Matrix Lie Group

Lie Algebra of a Matrix Lie Group

Let X be nxn real or complex matrix. The exponential of X is defined by the usual power series e^tX = ∑ X^m / m!  for m = 1 ... ∞. e^tX is a smooth curve in M_n(C).

d/dt (e^tX) = Xe^tX = e^tXX.  In particular, d/dt (e^tX) |_t=0= X.

A function A: R --> GL(n;C) is called a one-parameter subgroup of GL(n;C) if 1) A is continuous; 2) A(0) = I; 3) A(t+s) = A(t)A(s) for all t, s ∈ R. If A is a one-parameter subgroup of GL(n;C), then there exists a unique nxn complex matrix X such that A(t) = e^tX.

Recall that a matrix Lie group is any subgroup G of GL(n;C) with the following properties: If X_m is any sequence of matrices in G, and X_m converges to some matrix X then either X ∈ G, or X is not invertible. Every matrix Lie group is a smooth embedded submanifold of M_n(C) and is thus a Lie group.  But not every Lie group is isomorphic to a matrix Lie group. 

Let G be a matrix Lie group. The Lie algebra of G, denoted g, is the set of all matrices X such that e^tX is in G for all real numbers t. X is in g if and only if the one-parameter subgroup generated by X lies in G. It only require that e^tX be in G for all real numbers t. If e^X ∈ G but e^tX ∉ G for some real values of t. Then X is not in the Lie algebra of G.  

Every matrix Lie group is an embedded submanifold of GL(n;C). g is the tangent space to G at the identity. So a matrix X is in g if and only if there exists a smooth curve Υ in M_n(C) such that 1) Υ(t) lies in G for all t;  2) Υ(0) = I;  3) dΥ/dt |_t=0 = X. This means that g can alternatively be defined as the set of all derivatives of smooth curves through the identity in G. 

Let X and Y be nxn complex matrices, some further properties of the matrix exponential.

1) The Lie Product Formula,  e^X+Y = lim_m-->∞ (e^X/m e^Y/m)^m .

2) For any X ∈ M_n(C), we have det(e^tX) = e^trace(X).

General Linear Groups gl(n;C), gl(n;R)

The Lie algebra of GL(n;C) is the space of all nxn complex matrices, denoted gl(n;C). 

The Lie algebra of GL(n;R) is the space of all nxn real matrices, denoted gl(n;R). If G is a subgroup of GL(n:R), then the Lie algebra of G must consist entirely of real matrices. 

(If e^tX is real for all real numbers t, then X = d/dt e^tX |_t=0 will also be real. )

Special Linear Groups sl(n;C), sl(n;R)

The Lie algebra of SL(n;C) is the space of all nxn complex matrices with trace zero, denoted by sl(n;C).

(If X is any nxn matrix such that det(e^tX )=1 for all t, then e^{t trace(X)} =1 for all t. Therefore, t trace(X) is an integer multiple of 2πi for all t, which is only possible if trace(X) = 0.)

The Lie algebra of SL(n;R) is the space of all nxn real matrices with trace zero, denoted by sl(n;R).

Unitary Groups u(n), su(n)

The Lie algebra of U(n) is the space of all nxn complex matrices X such that X^* = -X, denoted u(n).

(Recall that a matrix U is unitary if and only if U^* = -U^-1. Thus, e^tX is unitary if and only if (e^tX )^* = (e^tX )^-1 = e^-tX . Since (e^tX )^* = e^tX* . Therefore e^tX* = e^-tX.)

The Lie algebra of SU(n) is the space of all nxn complex matrices X such that X^* = -X and trace(X) = 0, denoted u(n).

Orthogonal Groups so(n), so(n;C)

The identity component of O(n) is just SO(n). Since the exponential of a matrix in the Lie algebra is automatically in the identity component, the Lie algebra of O(n) is the same as the Lie algebra of SO(n). The Lie algebra of O(n) and SO(n) is the space of all nxn real matrices X with X^tr = -X, denoted so(n). Note that the condition X^tr = -X forces the diagonal entries of X to be zero, so necessarily the trace of X is zero. 

(An nxn real matrix R is orthogonal if and only if R^tr = -R^-1. So, given an nxn real matrix X, e^tX is orthogonal if and only if (e^tX )^tr = e^tX )^-1 or e^{t tr(X)} = e^-tX. If it holds for all t, then by differentiating at t = 0, we would have  X^tr = -X.)

The Lie algebra of SO(n;C) is the space of nxn complex matrices satisfying X^tr = -X, denoted by so(n;C).

Symplectic Groups sp(n;R), sp(n;C), sp(n)

Let J be the matrix in the definition of the symplectic groups. sp(n;R) is the space of 2n x 2n real matrices X such that JX^trJ = J. 

sp(n;C) is the space of 2n x 2n complex matrices satisfying the same condition. The elements of sp(n;C) are precisely 2n x 2n matrices of the form (A,C; B, -A^tr), where A is an arbitrary nxn matrix and B and C are arbitrary symmetric matrices. 

sp(n) = sp(n;C) ∩ u(2n).

Generalized Orthogonal Groups so(n;k)

The Lie algebra of O(n;k), which is the same as the Lie algebra of SO(n;k), consists of all (n+k) x (n+k) real matrices X with gX^trg = -X, is denoted by so(n;k).

(a matrix A is in O(n;k) if and only if A^trgA = g, where g is the (n+k) x (n+k) diagonal matrix with the first n diagonal entries equal to one and the last k diagonal entries equal to minus one. This condition is equivalent to the condition g^-1A^trg = A^-1. Since g^-1 = g, gA^trg = A^-1. Now, if X is an (n+k) x (n+k) real matrix, then e^tX is in O(n;k) if and only if ge^{t tr(X)}g  = e^{t gtr(X)g} = e^-tX. This condition holds for all real t if and only if gtr(X)g = -X.)

Heisenberg Group

The Lie algebra of the Heisenberg group is the space of all 3 x 3 real matrices that are strictly upper triangular. 

(Recall that H is the group of all 3 x 3 real matrices A = (1,0,0; a,1,0; b,c,1), the exponential of the matrix of the form (0,0,0; α,0,0; β,γ,0) was in H.)

Euclidean and Poincare Groups

The Lie algebra of E(n) is the space of all (n+1) x (n+1) real matrices of the form (Y⁽¹⁾,0 : Y⁽²⁾,0 : ... : y₁, ... , y_n,0) with Y satisfying Y^tr = -Y.

The Lie algebra of P(n;1) is the space of all (n+2) x (n+2) real matrices of the form (Y⁽¹⁾,0 : Y⁽²⁾,0 : ... : y₁, ... , y_n+1,0) with Y ∈ so(n;1).

Chevalier de Lascombes 2004 Margaux, Bordeaux

Chevalier is the second wine of Chateau Lascombes in the commune of Margaux.

It's great for early drinking, but can also be kept until about 2009 when it will have developed some more complex earthy tones.

Grape varieties: Cabernet Sauvignon / Merlot
Style : Full Bodied

Lie Group

A Lie group is a smooth manifold G together with a smooth map from G x G --> G that makes G into a group and such that the inverse map g -->  g^-1 is a smooth map of G to itself. 

A Lie group is a differentiable manifold G which is also a group and such that the group product G x G --> G and the inverse map g --> g^-1 are differentiable. 

Lie Algebra

A finite-dimensional real or complex Lie algebra is a finite-dimensional real or complex vector space g, together with a map [.,.] from g x g into g, with the following properties:

1. [.,.] is bilinear.

2. [X,Y] = - [Y,X] for all X,Y ∈ g (skew symmetry).

3. [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0 (Jacobi identity)

g can be any vector space and that the "bracket" operation [.,.] can be any bilinear, skew symmetric map that satisfies the Jacobi identity. Because g does not necessarily have a product operation defined on it. So [X,Y] is not necessary equal to XY-YX. For any Lie algebra, the Jacobi identity means that the bracket operation behaves as if it were XY-YX, even if it is not actually defined this way. We can think of the bracket operation as making g into an algebra in the general sense. This algebra is not associative. The Jacobi identity is to be thought of as a substitute for associativity.

If g and h are Lie algebras, then a linear map Φ : g --> h is called a Lie algebra homomorphism if ϕ([X,Y]) = [ϕ(X),ϕ(Y)] for all X, Y ∈ g . In addition, if ϕ is one-to-one and onto, then ϕ is called a Lie algebra isomorphism. A Lie algebra isomorphism of a Lie algebra with itself is called a Lie algebra automorphism.

Let g be a Lie algebra. For X ∈ g , define a linear map ad_X : g --> g by ad_X(Y) =[X,Y].  Thus, the map (X --> ad_X), "ad" can be viewed as a linear map from g into gl(g), where gl(g) denotes the space of linear operators from g to g.

ad_X(Y)² = [X,[X,Y]]. and ad_[X,Y] (Z) = [[X,Y],Z]. By Jacobi identity and skew symmetry, [[X,Y],Z] = [X,[Y,Z]] + [Y,[Z,X]] = [X,[Y,Z]] - [Y,[X,Z]] =  [ad_X ad_Y - ad_Y ad_X] (Z) =  [ad_X , ad_Y] (Z). Therefore, we have ad_[X,Y] =  [ad_X , ad_Y]. This means ad : g --> gl(g) is a Lie algebra homomorphism. 

Lie Algebra in Tangent Space

If G is a Lie group and g an element of G, define a map,
L_g : G --> G by L_g(h) = gh.
This is the "left multiplication by g" map, which is smooth since the product map of G x G to itself is assumed smooth. Then the differential (L_g)_* of L_g at a point h will be a linear map of T_h(G) to T_gh(G). A vector field X on G is called left-invariant if X satisfies (L_g)_* (X_h) = X_gh.

Let T_e(G) denote the tangent space at the identity. Then, given any  vector v  ∈ T_e(G) , there is a unique left-invariant vector field X^v with X_e^v  = v, which can be constructed by defining X_g^v  = (L_g)_* (v) .

(Note that L_g o L_h = L_gh , by chain rule, it follows that (L_gh)_*,e = (L_g)_*h, (L_h)_*,e . )

The set of all left-invariant vector fields is a real vector space whose dimension is the same as that of G, and it is isomorphic as a vector space to T_e(G) by means of evaluation at the identity. If vector fields are considered as first-order differential operators, then the commutator of two vector fields is a vector field. Therefore, the commutator of two left-invariant vector fields is, again, a left-invariant vector field. 

The Lie algebra g of a Lie group G is the tangent space at the identity with the bracket operation defined by [v, w] = [X^v, X^w ]_e. If we identify the space of left-invariant vector fields with T_e(G) by means of the map v --> X^v, then g is the space of left-invariant vector fields, which forms a Lie algebra under the commutator of vector fields.

If X^v is the left-invariant vector field, then let Φ^v_t be the associated flow. The exponential mapping is the map exp: g --> G defined by exp(v) = Φ^v₁(e). The exponential mapping is the time-one flow along a left-invariant vector field starting at the identity. This means that to compute exp(v), we first construct the left-invariant vector field X^v and we then find an integral curve Υ^t to X^v that starts at the identity. Then, exp(v) = Υ^t(1). 

Physicist's Convention
Physicists are accustomed to considering the map X --> e^iX instead of X --> e^X . Thus, a physicist would think of the Lie algebra of G as the set of all matrices X such that e^itX ∈ G for all real numbers t. In this case, physics literature does not always distinguish clearly between a matrix Lie group and its Lie algebra and the Lie algebra is frequently referred to as the space of "infinitesimal group elements."

Let g be a finite-dimensional real or complex Lie algebra, and let X₁, ..., X_n be a basis for g (as a vector space). Then for each i and j, [X_i, X_j] can be written uniquely in the form [X_i, X_j] = ∑ c_ijk X_k, for k = 1.. n. The constant c_ijk are called the structure constants of g, it determine the bracket operation on g. 

In the physics literature, the structure constants are defined as [X_i, X_j] = √-1 ∑ c_ijk X_k. The structure constants satisfy both the conditions of skew symmetry and Jacobi identity. 

Tangent Space & Vector Fields

Tangent Space

For a general manifold, not necessary embedded in R^m, we define the notion of tangent space by abstracting the notion of the directional derivative. The tangent space at m to M , denoted T_m(M ), is the set of all linear maps X from C^∞(M ) into R satisfying:

• "Product rule": X(fg) = X(f)g(m) + f(m)X(g) for all f and g in C^∞(M );


• "Localization" : if f is equal to g in a neighborhood of m, then X(f) = X(g).

This is a real vector space. An element of T_m(M ) is called a tangent vector at m.

If x₁, ..., x_n is a local coordinate system, then each tangent vector X at m is expressed uniquely as

X(f)  = ∑ a_k ∂f / ∂x_k(m), for k = 1, n and a₁, ..., a_n real constants. This means T_m(M ) is a real vector space of dimension n.

Vector Fields

A vector field is a map X that associates to each point m in M a tangent vector X_m ∈ T_m(M ). A vector field can be expressed 

X_m(f)  = ∑ a_k(m) ∂f / ∂x_k, k = 1, n

where a_k's are real-valued functions. A vector field is smooth if the coefficient functions a_k are smooth in each local coordinate system. 

If we apply a vector field to a smooth function f by applying X_m to f at each point m. The result X(f) is then another function, which will be smooth if X is a smooth vector field. A smooth vector field is a map from  C^∞(M ) -->  C^∞(M ) that satisfies the product rule : X(fg) = fX(g) + X(f)g. Here X(fg) is a function, not a number, and that on the right-hand side, we do not evaluate f or g at any point. The equation X(fg) can be restated as saying that a vector field is a derivation of the algebra of smooth functions.

Therefore, we can think of the vector field as a first-order differential operator (mapping the space of smooth functions to itself), the one obtained by differentiating a function at each point in the direction of the tangent vector at that point (we can also see a geometric picture of a vector field as a collection of arrows, one at each point in the manifold). If we multiply two vector fields, we will get a second-order differential operator; that is not a vector field. However, if X and Y are vector fields and we compute their commutator XY - YX, then the second-order items in XY will cancel with the second-order terms in YX and the result will again be first-order differential operator. The space of smooth vector fields then becomes an infinite-dimensional Lie algebra with the bracket defined by [X, Y] = XY - YX. 

If X is a vector field and ϒ : (a,b) --> M  is a smooth curve in M , then ϒ is called an integral curve for X if for each t ∈ (a,b), we have dϒ/dt = X_ϒ(t). In a smooth local coordinate system x₁, ..., x_n, ϒ(t) will be represented a family of functions x₁(t), ..., x_n(t) and the vector field X will be represented in the form of X_m(f) as above, with each a_k being a smooth function of x₁, ..., x_n.  Here, for all smooth functions f on M,

dϒ/dt (f) = df(ϒ(t))/dt. By chain rule,  df(ϒ(t))/d = ∑ ∂f / ∂x_k dx_k/dt.

Compare to X_m(f)  = ∑ a_k(m) ∂f / ∂x_k, k = 1, n, we have 

dx_k(t) / dt = a_k(x₁(t), ..., x_n(t)). This is a equation of first-order ordinary differential equations.

Flow Along a Vector Field

A vector field X is called complete if ϒ(t) can be defined for all t for all initial points m. If X is a complete vector field, then one can define the associated flow on M. This is a family of maps Φ_t: M  --> M  defined so that if ϒ is an integral curve for X which ϒ(0) = m, then Φ_t(m) =  ϒ(t). This means that Φ_t(m) is defined by starting at m and "flowing" along the vector field X for time t. (If X is not complete, one can still define a sort of flow, but then each Φ_t is defined only on part of M.) If X is a smooth complete vector field, then each Φ_t is a smooth map of M to itself, and the maps satisfy Φ_t o Φ_s = Φ_t+s.

Submanifolds of Vector Fields

Suppose V is a real vector space of dimension n. A subset M  of V is called a smooth embedded submanifold of dimension k if given any m₀ in M  , there exists a smooth coordinate system (Φ, U) defined in the neighborhood U of  m₀ such that for any m ∈ U, m is in U ∩ M if and only if  Φ(m) is in R^k ⊂ Rⁿ. Here, R^k is the subset if Rⁿ where the last n - k coordinates are zero. This says that locally, in a suitable coordinate system, M  looks like R^k sitting inside Rⁿ. If M  is a smooth embedded submanifold of dimension k, then we can make M  into a smooth manifold of dimension k as follows. 

If M  is a smooth embedded submanifold of V, then the inclusion map i of M  into V is a smooth map. The differential i_* : T_m(M ) --> T_m(V) is injective, and it is customary to identify T_m(M ) with its image  T_m(V ), which is a k-dimensional subspace of the n-dimensional space T_m(V ). This allows us to think of the tangent space to M  at m as a subspace of the tangent space to V at m. However, we are identifying the tangent space at m to V with V itself. Thus, the tangent space to M  at m is identified with a subspace of V. 

Manifold

Manifold and Coordinate Functions

A topological manifold M of dimension n is a topological space that is locally homeomorphic to Rⁿ. We may say that a manifold is an object that looks locally like a little piece of Rⁿ. An example would be a torus, the two dimensional surface of a "doughnut" in R³, which looks locally (but not globally) like R². This means that for each point m in M , there is a neighborhood of U of m and a one-to-one, continuous map Φ of U into Rⁿ onto some open set  Φ(U) in Rⁿ such that inverse map Φ^-1 : Φ(U) --> U is also continuous.

The map Φ can be considered as local coordinate functions x₁, ..., x_n, where each x_k is the continuous function from U into R given by x_k(m) = Φ(m)_k, be the k-th component of Φ(m). 

If Ψ is another homeomorphism of another neighborhood V of m, and y_k(m) = Ψ(m)_k is the associated coordinate system, then both coordinate systems are defined in the neighborhood U ∩ V of m. The "change of coordinates" map for m in U ∩ V is the map Ψ o Φ^-1 that maps the set Φ(U ∩ V) onto the set Ψ(U ∩ V), i.e.,

(y₁(m), ..., y_n(m)) = (Ψ o Φ^-1) (x₁(m), ..., x_n(m)). This is continuous since both Ψ and Φ^-1 are continuous.

Smooth Manifold

Smooth manifold is made by choosing a collection of local coordinate systems that cover the whole manifold. For any two coordinate systems that are defined in overlapping regions, the expression for one set of coordinates in terms of the other is always smooth. These coordinate systems were chosen in order to give a smooth structure to the topological manifold M . Some manifolds do not admit a smooth structure. When a smooth structure exists, it is not unique.  

Once a smooth structure is chosen, we define a smooth local coordinate system to be any local coordinate system (U, Φ) such that Φ o Φ_α^-1 is smooth for each (U_α, Φ_α). A function f : M --> R is called smooth if for each smooth coordinate system (U, Φ), the function f o Φ^-1 is a smooth function on the set Φ(U).

Complex Manifold

A complex manifold is a smooth manifold of dimension 2n such that the basic coordinate patches (U_α, Φ_α) have the property that the change-of-coordinates map Φ_β o Φ_α^-1 is holomorphic for each β and α. Here, R²ⁿ is identified with Cⁿ and holomorphic means the same as complex analytic.  

If V is a complex vector space, then a subset M of V is called an embedded complex submanifold of dimension k if, given any m₀ in M , there exists a holomorphic local coordinate system  (Φ, U) defined in the neighborhood U of m₀ such that for any m ∈ U, m is in U ∩ M if and only if Φ(m) is in C^k ⊂ Cⁿ. 

Chilean Red Wines

Phylloxera, a grape louse, kills vines. In 1870s, phylloxera attacked the French vineyards and killed almost all vines. Luckily, it was discovered that American Vitis labrusca vines are immune to this louse. So all the European vines were pulled up and grafted onto this american rootstocks.

Chilean wine producers imported their wines from France in the 1860s, before the phylloxera attack. The climate in Chile is somewhere between California and Bordeaux. The main red grapes (parentheses are names of the regions) grown in Chile are Cabernet Sauvignon (Maipo Valley), Carmenere (Rapel Valley), Merlot (Rapel Valley), and Syrah (It was recently discovered that 40% of the Merlot planted is not really Merlot, but is Carmenere). The 1997 and 2003 vintages were two of the best ever produced in Chile. Some of the top wine producers are (parentheses are names of the best wines):

• Concha y Toro (Don Melchor, Almaviva)
• Cousino Macul (Finis Terrae, Antiguas Reservas)
• Vina Errazuriz (Don Maximiano)
• Los Vascos (Reserva De Familia)
• Vina Montes (Alpha M)
• Santa Rita (Casa Real)
• Sena
• Veramonte (Primus)

Some of the foreign investment in Chile:

• Domaines  Lafite Rothschild (France) - Los Vascos
• Baron Philippe de Rothschild (France) - Concha y Toro
• Quintessa (California) - Veramonte

Bordeaux Classification

Bordeaux

There are 57 wine regions in Bordeaux produce high-quality wines that are allowed to carry the AOC (Appellation d'Origine Controlee) designation on the label. Some famous places:

• Medoc (left bank) - Cabernet-style, produces only red wines
  
• Haut Medoc, St-Estephe, Pauillac, St-Julien, Margaux, Moulis, Listrac
• Graves / Pessac-Leognan (left bank) - Cabernet-style, both red and dry white
• Pomerol (right bank) - Merlot-style, only red
• St-Emilion (right bank) - Merlot-style, only red
  

Major Bordeaux classifications

In 1855, Bordeaux officially classified the quality levels of some its chateaux. Brokers from the vine industry were asked by Napoleon III to select the best wines to represent France in the International Exposition in 1855. In the Medoc, the 61 highest-level chateaux are called Grand Cru Classe. There are also 247 chateaux in the Medoc that are entitled to be called Cru Bourgeois, a step below Grand Cru Classe. Other areas, such as St-Emilion and Graves, have their own classification systems.

• Medoc
• Grand Crus Classe - 1855; 61 chateaux
• First Growths - Premiers Crus (5)
• Second Growths - Deuxiemes Crus (14)
• Third Growths - Troisiemes Crus (14)
• Fourth Growths - Quatriemes Crus (10)
• Fifth Growths - Cinquiemes Crus (18)
• Crus Bourgeois - 1920, revised 1932, 1978, and 2003; 247 chateaux
• Graves (Grand Crus Classe) : 1959; 16 chateaux
• Pomerol : no official classification
• St-Emilion : 1955, revised 1996, revised 2006; 15 Premiers Grand Crus Classe and 46 Grand Crus Classe.

According to French law, a chateaux is a house attached to a vineyard having a specific number of acres, as well as having winemaking and storage facilities on the property. A wine may not called a chateaux wine unless it meets these criteria.

As read from the Kevin Zraly's Box Score of the 1855 classification:

• Pauillac has three of the five first growths and 12 fifth growths
• Margaux has most of the third growths in the left bank
• Margaux has the greatest number of classed vineyards in all of Medoc. It is also the only area to have a chateaux rated in each category.
• St-Julien has no first or fifth growths, but is very strong in the second and fourth.

Second-label wines are from the youngest parts of the vineyard and are lighter in style and quicker to mature but are usually a third the price of the chateaux wine.

First Growths

• Chateaux Lafite-Rothschild (Pauillac) - Carruades de Lafite Rothschild
• Chateaux Latour (Pauillac) - Les Forts de Latour
• Chateaux Margaux (Margaux) - Pavillon Rouge du Chateaux Margaux
• Chateaux Haut-Brion (Pessac-Leognan (Graves)) - Bahans du Chateaux Haut-Brion
• Chateaux Mouton-Rothschild (Pauillac) - Petit Mouton

Second Growths

• Chateaux Leoville-Barton (St-Juilen) -La Reserve de Leoville Barton
• Chateaux Leoville-Las-Cases (St-Juilen) - Clos du Marquis
• Chateaux Pichon Lalande (Pauillac) - Reserve de la Comtesse
• Chateaux Pichon Longueville (Pauillac) - Les Tourelles de Pichon

Third Growths

• Chateaux Palmer (Margaux) - Reserve du General

Fourth Growths

• Chateaux St-Pierre (St-Julien)

Fifth Growths

• Chateaux Lynch-Bages (Pauillac) - Chateaux Haut Bages-Averous

Other Quality Levels

Bordeaux - sometimes known as the "proprietary" wines. It is the least expensive AOC wines in Bordeau. Some famous brand name, e.g. Mouton-Cadet, Lauretan, Lacour Pavillon, Baron Philippe, Michel Lynch.

Bordeaux + Region - come from one of the 57 different regions. Only grapes and wines made in those areas can be called by their region names, e.g., Pauillac and St-Emilion. They are more expansive than those labeled simply Bordeaux. The major shippers of regional wines form Bordeaux are: Barton & Guestier (B&G), Cordier, Dourthe Kressmann, Eschenauer, Sichel, Yvon Mau, Ets J-P Moueix, Baron Philippe de Rothschild, Borie-Manoux, and Dulong.

Special Groups

Generalized Orthogonal and Lorentz Groups, O(n;k) and O(3;1)

Let n and k be positive integers, and consider R^n+k . Define a symmetric bilinear form: [• , •]_n,k on R^n+k by [x, y]_n,k = x₁y₁ + ... + x_ny_n - x_n+1y_n+1 - ... - x_n+ky_n+k

The set of (n+k) x (n+k) real matrices A which preserve this form, i.e., [Ax ,Ay]_n,k = [x, y]_n,k for all x, y ∈ R^n+k,  is the generalized orthogonal group O(n;k). It is a subgroup of GL(n+k;R) and a matrix Lie group.

If A is an (n+k) x (n+k) real matrix, let A⁽ⁱ⁾ denote the i-th column vector of A, that is A⁽ⁱ⁾ = (A_1,i ... A_n+k,i). Then, A is in O(n;k) iff the following conditions are satisfied:

[A^(l), A^(j)]_n,k = 0  , l ≠ j,

[A^(l), A^(l)]_n,k = 1  , 1 ≤ l ≤ n,

[A^(l), A^(l)]_n,k = -1  , n+1 ≤ l ≤ n+k.

For any A in O(n;k), det A = ±1. Of particular interest in physics is the Lorentz group O(3;1). 

Heisenberg Group H

The set of all 3 x 3 real matrices A of the form 

A = (1,0,0 : a,1,0 : b,c,1), 

where a, b, and c are arbitrary real numbers, is the Heisenberg group. The Lie algebra of H gives a realization of the Heisenberg commutation relations of quantum mechanics. 

The product of two matrices of this form is again of the same form. The inverse of  A is A^-1 = (1,0,0 : -a,1,0 : ac-b,-c,1)

H is a subgroup of GL(3:R). The limit of matrices of this form is again of the same form, and so H is a matrix Lie group.

Euclidean Group E(n)

By definition, it is the group of all one-to-one, onto, distance-preserving maps of Rⁿ to itself, i.e., f : Rⁿ --> Rⁿ such that d(f(x), f(y)) = d(x, y) for all x, y  ∈ Rⁿ. Here, d is the usual distance on Rⁿ : d(x, y) = |x - y|.  f need not be linear. The orthogonal group O(n) is a subgroup of E(n) and is the group of all linear distance-preserving maps of Rⁿ to itself. For x ∈ Rⁿ , define the translation by x, denoted T_x by,

T_x(y) = x + y.

The set of translations is also a subgroup of E(n).

Every element T of E(n) can be written uniquely as an orthogonal linear transformation followed by a translation in the form T = T_xR, with x ∈ Rⁿ and R  ∈ O(n).

We can write an element of T of E(n) as a pair of {x, R}. For y ∈ Rⁿ, {x, R}y = Ry + x. The product operation for E(n) is :

{x₁, R₁} {x₂, R₂}  =  {x₁+ R₁x₂, R₁R₂ } 

The inverse of an element of E(n) is given by {x, R}^-1 = {-R^-1x, R^-1}.

E(n) is not a subgroup of GL(n;R), since translations are not linear maps. However, E(n) is isomorphic to a subgroup of GL(n+1;R), via the map which associates to {x,R} ∈ E(n) the following matrix:

(R⁽¹⁾,0 : R⁽²⁾,0 : ... : x₁, ... , x_n,1)  with R  ∈ O(n). Because the limit of things of this form is in the same form. So Euclidean group can be expressed as a matrix Lie group.

Poincare Group P(n:1)

Similarly, we can define the Poincare group P(n;1) to be the group of all transformations of Rⁿ⁺¹ of the form T = T_xA, with x ∈ Rⁿ⁺¹ and A  ∈ O(n;1). This is the group of affine transformations of Rⁿ⁺¹ which preserve the Lorentz "distance" d_L(x, y) = (x₁ - y₁)² + ... + (x_n - y_n)² - (x_n+1 - y_n+1)² . An affine transformation is one of the form x --> Ax + b, where A is a linear transformation and b is a constant. The Poincare group P(n;1) is isomorphic to the group of (n+2) x (n+2) matrices of the form

(A⁽¹⁾,0 : A⁽²⁾,0 : ... : x₁, ... , x_n+1,1)  with A  ∈ O(n;1). The set of matrices of this form is a matrix Lie group. 

Classical Groups

Group

A group is a set G of which the product operation is a map of G x G into G (closure), denoted by g*h, has the following properties:

Associativity

For all g, h, k ∈ G,  g*(h*k) = (g*h)*k

Identity

There exists an element e in G such that for all g ∈ G, g*e = e*g = g

Inverse

For all g ∈ G, there exist h ∈ G with g*h = h*g = e.

If  g*h = h*g for all g, h ∈ G, then the group is said to be commutative (abelian).


The special and general linear group, the orthogonal and unitary groups, and the symplectic groups make up the classical groups.

General Linear Groups, GL(n;R) and GL(n;C)

The general linear group over the real numbers, GL(n;R), is the group of all nxn invertible matrices with real entries. The general linear group over the complex numbers, GL(n;C), is the group of all nxn invertible matrices with complex entries. 

A matrix Lie group is any subgroup G of GL(n;C) with the following property: If A_m is any sequence of matrices in G, and A_m converges to some matrix A then either A ∈ G, or A is not invertible, i.e., a matrix Lie group is a closed subgroup of GL(n;C).

Special Linear Groups, SL(n;R) and SL(n;C)

The special linear group is the group of nxn invertible matrices, with real or complex entries, having determinant one. Both of these are subgroups of GL(n;C). Furthermore, if A_m is a sequence of matrices with determinant one and A_m converges to A, then A also has determinant one, because determinant is a continuous function. Thus, SL(n;R) and SL(n;C) are matrix Lie groups.

Orthogonal Groups and Special Orthogonal Groups, O(n) and SO(n)

The set of all nxn real orthogonal matrices is the orthogonal group O(n). It is a subgroup of GL(n;C). An nxn real matrix A is said to be orthogonal if the column vectors that make up A are orthonormal, that is, if

∑_l A_ljA_lk = δ_jk ,  1 ≤ j, k ≤ n,  l=1 to n. Kronecker  delta, δ_jk =1 for j = k, δ_jk = 0 for j ≠ k.

Equivalently, A is orthogonal if it preserves the inner product, namely <x, y> = <Ax, Ay> = ∑_k x_ky_k for all vectors x, y in Rⁿ. Still another equivalent definition is that A is orthogonal if A^TA = I, i.e., A^T = A^-1. The limit of a sequence of orthogonal matrices is orthogonal, because the A^TA = I is preserved under limits. Thus, O(n) is a matrix Lie group. 

The set of nxn orthogonal matrices with determinant one is the special orthogonal group SO(n). Both orthogonality and the property of having determinant one are preserved under limits, so SO(n) is a matrix Lie group. Since elements of O(n) already have determinant ±1, SO(n) is "half" of O(n). 

Geometrically, elements of O(n) are either rotations or combinations of rotations and reflections. The elements of SO(n) are just the rotations.

Unitary and Special Unitary Groups, U(n) and SU(n)

The set of all nxn unitary matrices is the unitary group U(n), and it is a subgroup of GL(n; C). An nxn real matrix A is said to be unitary if the column vectors that make up A are orthonormal, that is, if ∑_l A^*_jlA_lk = δ_jk.

Equivalently, A is unitary if it preserves the inner product, namely <x, y> = <Ax, Ay> = ∑_k conj(x)_ky_k for all vectors x, y in Cⁿ. Still another equivalent definition is that A is unitary if A^*A = I, i.e., A^* = A^-1, here A^* is the adjoint of A. The limit of unitary matrices is unitary, so U(n) is a matrix Lie group. The set of unitary matrices with determinant one is the special unitary group SU(n). Note that a unitary matrix can have determinant e^iθ for any θ, and so SU(n) is a smaller subset of U(n) than SO(n) is of O(n). 

Complex Orthogonal Groups, O(n;C) and SO(n;C)

Consider the bilinear form (• , •) on Cⁿ defined by (x, y) = ∑_k x_ky_k. This form is not an inner product because it is symmetric rather than conjugate-symmetric. The set of all nxn complex matrices A which preserve the form, (Ax, Ay) = (x, y),  is the complex orthogonal group O(n;C), it is a subgroup of GL(n;C). The group SO(n;C) is defined to be the set of all A in O(n;C) with det A = 1 and it is also a matrix Lie group. 

Symplectic Groups, Sp(n;R), Sp(n;C), and Sp(n)

The set of all 2nx2n matrices A which preserves B (i.e., B[Ax, Ay] = B[x, y] for all x, y ∈ R²ⁿ) is the real symplectic group Sp(n;R), and it is a subgroup of GL(2n;R), here B is the skew-symmetric bilinear form on R²ⁿ defined as:

B[x, y] = ∑_k x_ky_n+k - x_n+ky_k. 

This group arises naturally in the study of classical mechanics. If J is the 2nx2n matrix  J = ( 0, -I : I, 0), then B[x, y] = (x, Jy), and a 2nx2n real matrix A is in Sp(n;R) if and only if A^TJA = J. This relation shows that det A = ±1, for all A ∈ Sp(n;R). In fact, det A = 1 for all A ∈ Sp(n;R).

We can define a bilinear form on C²ⁿ by the same formula. The set of 2nx2n complex matrices which preserve this form is the complex symplectic group Sp(n;C).  A 2nx2n complex matrix A is in Sp(n;C) if and only if A^TJA = J. It can show that det A = ±1, for all A ∈ Sp(n;R). This relation shows that det A = ±1, for all A ∈ Sp(n;C). In fact, det A = 1 for all A ∈ Sp(n;C).

Finally, the compact symplectic group Sp(n) is defined as

Sp(n) = Sp(n;C) ∩ U(2n)