have:
[E_alpha, E_beta] = N_alpha_beta E_(alpha_+beta) (B.20)
with N_alpha_beta not= 0.
Cf. [Carter 1972], pp. 52-57.
As a special case of (B.20)
[E_alpha, E_-alpha] = SUM alpha_i H_i = alpha . H (B.21)
is a linear combination of the H_i.
Nonisomorphic real Lie algebras may have isomorphic complex extensions, and complex Lie algebras generally have various real forms that are Lie algebras of Nonisomorphic Lie Groups. A complex semisimple Lie algebra admits one and only one, up to an isomorphism, real form that is compact. The idea of a complex Lie algebra can be introduced for the purpose of studying its various real forms. One can also then consider the concept of a complex Lie algebra for the purpose of introducing a complex Lie group, [Pontryagin 1966] p. 428.
All of the algebras su(n-k, k) for k = 0, 1, ..., n, complexify to sl(n, C)., while u(n-k, k) for k = 0, 1, ..., n, complexify to gl(n, C).
Malcev has considered the conditions under which a real Lie group can be embedded in a complex Lie group with the result that this cannot be done for a simple Lie group with infinite center.
is the set of linear mappings of L into itself that annihilates no element of L except the zero element. The Lie algebra gl( L ), of GL( L ) is then the vector space of all endomorphisms of L equipped with a bracket operation defined as commutator of the endomorphisms.
The Adjoint Group, Int( L )
Let L be a Lie algebra over C. The adj maps of L into itself are endomorphisms of L by commutator, and therefore adj( L ) is a subalgebra of gl( L ). This subalgebra has a group that is an analytic subgroup of GL( L ). Call this subgroup Int( L ).
Aut( L ), the group of all isomorphisms of L onto itself is a closed subgroup of GL( L ). Let PARTIAL( L ) be the Lie algebra of Aut( L ).
The inner automorphisms of L are the elements of the connected group Int( L ), which is a normal subgroup of Aut( L ). For s in Aut( L ), the inner automorphism
sigma: g -> s g s^(-1)
is an automorphism of Aut( L ), while the differential
in the neighborhood of the identity, (dsigma)_e
is an automorphism of PARTIAL( L ).
For s, g in G the Lie group of L, the inner automorphism
I(sigma): g -> s g s^(-1)
is an analytic isomorphism of G onto itself.
Then the differential at the identity
Adj(sigma) := dI(sigma)_e
is an automorphism of the Lie algebra L;
for sigma in G and X in L,
exp[ Adj(sigma) X] = sigma exp[ X ] sigma^(-1)
defines the adjoint action of the group on its algebra,
and the mapping
sigma -> Adj(sigma)
being a homomorphism (analytic) of G into GL(L),
is a representation of G, in fact,
the adjoint representation of G.
[Helgason 1962],, p. 117
An endomorphism D of L is called a derivation if
D([X, Y]) = [D(X), Y] + [X, D(Y)] (B.22)
adj(L) = PARTIAL(L) (B.23)
That is, every derivation is inner.
[Helgason 1962],, p. 122
[Helgason 1962],, p. 162ff If M is a C^(infinity) affinely connected space, with connection components GAMMA^r_(st) that are symmetric in the two lower indicies, (zero torsion), construct in the usual way the Riemann-Christoffel curvature tensor
R^q_(rst) = PARTIAL_s GAMMA^q_(rt) - PARTIAL_t GAMMA^q_(rs)
+ GAMMA^q_(.s) GAMMA^._(rt) - GAMMA^q_(.t) GAMMA^._(rs)
Then M is affine locally symmetric iff,
NABLA_p R^q_(rst) = 0
where NABLA_p is the covariant derivative.
This is generally regarded as a theorem and not the fundamental
definition.
Cf. [Helgason 1962],, p. 179,
"The Exponential Mapping and the Curvature"
Involutive Automorphisms
There are a number of representations that are fundamental in the sense that any finite dimensional representation can be found in the direct sum decomposition of some direct product of the fundamentals. The number of the fundamental representations is equal to the rank of the algebra. The fundamental representations for SU(n) are those that correspond to Young tableaux with one column and k boxes where k = 1, 2, ..., n-1, i.e., the fully antisymmetric tensors. It turns out that all these can, in fact be constructed from the 1 box representation, i.e., the defining representation.
If v^k are the contravariant components of a vector v of the carrier space of the defining IRREP of SU(n), and w^k is another such set, not collinear with v, then (v^k w^j - w^k v^j) is an antisymmetric contravariant tensor of rank 2. While the dimension of the carrier space is n, the space of these antisymmetric tensors n(n-1)/2 = (n 2), a binomial coefficient. Generally, for SU(n), there are fundamental IRREPs having dimension (n k), for k=1,2,...,(n-1). The defining IRREP is, of course, always the first fundamental IRREP. These are not the only IRREPS, but those fundamental IRREPS from which all other IRREPS are built.
For example, for SU(2), there is only one fundamental IRREP; it is the two dimensional "2" IRREP. For SU(3), there are two fundamental IRREPs, the 3 and 3*, where the * indicates the "Hodge star" involution that maps a tensorial quantity to its cognate pseudo tensor, in this case a vector to its pseudovector. For SU(4), the fundamental IRREPS are the 4, 6, 4*. For SU(5), there are the 5, 10, 10*, 5*, while for SU(6) there are the 6, 15, 20, 15*, 6*, and so forth, following the binomial pattern.
That there are more IRREPS for SU(n) than the fundamental IRREPS can be seen by constructing the Adjoint IRREP of SU(n),
(n) X (n*) = Adj(n²-1) + 1
If n=2, the adjoint REP (the action of the Lie group on its own Lie algebra) is 3 dimensional; the "1" is the trivial REP.
Repeating from above, the group manifold of any special orthogonal group SO(n), is not simply, but rather doubly connected; it has a simply connected covering group called Spin(n) whose representations provide the so called "double valued representations" of SO(n). [Chevalley 1946] The important fact of the so(n) and spin(n) Lie algebras is that they are structurally identical; the two valuedness arises only in the exp map of the algebras to their respective Lie groups. Below, we show the explicit construction of the defining IRREP of the so(n) algebra, and of the su(n) algebra, showing how so(n) is contained in the su(n). From this is can be seen that the spin(n) IRREP of so(n) extends to a Uspin(n) IRREP of su(n), so that every su(n) has a spin representation.
The spin IRREP of su(n) and all its fundamental IRREPS are pulled together in the context of a complex Clifford algebra since the spin representation acting on the generators of the Clifford algebra
S g_k(n) S! = Sum a^j_k g_j k induces a fundamental IRREP action on each of the subspaces of degree m in the Clifford algebra. The dimension of the subspace of degree m being, of course (n m).
Let L be semisimple, so the Killing form as an expression
of the Cartan metric on L as a vector space is non-degenerate.
The Cartan metric metric provides a map between L and its
dual space L^+, the space of linear functionals
acting on L.
The adjoint action of the group on L is then mapped by the
Cartan metric to an action of the group on L^+,
the coadjoint representation.
Every orbit in the coadjoint representation can be given a symplectic
manifold structure that is invariant under the action of the group:
Let X, Y be in L, and f be in L^+.
The adjoint and coadjoint representations are defined by
ad(X) Y = [X, Y]
and
<coadj(X)|Y> = - <f|adj(X)Y> = <f|[Y, X]>
If L_f is the subalgebra that is the stabilizer
of f, the elements of the coset space L/L_f
span the tangent space of the orbits at the point f.
A symplectic structure can be defined on L/L_f by setting
x V_f y = - y V_f x = <f|[y, x]>
for x, y in L/L_f.
The adjoint action of a Lie group on its algebra defined in equation (B.10) and in the preceding, exponentiates to the adjoint action of the group on itself. For the adjoint action of SU(n) on its algebra su(n), let h in su(n) and U in SU(n). The map
h -> U h U^(-1)
is the adjoint map. Any Hermitean matrix can be diagonalized by
a unitary transformation, so the adjoint orbits in su(n) are parametrized
by traceless diagonal Hermitean matrices. Consider the pencil of matrices
a G(n)
Its stability group in SU(n) is U(n-1). The corresponding adjoint
orbit is then the factor space SU(n)/U(n-1), the complex projective
space P_n(C).
If a group is semisimple, (Killing form is nondegenerate), coadjoint orbits can be identified with adjoint orbits by the Killing form. The adjoint-coadjoint structure defines a natural symplectic structure. This symplectic structure is invariant under SU(n). Also P_n(C) is symplectic since S^2 is symplectic. [Section XII],
Let a Lie group G act on some space M and let p in M.
G(x) in M, for x in M.
Then the stability subgroup H_p of G for the
action of G on M is defined as
H_p = {g : g(p) = p for g in G, p in M}
In an abstract Lie algebra, only the Lie product is defined; only in a specific representation does an associative product of algebra elements as the product of the representing matrices make sense. The enveloping algebra is the algebraic abstraction of making sense of an associative product of Lie algebraic elements.
If L is a Lie algebra with basis X_k, define the enveloping algebra A(L) as a factor algebra T(L)/J where T(L) is the tensor algebra over L and J is the two sided ideal in T(L) generated by elements of the form
(X DIRECT-PRODUCT Y - Y DIRECT-PRODUCT X) - [X, Y]
A(L) is then an algebra of infinite dimension, so
no finite representation is faithful.
The algebra Alg(Hilb(n)) is a nonfaithful representation
of the enveloping algebra A(U(G, n, C))
[Section XI],
The Casimir invariants are polynomials in the Lie algebra basis, (members of the universal enveloping algebra), that are invariants (i.e. scalars or multiples of the indentity) in any irreducible representation. Casimir discovered the invariant operator
c^(2) := g_kj X^k X^j = c_(kl)^i c_(ji)^l X^k X^j
which generalizes to
c^(m) := g_kj...st X^k X^j ... X^s X^t
where, with the Xs in the adjoint representation,
g_kj...st := Tr(X^k X^j ... X^s X^t)
in analogy to the definition of the Cartan Metric.[B.11]
There are m indicies, so that c_(m) is formally a polynomial
of "order" m.
Generally, all the Casimir invariants are not necessarily independent. The values of a number of of them, equal to the rank of the algebra, is sufficient to characterize an irreducible representation. Vector operators may also be constructed along similar lines by:
T^(m)_a = g_(akj...st) X^k X^j ... X^s X^t
so
c^(m) = T^(m)_a X^a = X^a T^(m)_a
The T^(m)_a are also not generally independent of one
another. [Okubo 1977]
The Euler decomposition of the group SO(3) or its covering group SU(2) that gives any rotation as a product of three rotations [Appendix C] can be generalized to semisimple Lie groups. See Cartan Canonical Decomposition above.
Let G be a connected Lie group with finite center, whose Lie algebra is L. For the canonical decomposition L = K DIRECT-SUM M, every element of G admits a decomposition
g = m k
where m in exp( M ), and k in exp( K ).
For the Lorentz covering group SL(2, C) any element g in SL(2, C) can be represented as a product,
g = KBK'
where K and K' are elements of the maximal compact
subgroup SU(2) and B = exp( A ) is a noncompact element of
the exponentiated Cartan subalgebra.
Similarly for g in U(n-1,1) where K and K' are elements of U(n-1) DIRECT-PRODUCT U(1) and B = exp( A ) is a diagonal matrix with
Among the low rank Lie algebras, the following isomorphisms occur: [Helgason 1962],, p. 351
su(2) = so(3)
= sp(1)
su(1, 1) = so(2, 1)
= sl(2, R)
so(4) = sp(1) DIRECT-PRODUCT sp(1)
= so(3) DIRECT-PRODUCT so(3)
= su(2) DIRECT-PRODUCT su(2)
so(3, 1) = sl(2, C)
so(2, 2) = sl(2, C) DIRECT-PRODUCT sl(2, C)
so*(4) = su(2) DIRECT-PRODUCT sl(2, C)
so(5) = sp(2)
so(3, 2) = sp(2, C)
so(4, 1) = sp(1, 1)
su(4) = so(6)
su*(4) = so(5, 1)
su*(3, 1) = so*(6)
sl(4, C) = so(3, 3)
su(2, 2) = so(4, 2)
so*(8) = so(6, 2)
The inhomogeneous groups attached to the orthogonal, unitary, pseudo-orthogonal and pseudo-unitary groups are groups of affine motions and can be realized as the semidirect products of the aforementioned groups with Abelian groups of translations.
A group G is a semidirect product of K by M iff:
i) K is a normal subgroup of G
ii) G = {km: k in K, m in M}
iii) K INTERSECT M = {e}
where e is the identity element of G.
A semidirect product of K by M is also an extension of K
by M, and is also called a split extension.
Topologically, the underlying space of U(n) is homeomorphic to T^1 CROSS SU(n), and its Poincare (covering) group is isomorphic to the additive group of integers. (The additive group of integers is the covering group of T^1 considered as an Abelian group) [Chevalley 1946], p. 61 SO(n), SP(n), U(n) and SU(n) are connected for n >= 1. [Chevalley 1946], p. 37 SP(n) and SU(n) are simply connected for n >= 1. [Chevalley 1946], p. 60 SO(n) is doubly connected for n >= 3. [Chevalley 1946], p. 37 For n >= 2: the coset spaces O(n)/O(n-1) and SO(n)/SO(n-1) are homeomorphic to the sphere S^(n-1); the coset spaces U(n)/U(n-1) and SU(n)/SU(n-1) are homeomorphic to the sphere S^(2n-1); [Chevalley 1946], p. 33
We can construct in Hilb(n), the defining representation of the Lie algebra so(n) by forming the basis for all real skew matrices:
Define, for k, j = 0, 1, ..., n-1
^e_(kj) := |n, k><n, j| - |n, j><n, k|
of which there are n(n-1)/2. It is clear that ^e_(kk) = 0
and that the ^e_(kj) for k not= j, are also skew-Hermitean.
Then,
[^e_(kj), ^e_(st)] =
^e_(kt) delta_(js) - ^e_(ks) delta_(jt)
+ ^e_(js) delta_(kt) - ^e_(jt) delta_(ks)
where delta_(ij) results from inner products
of the basis vectors,
hence the components of the tacitly assumed Euclidean metric
ground form on the carrier space, that is being preserved.
The Hermitean operator P(n) defined by equation
(2.7b) can be expressed
n-1
P(n) = (-i/sqrt( 2 )) SUM sqrt( k) ^e_(k-1)_k
k=1
The representations of U(n) are characterized by n integers that
satisfy
f_1 >= f_2 >= ... >= f_n
or sometimes more conveniently another set of integers
lambda_k := f_k + n - k
with
n >= k >= 1
which then satisfy
lambda_1 > lambda_2 > ... > lambda_n
Then the dimension of the representation is given by Weyl's formula
PI_(k<j) (lambda_k - lambda_j)
-------------------------------
1! 2! ... (n-1)!
[Weyl 1946], also, [Okubo 1975].
The Unitary Group SU(n)
Now enlarge the algebra so that the e\\^_kj become a subalgebra by adding the skew-Hermitean elements
^f_kj := i(|n, k><n, j| + |n, j><n, k|)
of which there are n(n+1)/2. The ^f_kk are diagonal and
nonvanishing. They will span an Abelian Cartan subalgebra.
These have the readily verifiable commutation relations
[^f_(kj), ^f_(st)] = - ^e_(js) delta_(kt) - ^e_(ks) delta_(jt)
- ^e_(jt) delta_(ks) - ^e_(kt) delta_(js)
and
[^e_(kj), ^f_(st)] = ^f_(kt) delta_(js) + ^f_(ks) delta_(jt)
- ^f_(jt) delta_(ks) - ^f_(js) delta_(kt)
Together the ^e_(kj) and ^f_(st) form a basis
for the defining representation of the algebra u(n)
with n^2 basis elements.
The commutation relations show that the ^e_(kj) span the subalgebra so(n)
and the ^f_(kj) span an invariant subspace of adj( so(n) ).
The real and Hermitean operator Q(n) defined by
equation (2.7a) can be expressed
n-1
Q(n) = (-i/sqrt(2)) SIGMA sqrt(k) ^f_(k(k-1))
k=1
Similarly one expresses the standard generators of the n-dimensional
UIRREP of su(2) by,
n-1
S_1(n) = (i/2) SIGMA [k(n-k)]^(1/2) ^f_(k(k-1))
k=1
n-1
S_2(n) = (i/2) SIGMA [k(n-k)]^(1/2) ^e_(k(k-1))
k=1
n-1
S_3(n) = (i/2) SIGMA [(1/2)(n-1) - k] ^f_(kk)
k=0
The Pseudounitary Group SU(n-1, 1)
If in the above construction, we make the replacements:
|n, k> -> gamma_(+|-)^(1/2)(n) |n, k>
<n, j| -> <n, j| gamma_(+|-)^(1/2)(n)
with gamma(n) as defined in [Appendix D], and define
gamma_(kj)(n) = <n, k|gamma(n)|n, j>
so that
e_(kj) = gamma_(+|-)^(1/2)(n) ^e_(kj) gamma_(+|-)^(1/2)(n)
we have a slightly different set of e_(kj) not all skew,
but instead gamma(n)-skew, that obey
[e_(kj), e_(st)] = e_(kt) gamma_(js)(n) - e_(ks) gamma_(jt)(n)
+ e_(js) gamma_(kt)(n) - e_(jt) gamma_(ks)(n)
which is the defining representation of the Lie algebra so(n-1, 1).
These extend as above to the generators of u(n-1, 1)
Similarly, make instead the replacements
|n, k> -> |n, k>_g := G_(+|-)^(1/2)(n) |n, k>
<n, j| -> g_<n, j| := <n, j| G_(+|-)^(1/2)(n)
(Note that complex conjugation of G_(+|-)^(1/2)(n)
in the map to the dual space is NOT taken.)
and define the matrix elements
G_kj(n) = <n, k|G(n)|n, j>
= <n, k| G_(+|-)^(1/2)(n) G_(+|-)^(1/2)(n) |n, j>
= g_<n, k|n, j>_g
then the resulting E_kj(n) are the generators of the Lie
algebra of the special "pseudo-orthogonal" subgroup SO(G, n) of SU(G, n)
which is conjugate to SO(n-1, 1) in GL(n, C). These obey the defining
commutation relations:
[E_kj(n), E_st(n)] = E_kt(n) G_js(n) - E_ks(n) G_jt(n)
+ E_js(n) G_kt(n) - E_jt(n) G_ks(n)
= (delta^a_k delta^._j - delta^a_j delta^._k) X
(delta^b_t G_.s(n) - delta^b_s G_.t(n)) E_ab(n)
by combining terms with aid of Kronecker deltas,
[E_kj(n), E_st(n)] = C_[kj][st]^[ab] E_ab(n)
with structure constants of the conjugate pseudo-orthogonal
group defined as,
C_[kj][st]^[ab] :=
(delta^a_k delta^._j - delta^a_j delta^._k) X
(delta^b_t G_.s(n) - delta^b_s G_.t(n))
where the skewsymmetric bracketed index set is a label for
the basis set of the algebra, and the dot is a dummy index of contraction.
The completely covariant structure constants can also be expressed by
a form quadratic in a completely covariant fourth rank tensor R:
C_[kj][st][ab] := G^(cd)(n) R_cakj(n) R_bdst(n)
where the contravariant defining carrier space metric is defined
as an inverse as usual by
G^(ab)(n) G_(bc)(n) := delta^a_c
so the G^(ab)(n) and G_(bc) can be used to raise and lower
tensor indicies; and the covariant fourth rank tensor R is
defined as
R_cakj(n) := G_(cj)(n) G_(ak)(n) - G_(ck)(n) G_(aj)(n)
= [2(n-1)]^(-1) g_[ca]_[kj](n)
The Cartan metric that is the ground form for the inner
product on the algebra can then also be expressed in terms
of the inner product ground form on the carrier
space of the defining representation:
g_[kj]_[hi](n) = 2(n-2) (G_(ki)(n) G_(jh)(n) - G_(kh)(n) G_(ji)(n))
with inverse
g^([kj]^[hi](n) =
[2(n-2)]^(-1) (G^(ki)(n) G^(jh)(n) - G^(kh)(n) G^(ji)(n))
showing that geometrically, the linear space of the
adjoint representation is the bivector space associated
to the defining representation carrier space.
The bivector space, in turn, is a tangent space of the group manifold.
This also shows that the group manifold can be seen as a space of
constant curvature with a natural symplectic structure defined by the
bivectors.
The Cartan metric expresses the metrization of the bivector space
induced by the metric of the tangent space.
The tensor R_abcd(n) automatically, by its structure,
has the symmetry properties:
R_abcd(n) = - R_bacd(n)
R_abcd(n) = - R_abdc(n)
R_abcd(n) = + R_cdab(n)
which are the typical symmetries for a Riemann-Christoffel
curvature tensor derived from a symmetric (i.e. torsionless)
affine connection that is in
turn derived from a Riemannian metric or pseudometric.
More generally the cyclic identity
R^d_abc(n) + R^d_bca(n) + R^d_cab(n) = 0
also holds
[Schroedinger 1963], pp. 50, 73.
These results are all fairly
general for the case of semisimple Lie algebras where the Cartan
metric of the algebra is not degenerate.
[F_kj(n), F_st(n)] = - E_js(n)G_kt(n) - E_ks(n)G_jt(n)
- E_jt(n)G_ks(n) - E_kt(n)G_js(n)
= - (delta^a_k delta^._j + delta^a_j delta^._k) X
(delta^b_t G_.s(n) + delta^b_s G_.t(n)) E_ab(n)
and
[E_kj(n), F_st(n)] = F_kt(n)G_js(n) + F_ks(n)G_jt(n)
- F_jt(n)G_ks(n) - F_js(n)G_kt(n)
= (delta^a_k delta^._j - delta^a_j delta^._k) X
(delta^b_t G_.s(n) + delta^b_s G_.t(n)) F_ab(n)
Collecting the complete set of commutation relations for the
generators of su(G, n),
[E_kj(n), E_st(n)] = + C_[kj]_[st]^[ab] E_ab(n)
[E_kj(n), F_st(n)] = + C_[kj]_[st]^[ab] F_ab(n)
[F_kj(n), F_st(n)] = - C_[kj]_[st]^[ab] E_ab(n)
with the structure constants defined as before.
For convenience, also define expressly, the elements
of the Cartan subalgebra,
F_k(n) := (1/2)F_kk(n)
so that
[E_kj(n), F_s(n)] = + F_ks(n) G_sj(n) - F_js(n) G_sk(n)
[F_kj(n), F_s(n)] = - E_ks(n) G_sj(n) + E_js(n) G_sk(n)
and
[F_k(n), F_j(n)] = 0
The operators E_kj(n) and F_st(n) are
right G(n)-skew and form a basis for the defining representation
of the algebra su(G, n).
Using G_kj(n) = 0, for k not= j,
(E_kj(n))^2 = + (i/2) G_jj(n) F_kk(n) + (i/2) G_kk(n) F_jj(n)
(E_kj(n))^3 = + G_kk(n) G_jj(n) E_kj(n)
(E_kj(n))^4 = - (i/2) G_kk(n) G_jj(n) ( F_kk(n) + F_jj(n) )
(E_kj(n))^5 =
(1/2) G_kk(n) G_jj(n)[ ( G_kk(n) + G_jj(n) ) E_kj(n)
- i ( G_kk(n) - G_jj(n) ) F_kj(n) ]
(F_kj(n))^2 = + (i/2) G_jj(n) F_kk(n) + (i/2) G_kk(n) F_jj(n)
(F_kj(n))^3 = - G_kk(n) G_jj(n) F_kj(n)
(F_kj(n))^4 = - (i/2) G_jj(n) G_jj(n) ( F_kk(n) + F_jj(n) )
(F_kj(n))^5 =
(i/2) G_kk(n) G_jj(n)[ ( G_kk(n) - G_jj(n) ) E_kj(n)
- i ( G_kk(n) + G_jj(n) ) F_kj(n) ]
For k, j, not= n-1,
the operators E_kj(n) and F_kj(n) generate
by the exp map, genuine rotations of the complex space, while
the operators E_k_(n-1)(n) and F_k_(n-1)(n)
generate the noncompact one parameter subgroups of hyperbolic
rotations.
Using the above formulae and their obvious extensions to
higher powers to evaluate the exponential series expansions, obtain:
For k not= (n-1) and j, not= (n-1),
exp( theta E_kj(n) ) =
I(n) + i (1 - cos(theta)) ( F_k(n) - F_j(n) ) + sin(theta) E_kj(n)
and
exp( theta F_kj(n) ) =
I(n) + i (1 - cos(theta)) ( F_k(n) - F_j(n) ) + sin(theta) F_kj(n)
while For k not= (n-1),
exp( theta E_(n-1)k(n) ) =
I(n) + i(1 - cosh(theta sqrt(n-1))) ( F_k(n) - F_(n-1)(n)/sqrt(n-1) )
- sinh(theta sqrt(n-1))/sqrt(n-1) E_(n-1)k(n)
and
exp( theta F_(n-1)k(n) ) =
I(n) + i(1 - cosh(theta sqrt(n-1))) ( F_k(n) - F_(n-1)(n)/sqrt(n-1) )
- sinh(theta sqrt(n-1))/sqrt(n-1) F_(n-1)k(n)
Finally,
exp( theta F_(n-1)(n) ) =
I(n) + i(exp( -i(n-1)theta) - 1)/(n-1) F_(n-1)(n)
Relation of Pseudo-orthogonal and Pseudo-unitary
Algebras to Algebras of their Special counterparts
For the Lie algebras u(p, q) of unitary groups U(p, q), where U(n) := U(n, 0), and U(p, q), are non-compact forms of U(n) preserving an indefinite form with p "+1's" and q "-1's", the defining commutation relations can be specialized to
[A_(mu nu), A_(lambda sigma)] =
g_(nu lambda) A_(mu sigma) - g_(mu sigma) A_(lambda nu)
for all mu, nu, lambda, sigma = 0, 1, ..., n-1,
where g_(mu sigma) is the metric of the defining space.
The Lie algebra generators for the associated SU(n) can be
defined in terms of those for U(n), by mapping each to their
traceless counterpart.
B_(mu nu) := A_(mu nu) - (1/n) delta_(mu nu) Tr(A_(mu nu))
Explicitly in Hilb(n), define, for k, j = 0, 1, ..., n-2
^e_kj := |n, k><n, j| - |n, j><n, k|
^f_kj := i|n, k><n, j| + i|n, j><n, k|
Further define for k = 0, 1, ..., n-1
^e_k := |n, n-1><n, k| - |n, k><n, n-1|
^f_k := i|n, n-1><n, k| + i|n, k><n, n-1|
These constitute n^2 generators for U(n-1,1), and the
"normal" basis for the algebra u(n) due to Cartan.
[This basis generalizes the spin matrices for su(2).]
The ^e_kj and ^f_kj generate the compact
subgroup U(n-1), while the ^e_k and ^f_k generate
the hyperbolic "boost" elements.
This is clearly related to the Cartan canonical
decomposition in that the pure boosts are elements of the invariant
subspace of the decomposition.
|u |xi>| |u! |xi>|
~u = | | ~u! = | |
|<xi| e^(ilambda)| |<xi| e^(-ilambda)|
|sqrt2 Q(n-1) sqrt(n-1)|n, n-2>|
Q(n) = (1/sqrt2)| |
|<n, n-2|sqrt(n-1) 0 |
~u! Q(n) ~u =
| u! sqrt2 Q(n-1) u + sqrt(n-1) [|xi><n, n-2|u + u!|n, n-2><xi|] 0|
| |
| 0 sqrt2 <xi|Q(n-1)|xi> + 2sqrt(n-1) Re(<xi|n, n-2>e^(ilambda) |
|0 sqrt2 u!Q(n-1)|xi> + sqrt(n-1)[u!|n, n-2>e^(ilambda) + <n, n-2|xi>|xi>]|
+ | |
|sqrt2 <xi|Q(n-1)u + sqrt(n-1)[<n, n-2|ue^(-ilambda) + <xi|n, n-2><xi|] 0 |
SU(n) possesses a discrete center
(the set of elements which commute with all members of the algebra)
Z_n,
which is isomorphic with the cyclic group of order n, where the
elements are in the defining representation,
exp(i2 pi k/n) I(n)
for k = 0, 1, ..., n-1. While I(n) is not a generator for SU(n),
it is a member of the algebra and is a generator for U(n).
Clifford Algebras
See, also Construction from the Hilbert Space V (of quantum set theoretic Clifford algebras) Clifford algebras over the real and complex fields must be distinguished; over C the structure theory is more complicated, concern here, however, is with algebras over the complex field. Denote a Clifford algebra associated with a quadratic form on C^n by C(C, n) It is a standard theorem of Clifford algebra representation theory that every representation of a Clifford algebra over C is completely reducible. Cases n even and n odd must be distinguished: For even n = 2(nu), there is a single faithful, irreducible representation isomorphic to the complete complex algebra of 2^(nu) x 2^(nu) matrices; it is thus a simple algebra. For odd n = 2(nu) + 1, there is a single irreducible representation of dimension 2^(nu), which is not faithful; A faithful but reducible representation is obtained, however, with the direct sum of two inequivalent irreducible representations. In fact,
C(C, 2(nu)+1) = C(C, 2(nu)) DIRECT-SUM C(C, 2(nu))
and the direct summands C(C, 2(nu)) are not only subalgebras,
but mutually annihilating two-sided ideals in C(C, 2(nu)+1).
[Riesz 1958], pp. 15-19
Let <n, k|G(n)|n, j> be the invariant quadratic form or
inner product on Hilb(n). The Clifford algebra generators G_k(n)
for k = 0, 1, ..., n-1 are defined by
(1/2) {G_k(n), G_j(n)} = <n, k|G(n)|n, j> I(m)
(1/2) {G^+_k(n), G^+_j(n)} = <n, k|G^(-1)(n)|n, j> I(m)
where I(m) is the unit element of the Clifford algebra C(Hilb(n)).
We associate G_j(n) with the basis
( G^+_(+|-)(n) )^(1/2)|n, j>
and G^+_k(n) with the G(n)-dual basis
<n, k| G^(-1/2)_(-|+)(n)
so that for |psi> in Hilb(n) represented by
|psi> = SIGMA alpha_k |n, k>
k
and
<psi| = SIGMA alpha_k^* <n, k|
k
we define the associated Clifford algebra elements
PSI = SIGMA alpha_k G_k(n)
k
and
PSI^+ = SIGMA alpha_k^* G^+_k(n)
k
Then,
(1/2){PSI, PHI^+} = <psi|phi>
and
(1/2){PSI, PHI} = <psi|G(n)|phi>
are the Clifford algebra expressions for the Euclidean and G(n)
inner products respectively, on Hilb(n).
The commutators
[G_k(n), G_j(n)] = G_kj(n)
then obey the commutation relations for the Lie algebra u(G, n).
Construction of Clifford Algebra Representations
From the decomposition of the algebra for n odd, it is clear that we can consider representations for all n by first constructing the IRREP for even n. Following Brauer and Weyl [Brauer 1935], let
| sigma_3 X sigma_3 X ... X sigma_1 X sigma_0 X ... X sigma_0,
|
| for k = 0, 1, ..., nu-1
|
gamma_k(n) = |
|
| sigma_3 X sigma_3 X ... X sigma_2 X sigma_0 X ... X sigma_0,
|
| for k = (nu), (nu)+1, ..., 2(nu)-1
where in the direct products, there are (nu) factors,
and sigma_1 and sigma_2
appear in the products at the k-th place.
The sigma_k are the generators
for the Clifford algebra C(C, 2).
defined at the beginning of
[Appendix C],
but without the factors of "1/2".
sigma_0 is the 2x2 identity. Then one can verify that,
{gamma_k(n), gamma_j(n)} = 2 delta_(kj)
for k, j = 0, 1, ..., 2(nu)-1.
Showing that gamma_k(n) are the generators
of the Clifford algebra associated with a Euclidean inner product.
For an inner product of Lorentzian rather than Euclidean signature,
where the single minus sign is associated with the index value n-1,
replace the above defined gamma_(n-1)(n)
with igamma_(n-1)(n), which is not Hermitean.
Then also, if the inner product is G(n), replace gamma_(n-1)(n)
with isqrt(n-1) gamma_(n-1)(n), and in so doing
define the IRREP of the generators G_k(n), which obey:
(1/2) {G_k(n), G_j(n)} = <n, k| G(n) |n, j> I(m)
The complex conjugates and the transposes then also obey
(1/2) {G^*_k(n), G^*_j(n)} = <n, k| G(n) |n, j> I(m)
(1/2) {G^t_k(n), G^t_j(n)} = <n, k| G(n) |n, j> I(m)
and there is a similarity transformation
T G^*_k(n) T^(-1) = - G^t_k(n)
for all k, with T explicitly given by
T = i G_n-1(n)
so T is in fact, Hermitean.
Moreover,
(1/2) {G^*_k(n), G^*_j(n)} = <n, k| G(n) |n, j> I(m)
Furthermore, define the G(n)-conjugate generators,
G^(+S1k(n) := G!_k(n) <n, k|G^(-1/2(n)|n, k>
then
(1/2) {G^(+S1k(n), G^(+S1j(n)} = <n, k|G^(-1)(n)|n, j> I(m)
and
(1/2) {G^(+S1k(n), G_j(n)} = delta_kj I(m)
The Spin Representations of the Orthogonal Groups
All the groups SO(p, q), p+q = n, are doubly connected for n >= 2. If g_kj is the metric of the defining space with signature (p, q), the generators GAMMA_k(n) of the associated real Clifford algebra are defined by
GAMMA_k(n) GAMMA_j(n) + GAMMA_j(n) GAMMA_k(n) = 2 g_kj
The dimensions of the GAMMA_k are 2^(nu)
for n = 2(nu) and for n = 2(nu) + 1.
s_kj := i [GAMMA_k(n), GAMMA_j(n)]
where the generators GAMMA_k(n) span the vector subspace
of the Clifford algebra.
Then s_kj are the basis elements of a representation of
the Lie algebra so(p, q), and
also span the bivector subspace of the Clifford algebra.
For g in SO(p, q)
S^(-1)(g) GAMMA_k(n) S(g) = a_kj(g) GAMMA_j(n)
then S(g) is a spinor rotation associated with the matrix elements a_kj(g)
of a rotation in the defining representation that rotates the vectors of the
Clifford algebra. Further S(g) is unitary, and can be consistently
normalized so that.
[Det( S(g) )]^2 = 1
leaving the determinant ambiguous by sign. That the sign cannot
be set consistently over the over the group indicates that the
spin representation is double valued.
[Brauer 1935],
[Chevalley 1946], pp. 61-67.
For a pseudoorthogonal group, of course, S(g) cannot be unitary.
Email me, Bill Hammel at