The Heisenberg Algebra In An m Dimensional Space
And Its Bilinear Extensions [1]

A Note: Hermiticity [2]

  1. Introduction
  2. Defining a Heisenberg Algebra In An m Dimensional Space
  3. Commutative Lie Algebras
  4. Lie Subalgebras
  5. Lie algebra Structure Theory
  6. Solvable & Nilpotent Lie algebras
  7. Simple vs. Semisimple Lie algebras
  8. Symplectic Structure
  9. Semidirect Products
  10. Lie Algebra Extensions
  11. The Question of Extensibility of a Lie Algebra
  12. Extension By Bilinear Elements
  13. Extension By Multilinear Elements of Higer Order
  14. Weyl Algebras
  15. Symmetric Bilinear Extensions Of H
  16. The Fourier Transforms of H
  17. Symplectic Structure Of H
  18. Antisymmetric Bilinear Extensions
  19. Structures of Lie algebras of Dimension r = 2, 3
  20. Foot Notes


Classical mechanics can be cast in various inequivalent forms, the most general being that given by Newton over 300 years ago. The formulation by Lagrange (a.k.a. analytical formalism) is slightly restrictive relative to Newton's original formulation, and that given by Hamilton (a.k.a. canonical formalism) is even more restrictive. Hamiltonian systems, however, seem to be those which lend themselves to quantization most readily by the "canonical quantization" that preserves the symplectic structure of Hamiltonian Poisson brackets, mapping it to the symplectic structure of certain abstract nilpotent Lie algebras often called Heisenberg algebras. These algebras are the subject of this modest elementary essay.

It is rarely understood that QM delays are related to relativistic delays; but that is another story for a different time.

A point that will become apparent is that the structure of a Heisenberg algebra for a Euclidean Q-space of m dimensions is considerably more complicated than one might be led to expect from the standard textbooks on quantum theory.

Certain classical mechanical systems, a subset of those that can be cast in canonical formlism can be "breezily quantized" by replacing the fundamental Poisson brackets of canonical formalism with with an algebraic analog wherein the pairs of canonical variables (qi, pk) no longer commute, but instead obey the Canonical Commutation Relations (CCR).

In this sense, Quantum Mechanics (QM) (of a restricted sort) can be seen as Classical Mechanics plus a funny kind of constraint imposed on the classical phase space; that is to say that the constraint is a specifically kinematical constraint, something before dynamics, or time evolution of the physical system.

Most other classical systems are not so breezily "quantized".

A general algorithmic procedure for quantization of an arbitrary classical system simply does not exist yet.

The major obstacle is the ordering of physical variables in products, which is not a problem in Classical Mechanics (CM), but becomes one in QM.

CCR as the fundamental kinematical postulate of QM and Quantum Theory (QT), generally, is of physical importance, but also turns out to be of a more abstract mathematical importance since it happens to define the simplest of all nilpotent Lie algebras, and its representation theory is deeply intertwined with the as yet unsolved problem of the representations of nilpotent Lie algebras generally.

The Heisenberg Algebra In An m Dimensional Qk Space

For a kinematic Heisenberg algebra H(m) of QM in m [3] dimensional space, usually given as (generally unbounded) operators CCR acting on a specifically constructed Hilbert space, there is, axiomatically, and more abstractly, a nilpotent Lie algebra of 2m+1 dimensions given by the defining Commutation Relations (CRs). This is a more specific way of saying that such an Heisenberg algebra is a Lie algebra. The fundamental commutator stucture is defined as follows.

		[Qa, Pb]  =  +i δab ħ I

		[Qa, I]    =  0   [Pa, I]    =  0

		[Qa, Qb]  =  0,  [Pa, Pb]  =  0

   That said, take ħ = 1 from here on, subject to change with notice.

There are four natural involutions of order two in the algebra: one exchanges indicies, and the second exchanges Q and P, the third can be instituted by reversing the sign of ħ, and the fourth is complex conjugation. Performing the first leaves the algebraic structure unchanged. Performing the other changes the first CR to: [4]

		[Qb, Pa]  =  -i δab I
		[Qa, Pb]  =  -i δab I

The first thing to remark on is that Lie algebras, when they are not being considered as structures derived from algebraic axioms alone, are related to Lie groups through the exp map. [Chevalley 1946]

		exp: Algebra  →  Group

		        a     →  exp( i x a )

where 'x' is a parameter taking values in the field of the algebra, and 'a' represents an element of any Lie algebra A, thus mapping Lie algebra elements to one parameter Lie subgroups.

This idea, however, becomes less than simple in the context of representations, because it turns out that there exist representations of the Heisenberg algebra that do not exponentiate to a representation of the Heisenberg group. This problem stems from the necessity of considering unbounded operators in *any* representation of H, that being a consequence of the results of [Wielandt 1949], [Wintner 1947] and Taussky in [Cooke 1950].

The existence of even one unbounded operator requires that the operators in any given representation be restricted to a common domain of definition that is less than the full Hilbert space on which they are expected to be defined. There are representations for which such a common domain is not dense in the Hilbert space and this obstruct passage through the exp map to an associated Heisenberg group representation.

In the case of the Heisenberg algebra, there is an interesting cross relationship within a q-p pair familiar from classical canonical mechanics stated as, momentum is the generator of spatial translations and position is the generator of momentum translations in the context of phase space.

For the adjoint action of the Heisenberg group on the Heisenberg algebra, calculated by way of the BCH formula, using the algebra commutator relations.

              exp( +i α P ) Q exp( -i α P )  =  Q + α I

              exp( +i β Q ) P exp( -i β Q )  =  P + β I

Digressions follow which seek to enclose Heisenberg algebras within the more general context Lie algebras. The fundamental Heisenberg algebra H(1) is a Lie algebra of dimension 3, and it makes sense then to be able to see where it fits in the structure theory of all Lie algebra structures of dimension 3. For ease of comparison, this ancillary information is relegated to footnote [5].

Commutative Lie Algebras

   A Lie algebra is said to be commutative if for a basis {Xk}
   of the Lie algebra L as a vector space,

	                 [Xk, Xj]  =  0

   for all pairs of basis elements.  This condition is easily understood
   to be invariant under the general linear group of basis changes.

Lie Subalgebras

   A Lie subalgebra K of a Lie algebra L is a closed linear
   subspace of the algebra that is also closed under commutation

	                 [K, K]  ⊂  K.

   An invariant Lie subalgebra is a subalgebra left invariant
   by commutation with the full algebra

	                 [L, K]  ⊂  K,

   and is actually then a two sided ideal of L.

H(m) has two obvious commutative Lie subalgebras, one spanned by the {Qa}, the other by the {Pb}. Commutative Lie algebras exponentiate either to translation or toriodal groups. Toroidal groups are compact, possessing only finite dimensional Unitary Irreducible Representations (UIRREPS), while translation groups, possessing only infinite dimensional UIRREPS, are not.

The matter of the group being toroidal or translational is a matter of the group's global topology; the algebra, however, speaks only to a local topology.

One aspect of life in the quantum lane is made somewhat mercifully simple in QM by the fundamental theorem of Stone and von Neumann [Neumann 1931] which tells that for the standard Schrödinger representation of CCR on a separable Hilbert space, all other other representations on a separable are unitarily equivalent, thus there is effectively only one such representation.

This is not to say that it is uninteresting to study the various realizations of this reprepresentation, on different basis vectors of the Hilbert space together with their specific unitary equivalences; that is, in fact, a rather interesting and important area of study, not entered into here.

It is worth remarking that v. Neumann's theorem on uniqueness assumes that the value of ħ is fixed; if variable, the values of ħ parametrize a one parameter class of UIRREPS that are unitarily *inequivalent*. Thus, passing through the values of ħ in a "classical" limit where ħ → 0 as in the obvious Inönü-Wigner contraction [Inonu 1953] will necessarily pass through an uncountble number of unitarily inequivalent representations. Mathematically, one should should not take such operations lightly.

The requirement that the Hilbert space be separable cannot be relaxed here; nonseparable Hilbert spaces do in fact appear in quantum field theory, where H(∞) is an infinite dimensional Lie algebra (which beasts are often called "pseudo Lie algebras" because of their additional topological requirements), so the issue of separability is not trivial because in nonseparable Hilbert spaces there are an infinite number of unitarily inequivalent representations of H, and the life of the "pseudo" Heisenberg algebra then possesses a charmingly sophisticated complication that will not be dealt with here explicitly. The conditions for Stone-Neumann theorem fail in the context of nonseparable Hilbert spaces. The cognate infinite dimensional groups are a bit strange regarding the requirements of a binary operation, and are often called groupoids to convey this.

The Q := {Qa} and P := {Pa} commutative subalgebras are clearly not invariant subalgebras, since set theoretically,

		[H, Q]  =  {I}, and [H, P]  =  {I}.

However, one can dissect H differently by looking at the subalgebras Q1 := {Qa, I} and P1 := {Pa, I}. Then,

		[H, Q1]  ⊆  Q1, and [H, P1]  ⊆  P1.

   so Q1 and P1 are both commutative invariant subalgebras of H, and

                          Q1 ∩ P1  =  0.

The 2m+1 dimensional Lie algebra H(m) is also clearly a direct sum of mutually commuting CCR subalgebras {Qa, Pa, I} for a = 1, ..., m.

Denote such a subalgebra by {a}. Then, for {a},

		[H, {a}]  =  i I  ∈  {a}


		[H, {a}]  ⊆  {a}

without any imposed dynamics, i.e., without any definition of any system beyond its degrees of classical freedom.

So, for every a, {a} is an invariant subalgebra of H. The complement of {a} in H is also a subalgebra of H which is also invariant in H, implying that for any m, H is a direct sum of m of the {a}.

Lie algebra Structure Theory

The fundamental theorem on Lie algebras due to Levi and Malcev says that any Lie algebra L has the structure of a semidirect product of a solvable Lie algebra N by a semisimple Lie algebra M.

The semidirect product structure is equivalent to saying that N is an ideal in L. N being an ideal in L is equivalent N being an invariant subalgebra of L.

From the Levi-Malcev theorem it is then clear that the two classes of Lie algebras to distinguish and to study, in order to understand the structure of all of them are those Lie algebras that are solvable and those that are semisimple.

Solvable Lie algebras are most like commutative algebras, while semisimple Lie algebras are the least like commutative algebras.

It is perhaps a counterintuitive fact of life that determining the structures of semisimple Lie algebras was a solved problem half a century ago, while the structures of all possible solvable Lie algebras remains unsolved, which says something about the relative difficulties of the associated structure problems. The idea that commutativity is somehow simpler than noncommutativity is the deceiver.

Among the solvable algebras, the nilpotent algebras, which include the commutative ones, have a representation theory where interestingly, the Heisenberg algebra H(1) has a seminal role. [Kirilov 1964]

Semisimple algebras can be understood as constructed as a direct sum of simple Lie algebras, where the representation theory has the representations of su(2) or so(3) playing a seminal role. The roles of su(2) in semisimple algebras and H(1) is nilpotent algebras are remarkably similar.

Solvable & Nilpotent Lie algebras

Simple vs. Semisimple Lie algebras

A Lie algebra is simple if it contains no nonzero invariant Lie subalgebras A Lie algebra is semisimple if it contains no nonzero, nontrivial *commutative*, invariant Lie subalgebras. As the words might suggest, simplicity implies semisimplicity.

From this one concludes that studying the representations of H(m) for any m can be reduced to studying those for m=1, and furthermore then that the properties of the representations for any m can be studied from a viewpoint of the abstract algebras.

Notice that as in classical mechanics, the model of a single particle in an m dimensional space is mathematically equivalent to the model of m particles in a 1 dimesnional space and that more generally, the model of n particles in an m dimensional space is mathematically equivalent to the model of n' particles in an m' dimensional space so along as mn = m'n'.

Symplectic Structure

   Consider elements A and B of R2n with Cartesian coordinates
   e.g., [Chevalley 1946]

                  A  :=  (x1, y1, x2, y2, ..., xn, yn)
                  B  :=  (u1, v1, u2, v2, ..., un, vn)

   A bilinear form f(A, B)  :=

          f(A, B)  :=   Σ (xk vk - yk uk)

   is a symplectic form or skew form on R2n:

	f(B, A)  =   Σ (uk yk - vk xk)  =   Σ (yk uk - xk vk)

	         =  - f(A, B)

   The group of transformations in GL(2n, R) that preserve this form is
   called a symplectic group, denoted Sp(n).  For sp(1) write the
   skew form as

                           │ 0   1 │ │u1│
                 |x1 y1| │       │ │   │
                           │-1   0 │ │v1│

   Allowing the associated Lie algebra sp(n) to be complex, or the
   parameters of Sp(n) to be analytically continued results in the
   complex symplectic group and algebra, Sp(n, C) and sp(n, C)

   Then, restricting the group elements of Sp(n, C) to be also unitary
   yields the unitary symplectic group, USp(n, C), or more
   simply USp(n), and its Lie algebra Usp(n).

	Real dimension: 2 n² + n
	Manifold Topology:

   Sp(n, C):
	Real dimension: 2(2 n² + n)
	Manifold Topology:

	Real dimension: 2 n² + n
	Manifold Topology:

Semidirect Products

A prototypical, and motivating geometric example of semidirect product of groups is how translation isometry groups and rotational isometry groups of a Euclidean space combine to form the full group of Euclidean isometries. Cf. F. Klein's abstraction of geometry

The semidirect product is, more generally, a possible answer to the group extension problem:

If G is a group and H ⊂ G, a subgroup of G, G/H := {gH: g ∈ G} forms the space of the Left Cosets of G w.r.t. H, which partition G into cosets. If H is a Normal Subgroup, i.e., a subgroup of G invariant under all conjugation maps H → g-1Hg, then the space of left cosets is isomorphic to the space of similarly defined Right Cosets, and the space of Cosets G/H = M is a (factor) group isomorphic to a subgroup of G, a kind of complement of H w.r.t. G.

Then, G is said to be an extension of the group H by the group M. The problem "extension problem" is determining all groups G that are extensions of H. Decomposition of G as a semidirect product is not the only solution to the extension problem.

Example: The Poincaré group is an extension of the Lorentz group by the translation group of Minkowski space.

   A Lie algebra A is a semidirect sum of an algebra K by M iff:

	i)   K is an invariant Lie subalgebra of A, i.e.,

	             K ⊂ A,
	             [K, K] ⊆ K
	             [A, K] ⊆ K

	ii)  A = {k+m: k ∈ K, m ∈ M}, i.e.,
		K and M are closed subspaces of A.

	iii) K ∩ M = ∅

   The semidirect sum of Lie algebras exponentiates to the semidirect
   product of Lie groups.

Lie Algebra Extensions

Central extension

Split extension

The Question of Extensibility of a Lie Algebra (viz. FCCR)

   For m=1, in physics speak, using formally Hermitean q and p,

		qp - pq  = i I

   where i = √-1, and its bilinear forms,

		q², p², (1/2){q, p}

   form a Lie algebra that has the structure of any of the Lie algebraic

		so(2, 1) = su(1, 1) = sl(2, R).

   For any representation of H(1)

	[q², p²]  =  {q, [q, p²} = 2{q, p}

	[qp, q²]  =  {q, [qp, q]}  =  {q, q}  = 2q²
	[pq, q²]  =  {q, [pq, q]}  =  {q, q}  = 2q²

   Then, there is a Lie structure isomorphic to

	so(2, 1):

		[q², p²]      =  + 2{q, p}
		[{q, p}, q²]  =  + 4q²
		[p², {q, p}]  =  - 4p²

   which is almost the set of CRs of so(3) = su(2), but for the one
   sign difference that is typical of associated noncompact groups.

Notice that an another product has been assumed to exist that is outside of the Lie algebraic structure. Think of this as being constructed by the tensor product H x H, where a possible basis is I, q, p, qp, pq, q², p², a 7 dimensional Lie algebra within which one can find both H itself and the bilinear so(2, 1) algebra as subalgebras. We have found a Lie algebra which contains H as an invariant subalgebra, and which is therefore an extension of H. The dimension of the algebra is reduced from 7 to 6, by the relation of linear dependence qp = pq + iI.

Notice that sl(2, C) has real dimension 6, the inhomogeneous isl(2, C), a semidirect of sl(2, C) with the translation group T²(C) in C² has real dimension 10.

Can one extend any Lie algebra in this manner? No.
For example, it turns out that in Lie algebras su(n), any quadratic expression in the generators can be expressed as a linear combination of the generators, plus the identity so extending the algebra by quadratics in the generators cannot extend the algebra.

The key to this is that the algebra su(n) with the identity spans the full space of nxn Hermitean operators. Any power of an Hermiteam operator is always Hermitean, and so can be expanded on the algebra basis.

There is apparently no guarantee for an arbitrary Lie algebra that its simple tensor product with itself will close on a Lie algebra structure. So, the question begging to be asked is what the necessary and sufficient conditions are for this to happen. In particular, what is there about the simple (m=1) Heisenberg algebra that facilitates this closure?

A first clue is the nilpotency of H.

Extension By Bilinear Elements

Extension By Multilinear Elements of Higer Order

Symmetric Bilinear Extensions Of H

   One can extend H to various dynamical Lie algebras that include,
   separately and in combinations, the formally Hermitean bilinear
   anticommutator forms:

	 Qab  :=  (1/2){Qa, Qb},
	 Pab  :=  (1/2){Pa, Pb},
	 Wab  :=  (1/2){Qa, Pb},

NEW - redefining M, and complementing Nab below,
Mab  :=  (1/2) (Qa Qb - Pa Pb)
       =  Qab - Pab

Each of these forms are symmetric in exchange of their generating
operators and also symmetric in their indicies.  They each
span closed Lie algebras Q2, P2 and W, each of dimension m(m+1)/2, since

	[Qab, Qcd]  =  0
	[Pab, Pcd]  =  0

	[Wab, Wcd]  =  + i δad Wbc - i δbc Wad

Then Q2 and P2 are commutative Lie algebras as are Q and P, while
W is not.


	[Wab, Qc]  =  - i δbc Qa

	[Wab, Pc] =   + i δac Pb

	[Qab, Qc]  =  0

	[Pab, Qc]  =  - i Pa δbc - i δac Pb

	[Qab, Pc]  =  + i Qa δbc + i δac Qb

	[Pab, Pc]  =  0


	[W, W]   ⊂ W

	[Q2, H]  ⊂ H
	[P2, H]  ⊂ H
	[W, H]   ⊂ H

Meaning that three Lie algebras each of dimension m(m+1)/2 + 2m + 1 exist,
each containing H as an invariant Lie subalgebra:

	KQ  :=  Q2 + H
	KP  :=  P2 + H
	KW  :=   W + H

These can be combined (added) in pairs to produce three more Lie algebras,
each of dimension m(m+1) + 2m + 1  =  m² + 3m + 1, that also have H as
an invariant Lie subalgebra:

	KQP  :=  Q2 + P2 + H
	KQW  :=  Q2 +  W + H
	KWP  :=   W + P2 + H

There is then also another Lie algebra K0 of dimension 3m(m+1)/2,
containing each of the Q2, P2 and W algebras as (noninvariant) subalgebras:

	K0  :=  W + Q2 + P2

The algebra W is actually generated by Q2 and P2 together since:

	[Qab, Pcd]  =  + i Wac δbd
		         + i Wad δbc
		         + i Wbc δad
		         + i Wbd δac

These CRs and the CRs from above, together with

	[Wab, Wcd]  =  + i δad Wbc - i δbc Wad

	[Wab, Qcd]  =  - i δbc Qad - i δbd Qac

	[Pab, Wcd]  =  - i δad Pbc - i δac Pbd

show that K0 is indeed a closed Lie algebra, with noninvariant
subalgebras, W, Q2 and P2, the last two being commutative, and
physically interpretable as algebras of translations in a phase
space of symmetrized tensor products (Q x Q + P x P) of m(m+1)
dimensions.  Since K0 posesses no invariant subalgebras, it is     (no? - E)
simple, and should then fit one of classical forms, i.e., unitary,
orthogonal or symplectic.  While W, observing the signs in the CRs,
has CRs of the form of the symmetric set of compact rotational
generators of a unitary group, the Q2 and P2 have CRs of a
noncompact nature.  A bit of guesswork is in order.  First, look
at the compact unitary form to try to get the dimensionality right.

Since we have not yet added the angular momentum operators in an
m dimensional space, and also since there are no more bilinears to
add, and since the angular momentum operators will constitute an
special orthogonal algebra so(m) for which there are m(m-1)/2 generators,
which angular momentum operators added to the K0 algebra would
create an algebra of dimension

	3m(m+1)/2 + m(m-1)/2  =  m(m+1) + m(m+1)/2 + m(m-1)/2
	                      =  m(m+1) + m²
	                      =  2m² + m
	                      =  m(2m + 1),

it is clear that K0 must be isomorphic to some noncompact form of
a subalgebra of either an orthogonal algebra, so(2m+1) or a symplectic
algebra, sp(2m), for m > 3, or su(2m) of dimension 4m² - 1.

	GUESS sp(m, m) ?  so(m+1, m) ?  The second is difficult to figure,
	yet both fit the m=1 special case for so(2, 1) = su(1, 1).
	The generation of W from [Q2, P2], suggests the symplectic form.
	But su(n, n), with dimension 4n² - 1 = 2n(2n+1)+2n(2n-1)
	looks more like what should be expected.  That might be,
	respectively, an so(2n+1) subalgebra type added to an so(2n)
	subalgebra type.  The so(2n+1) type is then the noncompact K0.
	Then, can we get n from m as a Diophantine equation?

		3m(m+1)/2 = 2n(2n+1)
		3m(m+1) = 4n(2n+1)

	If m=2
		18 = 4n(2n+1)
		 9 = 2n(2n+1)
	         n = none since 9 = 3² is 9's prime decomposition.

W acts on both commutative (noninvariant) subalgebras irreducibly.

Adding K0 to H, obtain a "nonrotational" dynamical Lie algebra H1
of dimension (2m + 1 + 3m(m+1)/2):

	H1  :=  K0 + H

This is the largest Lie algebra that can be obtained that includes H
as an invariant Lie algebra without including angular momentum, i.e.,
rotation operators.
The Lattice Of Nonrotational Invariant Lie Subalgebras of H1


The Fourier Transforms of H

The natural existence of Fourier transforms in H algebras makes them interesting and important to mathematicians engaged with general theories of harmonic analysis, where once again the Heisenberg algebra appears naturally. It is also the q-p Fourier relationship that leads to the fundamental uncertainty relationships of quantum mechanics - which is not say that these are the only Lie algebras (groups) that naturally define Fourier transforms.

Abstractly speaking, a Fourier transform is simply a unitary operator
that is idempotent of order 4.  There are members of KQP,

	Na  :=  (1/2)( Qa Qa + Pa Pa - I )

which are formally Hermitean iff Qa and Pa are,

	Na  =  Na.

These can easily be shown to obey,

	[Na, Qb]  =  -i Pa δab

	[Na, Pb]  =  +i Qa δab

	[Na, Nb]  =  0

demonstrating that Na is a generator of rotations in the (Qa, Pa) plane
of H.  Explicitly, the
Baker-Campbell-Hausdorff (BCH) formula
shows that these commutation relations imply for complex ω:

		    ‖ Qa ‖                     ‖ cos ω  -sin ω ‖ ‖ Qa ‖
    exp( i ω Na )  ‖     ‖ exp( -i ω Na )  =  ‖               ‖ ‖     ‖
		    ‖ Pa ‖                     ‖ sin ω   cos ω ‖ ‖ Pa ‖

Taking ω = (π/2), defines a Fourier transform Fa for the (Qa, Pa) plane.
A full Fourier transform can then be defined as the product of all such
commuting transforms,

	F  :=  Π exp( i ωa Na ),

	ωa  :=  π/2, for all a,

a product of unitary operators, or equivalently, since the mutual
commutivity, [Na, Nb] = 0 => [Fa, Fb] = 0, as,

	F  :=  exp( i (π/2) N ),

	N  :=  Σ Na

F then expresses a Fourier duality in H between the two commutative
Lie subalgebras of Q and P, treated as subspaces.  Since the
exponential map, maps, as above, Lie algebras to Lie groups, the above
rotation is a specific adjoint action of the associated Heisenberg
group on its algebra.

Then, where "t" is a transpose operator, F acts on a vector whose
components are Qa, and a vector whose components are Pa as:

	F (Q1, Q2, ..., Qm)t F  =  + (P1, P2, ..., Pm)t

	F (P1, P2, ..., Pm)t F  =  - (Q1, Q2, ..., Qm)t


	F Qk F  =  + Pk

	F Pk F  =  - Qk

which expresses the Fourier dual structure of H(m) explicitly.

	F Lab F  =  Lab


Symplectic Structure Of H

The Fourier duality just expressed is also an expression of
symplectic structure.

		       │ Qk │        │ 0  -1 │ │ Qk │
		    F  │     │ F  =  │       │ │     │,
		       │ Pk │        │ 1   0 │ │ Pk │

		       │ Q │        │ 0  -1 │ │ Q │
		    F  │   │ F  =  │       │ │   │.
		       │ P │        │ 1   0 │ │ P │

The symplectic structure of classical mechanics is expressed in
the Hamiltonian formalism similarly through Poisson brackets, or
the structure of the canonical equations of motion, indicating
that the Heisenberg algebra structure is itself the primary expression
of quantum mechanical symplectic structure.  This is at a deeper
level than that of the natural symplectic structure of unitary Lie
groups that may be associated with UIRREPS, which is contained in
the construction of commutators.

F is then a symplectic, and formally unitary transformation of H
which preserves CCR; preservation of the CCR implies preservation of
the symplectic structure of commutators.


Antisymmetric Bilinear Extensions

Now, let us return to include the "angular momentum" operators
for a,b = 1, 2, ..., m,

	Lab  :=  i (Qa Pb - Qb Pa)

If the Qa and Pb are formally Hermitean, then so is Lab which is
also as a collection of tensorial components,
antisymmetric in its indicies, and
therefore from standard counting techniques, possessed of m(m-1)/2
linearly independent elements.

These close amomg themselves under commutation, realizing the Lie
algebra CRs of the special orthogonal, so(m) Lie algebra:

	[Lab, Lcd]  = + i Lbc δad
	                - i Lac δbd
	                + i Lad δbc
	                - i Lbd δac

with dimension m(m-1)/2.

The full bilinearly extended Heisenberg algebra H2, is then given as

		H2  :=  H1 + L

This H2 algebra has dimension,

	3m(m+1)/2 + m(m-1)/2  =  m(m+1) + m(m+1)/2 + m(m-1)/2
	                      =  m(m+1) + m²
	                      =  2m² + m

which is the correct algebra dimension for so(2m+1), m > 2 and
for sp(2m), for m > 3.


3(m(m-1)/2 + m) - (m(m-1)/2 + m)  =  2(m(m-1)/2 + m) = m(m+1) left over.
These left overs are the (1/2){Xa, Xb}, X = Q, P and a =/= b.
But, if [Xa, Xb] = 0, the {} are not necessary.

For m=1, there are 3 + 3 = 6 elements of the algebra.        0 ang. mom.
                                                             1 nrg. op.
pack# = 2

For m=2, there are 5 + 9 = 14 elements of the algebra.       1 ang. mom.
                                                             2 nrg. op.
pack# = 6

For m=3, there are 7 + 18 = 25 elements of the algebra.      3 ang. mom.
                                                             3 nrg. op.
pack# = 12                  25 = 5²

For m=4, there are 9 + 30 = 39 elements of the algebra.      6 ang. mom.
                                                             4 nrg. op.
pack# = 24  > 20 = (30 - (6+4))

For m=5, there are 11 + 45 = 56 elements of the algebra.    10 ang. mom.
                                                             5 nrg. op.

	(all bilins - ang. mom. - nrg. op) = m(m+1)

For m=6, there are 13 + 63 = 76 elements of the algebra.
For m=7, there are 15 + 84 = 99 elements of the algebra.
For m=8, there are 17 + 108 = 125 elements of the algebra.
For m=9, there are 19 + 135 = 154 elements of the algebra.

Within the bilinear subalgebra, the n(n-1)/2 (bivector) angular momentum
operators can be found, as well a small collection of possible
energy operators.

When the abstract Heisenberg algebra is realized in an associative
algebra of linear operators, necessarily including unbounded
operators, the bilinear also close on themselves.  One can
also see the bilinear as elements of the universal enveloping
algebra, thus avoiding the selection of a specific representation.

This construction
actually gives the Levi decomposition of the Dynamical algebra.

[Levi's theorem says that any Lie algebra is the semidirect product
of a semisimple Lie algebra with a solvable Lie algebra.]

A nilpotent Lie algebra
is automatically solvable, and the bilinear form a semisimple Lie
algebra.  Specifically, for m=1, the dynamical algebra D of dimension 6
is the semidirect product of the kinematic Heisenberg algebra H with 3
elements and a semisimple Lie algebra isomorphic to su(1, 1), K again
of dimension 3.  The kinematic Heisenberg algebra is an invariant Lie
subalgebra of the dynamical Lie algebra.  Symbolically:

	D  =  H + K

	[D, D]  ⊆  D,  [H, H]  ⊆  {I},  [K, K]  ⊆  K

	[D, H]  ⊆  H, subsuming [K, H] ⊆ H

This extends to any m and is the basis of the semidirect product of the
Levi decomposition.  Is there an easy characterization of the structure
of K when m > 1?

Define the tensor operators:

	Qab  :=  (1/2){Qa, Qb},                   m(m+1)/2 of them l.i.
		Qab  =  Qba

	Wab  :=  (1/2){Qa, Pb},                   m(m+1)/2 of them l.i.
		Wab  =  Wba

	Pab  :=  (1/2){Pa, Pb},                   m(m+1)/2 of them l.i.
		Pab  =  Pba

NEW replacing old M's
Mab  :=  (1/2)(Qa Qb - Pa Pb)
	Mab  =  Mba

in K.

The associated (i/2)[.,.] forms all are either I or 0, except the
generalized angular momentum operators, which are bilinear tensor
(bivector) operators.

	Lab  :=  (i/2)(Qa Pb - Qb Pa)           n(n-1)/2 of them
		Lab  =  - Lba

old M

	Mab  :=  (i/2)(Pa Qb - Pb Qa)           n(n-1)/2 of them
		Mab  =  - Mba

		Lab - Mab  =  (i/2)(Wab - Wba)   (linear dependency)

		Lab + Mab  =  (i/2)[Qa, Pb] - (i/2)[Qb, Pa]
		             =  0

	Sab  :=  (1/2)(Qa Pb + Qb Pa)
		Sab  =  Sba

	Tab  :=  (1/2)(Pa Qb + Pb Qa)
		Tab  =  Tba

		Sab - Tab  =  (1/2)[Qa, Pb] + (1/2)[Qb, Pa]
		             =  i δab

		Sab + Tab  =  (1/2)(Wab + Wba)

old M
Thus, L, M, S and T tensor operators can be reconstructed from the
Heisenberg algebra CR relations and W.

For n=3, the bivector generators of rotations are hodge-* dual to
pseudovectors having three independent components, so angular momentum
is "simplified" to appear as if it were a vector in order to rot the minds of
high school children.  A "rotation" is generally *not* about an axis, but
rather in a 2-plane.  Pulling an ad hoc pseudovector then out of thin
air is less than helpful when it comes to understanding the fundamental
nature of rotations.  That silly, and fundamentally misleading
"cross product" of two vectors,
sensible only for a three dimensional space, is really a Grassmann
product of vectors forming a bivector followed by a Hodge-* operator,
mapping the bivector to a pseudovector (only in three space).
The Hodge-* operation, in n space generally, maps a bivector (rank 2)
to a tensor of rank n-2.


	[Qab, Qcd]  =  0,           [Pab, Pcd]  =  0,

	[Qab, Pcd]  =  + i Wac δbd
		         + i Wad δbc
		         + i Wbc δad
		         + i Wbd δac

	[Wab, Qcd]  =  - i δbd Qac - i δbc Qad

	[Wab, Pcd]  =  + i δad Pcbb + i δac Pdb

	[Wab, Wcd]  =  + i δad Wbc - i δbc Wad

A u(n), orthogonal or symplectic algebra?

	[Wab, Qc]  =  - i δbc Qa

	[Wab, Pc] =   + i δac Pb

	[Pab, Qc]  =  - i Pa δbc - i δac Pb

	[Qab, Pc]  =  + i Qa δbc + i δac Qb

	[Lab, Qc]  =  - i Qa δbc + i δac Qb

	[Lab, Pc]  =  + i Pb δac - i δbc Pa

old M
[Mab, Qc]  =  + 1/2 Qb δac - 1/2 δbc Qa

[Mab, Pc]  =  - 1/2 Pa δbc + 1/2 δac Pb

	This has to turn out to be the CR's for an SO(n)
	[Lab, Lcd]  = + i Lbc δad
	                - i Lac δbd
	                + i Lad δbc
	                - i Lbd δac
which is the so(n) CRs

	[Lab, Qcd]  =  (1/2)(+ Qca δbd - δad Qcbb)
	               + (1/2)(+ Qda δbc - δac Qdb) 

which is the so(n) CRs of a vector operator

	[Lab, Pcd]  =  (1/2)(- Pcbb δad + δbd Pca)
	               + (1/2)(- Pdb δac + δbc Pda)

	[Lab, Wcd]  =  (1/2)(- Wcbb δad + δbd Wca)
	               + (1/2)(+ Wad δbc - δac Wbd)

old M
	[Mab, Mcd]  = + (1/2)( Mcbb ) δad
	                + (1/2)( Mac ) δbd
	                + (1/2)( Mda ) δbc
	                + (1/2)( Mbd ) δac

From Q-P duality
old M
Lab   →  Mab
Wab   →  Wba

	[Mab, Qcd]  =  (1/2)(- Qcbb δad + δbd Qca)
	               + (1/2)(- Qdb δac + δbc Qda)

	[Mab, Pcd]  =  (1/2)(+ Pca δbd - δad Pcbb)
	               + (1/2)(+ Pda δbc - δac Pdb) 

	[Mab, Wcd]  =  (1/2)(- Wbd δac + δbc Wad)
	               + (1/2)(+ Wca δbd - δad Wcbb)

Dually, when Q → P

	Sab <→ Tab

old M
	[Lab, Mcd]  =  (i/2)Tca δbd
	               - (i/2)Tbd δac
	               - (i/2)Tda δbc
	               + (i/2)Tbc δad


	[Sab, Qc]  =  - (i/2)Qa δbc - (i/2)Qb δac

	[Tab, Qc]  =  - (i/2) δac Qb - (i/2) δbc Qa
	             =  [Sab, Qc]
	[Sab, Pc]  =  + (i/2) Pa δbc + (i/2) Pb δac

	[Tab, Pc]  =  + (i/2) δac Pb + (i/2) δbc Pa
	             =  [Sab, Qc]

	[Sab, Scd]  =  (1/2)(+ Lca) δbd
	               + (1/2)(- Lad) δbc
	               + (1/2)(+ Ldb) δac
	               + (1/2)(- Lbc) δad

From Q-P duality

old M
	[Tab, Tcd]  =  (1/2)(+ Mca) δbd
	               + (1/2)(+ Mda) δbc
	               + (1/2)(+ Mdb) δac
	               + (1/2)(+ Mcbb) δad

	[Sab, Tcd]  =  (1/2)(+ Mac ) δbd
	               + (1/2)(+ Mad ) δbc
	               + (1/2)(+ Mbd ) δac
	               + (1/2)(+ Mbc ) δad

SHO energy:

Symmetric energy operator tensor

	Eab  :=  Pab + Qab

with indicial symmetry

	Eab  =  Eba

is also a vector operator of so(n) (transforming according to the
adjoint representation)

	[Lab, Ecd]  =  + i Eca δbd - i Ecbb δad
	                 + i Eda δbc - i Edb δac

which also, in this consistent interpretation, obeys,

	[Eab, Qc]  =  [Pab, Qc]
                     =  - i Pa δbc - i δac Pb

	[Eab, Pc]  =  [Qab, Pc]
                     =  + i Qa δbc + i δac Qb

Eab is a vector operator of so(n) because under an adjoint action of the
Lie group SO(n) on its algebra so(n), the collection of its components
transforms according the adjoint (vector) representation whose matrices
Xa are similarity transforms of the structure constants:

	s Xc s-1  := (Cab)c

Trace (contraction) of the energy operator tensor

	E  :=  (Qab + Pab) δab  =  Σ (Qaa + Paa)

	[E, Qc]  =  - i Pc
	[E, Pc]  =  + i Qc

For the customarily defined uncertainties
[Section XIII]
U(.) in quantum theory as
standard deviations:

	U(E) U(Qc)  >=  (const.) <(Pc)>²
	U(E) U(Pc)  >=  (const.) <(Qc)>²

CRs of F and Eab and a Dab := (1/4)(Qab - Pab).
[Dab is the new Mab above]
Do Eab, Dab and Wab close? Probably.

Similarly define

	E1  :=  (1/4)(Qab - Pab) δab  =  Σ (1/4)(Qaa - Paa)

Need to define symbols for index symmetric quantities
(Qa Qb - Pa Pb).  Going to ditch M, so, ... ?

	E2  :=  (1/4) {Qa, Pb}  =  (1/2) Wab δab

	E3  :=  (1/4) E  =  Σ (Qaa + Paa)


	[E1, E2]  =  -i E3,        [E2, E3]  =  +i E1,

	             [E3, E1]  =  +i E2

which are the CRs for an so(2, 1) or su(1,1) Lie algebra, being a noncompact
form of the well known so(3) or su(2) algebras respectively with their
relationship by local group homeomorphism of the respective Lie groups
exhibited, e.g., by the classic Cayley-Klein parameters for rotations.

This so(2, 1) algebra is then an algebra associated with collective energetic
modes of m particles that are simple harmonic oscillators.  Formally,
E1 is an SHO Lagrangian, E2 is a q-p correlation coefficient and E3 is
an SHO Hamiltonian for such a system.

To be perhaps unnecessarily emphatic, the emergence of this so(2, 1)
algebra has nothing to do with the value of m for Hm.

This little summation trick also shows a further relevence for the very same
algebra that appears when m=1.  Note that the existence of this algebraic
substructure does not depend on whether there is a energetic significance
to the Ek.

	[E1, Qc]  =  + (i/2) Pc
	[E1, Pc]  =  + (i/2) Qc

	[E2, Qc]  =  - (i/2) Qc
	[E2, Pc]  =  + (i/2) Pc

	[E3, Qc]  =  - (i/2) Pc
	[E3, Pc]  =  + (i/2) Qc

So, with E a Lie subslagebra of K0, and the specific CRs just given,
set theoretically,

	[E, H]  ⊆  H

	H is an invariant Lie subalgebra of (E + H), which can be
	expressed as a semidirect product of Lie algebras H by E:
	E acts on (transvects) H, but not conversely.

Is E invariant in K0 ?
	[E, K0] is in E?


        |           |           |       |       |
        |     L     |           |       |       |
        | so(m)     |           |       |       |
        ------------|           |  Q2   |   Q   |
        |                       |       |       |
        |                W      |       |       |
        |                       |       |       |
        ------------------------|       |       |
        |                        \      |       |
        |         P2               \    |       |
        |                            \  |       |
        -------------------------------\|       |
        |                                \      |
        |          P                       \    |
        |                                    \  |

Transition amplitudes and products of operators

	<a| X Y |b>  =   Σ <a| X |c><c| Y |b>

Any product implies a succession of two events; more products,
more successive events.  In particular, X² traces two events (processes),
and this is what is necessary to define the derivatives that
enter differential equations.
A commutator takes the difference of orderings of process types.
What is interesting is that the intermediate states summed over in
the resolution of the identity are only an arbitrary collection of
a mutually distinctive collection of an orthonormal basis, and
not an intergal over the Hilbert ball - actually an intergal over
an open semiball, thinking of the geometry of the projective Hilbert
space.  The orthonormal set is an analog of the singletons on a
measure space of outcomes as discrete set.

          All possible |c>  in I = Σ |c><c|
	<a|   X   x   Y    |b>

That a light sphere truly has a thickness is the expression of at least
a fifth dimension of physical reality. (?)
The thickness of the sphere is inversely proportional to the energy of
the pulse - approximately.
These collectively give the commutation relations for the bilinear
extended dynamical Heisenberg algebra D.  If the set of Wab is denoted by
W, similarly Lab, Mab, Sab and Tab; Qab is denoted simply by Q2, and
Pab by P2, while Qa is denoted by Q, and Pa is denoted by P,
symbolically as before,

	H  =  I + Q + P  (dimension = 2m + 1)
		[Q, Q]  =  0,      [P, P]  =  0,      [Q, P]  =  I
		[Q, I]  =  0,      [P, I]  =  0
	[H, H]  =  I

Algebra of bilinears
	K  =  Q2 + P2 + W  (dimension = n² + n(n+1)  =  2n² + n)
		[W, W]  =  W                         W subalgebra of K
		[Q2, Q2]  =  0,    [P2, P2]  =  0,   Q2, P2 Abelian subalgebras
		[W, Q2]  =  Q2,    [W, P2]  =  P2,   Q2, P2 ideals in W

			[K, W]  =  K                 W not invariant in K

		[W, Q]  =  Q,      [W, P]  =  P,

			[W, H]  =  H                 W acts invariantly on H

Full Dynamic algebra  (dimension = (2n+1) + n(2n + 1)  = 2n² + 3n + 1)
n=1: dim = 6 =    3 +  3
n=2: dim = 15 =   5 + 10
n=3: dim = 28 =   7 + (21 =  3*7) - room for 4 Heisenberg algebras
n=4: dim = 45 =   9 + (36 =  4*9) - room for 5 Heisenberg algebras
n=5: dim = 66 =  11 + 55
	D  =  H + K

	[D, D]  =  D,              [H, H]  =  H,      [K, K]  =  K

	[D, H]  =  H, subsuming [K, H] = H subsuming [W, W]  =  W
	                                             [W, H]  =  H

	Graph of subalgebras (arrows point to subalgebras, and "i"
	marks invariant subalgebra relation.)

	    Q ← H → P
		│ i
	        D (D is a semidirect product of K and H)
	      i ↓ i
	 Q2 ← K → P2 (K a semidirect product of W and Q2 + P2)
For m=1, there is a 3 dimensional algebra

	qp - pq  = i I

of formally Hermitean elements, and its linearly independent Hermitean
bilinear forms,

	q², p², (1/2){q, p}

form a Lie algebra that has the structure of any of the homomorphs

	so(2,1) = su(1,1) = sl(2,R).


	[q², p²]  =  {q, [q, p²]} = 2{q, p} = 2qp + 2(qp - I)

	[qp, q²]  =  {q, [qp, q]}  =  {q, q}  = 2q²
	[pq, q²]  =  {q, [pq, q]}  =  {q, q}  = 2q²

Then, there is a Lie structure isomorphic to

	so(2, 1):

		[q², p²]      =  + 2{q, p}
		[{q, p}, q²]  =  + 4q²
		[p², {q, p}]  =  - 4p²

which is almost that of so(3) = su(2), but for the one sign difference.

A guess that so(2, 1) generalizes to so(?, m)
Dim W  = m², so guess so(m(m-1), m), then for m=2 K would be so(2, 2),
which has dimension 6, and that happens to be correct, as can be
computed directly.

NB the complexification of su(1,1) is sl(2, C)
   the complexification of su(2) in FCCR(2) is sl(2, C)
   the complexification of su(2) in FCCR(n) 2nd order CRs is sl(2, C)

	su(2)    = so(3)     = Usp(1)
	sl(2, R) = sp(1)
	so(2, 2) = sl(2, C) x sl(2, C)
	so(4, 1) = sp(1, 1)
	su(2, 2) = so(4, 2)
		Dim su(2,2) = 15
	sl(4, C) = so(3, 3)

If H is seen as an algebra of translations, and D := K x H as a
semidirect product, the D is isomorphic to the inhomogeneous iso(2, 1),
or isu(1, 1), of real dimension 3+3=6.  But this inhomogeneous group
acts naturally on the noncommutative phase space (q, p); it is not
a commutative geometry because the "coordinates" q and p to not commute.

polynomial functions on phase space
homogeneous polynomials

actions on spaces of homogeneous polynomials

For m=2
	[qa, pb]  =  i I δab
with 5 elements

q1q1, p1p1 and q1p1 form an so(2, 1) = su(1, 1) algebra
q2q2, p2p2 and q2p2 form an so(2, 1) = su(1, 1) algebra

Additionally, consider the 6 linearly independent cross products,

p1q1 p1q2	q1p1 q1p2
     p2q2            q2p2

Only p1q2 q1p2 are not duplicated in the two su(1, 1) algebras,
and these are linearly dependent: q1p2 = p2q1.
So, for m=2, K has dimension 8 with these two and I added.
So K for spatial dim 2 contains 2 distinct su(1, 1) subalgebras
each associated with one of 2 (p & q) degrees of freedom.

There is only one (2(2-1)/2) angular momentum operator:

	(q1 p2 - q2 p1)

The sets W: q1p1 → 1/2 (q1 p1 + p1 q1)
            q1p2 → 1/2 (q1 p2 + p2 q1)
            q2p1 → 1/2 (q2 p1 + p1 q2)
            q2p2 → 1/2 (q2 p2 + p2 q2)
	Q2: q1q1 → 1/2 (q1 q1 + q1 q1)
	    q1q2 → 1/2 (q1 q2 + q2 q1)
	    q2q2 → 1/2 (q2 q2 + q2 q2)

	P2: p1p1 → 1/2 (p1 p1 + p1 p1)
	    p1p2 → 1/2 (p1 p2 + p2 p1)
	    p2p2 → 1/2 (p2 p2 + p2 p2)

Give a total of 10 operators, so the semidirect product D := K x H
gives an algebra of 10 + 5 = 15 dimensions.

Dim ( su(2,2) = so(4,2) ) = 15
heis(m-1) is an invariant subalgebra of heis(m)

	[heis(m), heis(m-1)]  =  [heis(m-1)]

Is it true that for a heis(1) subalgebra that heis(1) is an
invariant subalgebra of heis(m)?

Is it true that heis(m)  =  heis(m-1) ⊗ heis(1)
where '⊗' is a direct product?

If the first is true, the second follows, since then inductively

	heis(m)  =  ⊗ heis(1)

a direct product of m copies of heis(1).
and every heis(1) subalgebra is an invariant subalgebra of heis(m).

Then K0(2) = so(2, 1) ⊗ so(2, 1) =? so(3, 1)
	Dimensions are right.

In low dimensional Lie algebras, there are the algebra isomorphisms
	su(1, 1) = so(2, 1) = sl(2, R),         su(2) = so(3) = sp(1)
	           so(4, 1) = sp(1, 1),                 so(5) = sp(2)
	su(2, 2) = so(4, 2)

m=1: (2n+1) =  3
m=2: (2n+1) =  5
m=3: (2n+1) =  7
m=4: (2n+1) =  9
m=5: (2n+1) = 11
Schrödinger representation [v. Neumann's theorem]

Weyl Algebra
Representations in QFT
Kurt Friedrichs, myriotic representations.
Emch and C*-algebras.
Quantum algebras
Quantum Weyl-Heisenberg Algebra in 1 Dimension

		[a, a]  =  qN

	[N, a]  =  -a,	[N, a]  =  +a

with q a complex "deformation parameter"
Representation theory of nilpotent Lie algebras
Gelfand-Kirilov Conjecture - generally now proved wrong.
Yutze Chow

Foot Notes

1. One might expect the material here would be standard in any decent
   book on quantum mechanics.  I have never seen it included.  It is
   written here primarily for the more mathematically inclined student
   of quantum theory.

   Note that this exposition is not pretending to be a textbook devoted
   to the theorem-proof school of writing.  It is intended merely to
   give an entry and sketchy overview of an area that is important to
   mathematical and theoretical physicists and to some mathematicians,
   regarding physical applications.

2. A Note:  Hermiticity vs. Skewhermiticity
   In the area of Lie algebras and groups, mathematicians are prone to
   standardize discussion using skewhermitean operators in representations
   of Lie algebras, while physicists, being heavily influenced by the
   idea of "Hermitean operators representing physical observables", tend
   to think in terms of Lie algebra elements as being represented by
   Hermitean operators.  This latter choice uniformly introduces the
   messy "i" factors into commutator formulas, and so is notationally
   less economical.  Nevertheless, the following notations are mostly
   those of a physicist.  I will, however, for convenience, slip back
   and forth between the Hermitean and skewhermitean viewpoints, hoping
   this causes no great confusion.

3. The dimension m is assumed finite thoughout.  When any mathematical
   structure is taken in limit from finite to infinite dimension, there
   are certain natural metric structures that are lost and that must be
   replaced, nonuniquely, with topological notions and definitions of
   convergence.  Work in this area, necessary for quantum field theory
   has been done; it happens to be discussed here only briefly.

4. In order to keep notation as uncluttered as possible, this is the
   last time the grouping or symmetry of a pair of tensorial indices
   will be emphasised; the subscript marking "_{ab}", will hereafter
   be written simply as "ab".

STRUCTURES OF LIE ALGEBRAS of dimension r = 2, 3:


        [Petrov 1969], pp. 62-65 (goes up to order 4)
        [Jacobson 1962], pp. 11-14 (goes up to order 3)
        [Hammermesh 1964], p. 306 (almost finishes 3 and
		 a constructive and analytical method is described)
        [Miller 1968], p. 66.

   For r=2
   According to Jacobson the 2 structures for complex algebras are:

        [X1, X2] = 0  translational or toroidal

        [X1, X2] = X1 rotational

   For r=3
   According to Jacobson the 6 structures for complex algebras are:

  (a)   [X1, X2] = 0   [X2, X3] = 0   [X3, X1] = 0        trans or tor

  (b)   [X1, X2] = X3  [X2, X3] = 0   [X3, 1] = 0   Heisenberg

  (c)   [X1, X2] = X1  [X2, X3] = 0   [X3, X1] = 0   Affine group

  (d)   [X1, X2] = 0   [X2, X3] = a X2 [X3, X1] = -X1

        [X1, X2] = 0   [X2, X3] = X2   [X3, X1] = -X1 + b X3

        Above are nonisomorphic for different a & b not zero;
        (d) with a=-1 is E(2), i.e., iso(2) in terms of raising and
        lowering operators.  [Miller p. 66]

  (e)   [X1, X2] = X3  [X2, X3] = -2 X2  [1, X3] = 2 1   so(3) so(2, 1)
                                                                su(2) su(1,1)

        In case (e) think raising and lowering operators.  Jacobson calls
        it the "split three dimensional simple Lie algebra".  The complex
        extension of these algebras is sl(2, C).

        [X1, X2] = 0   [X2, X3] = X1  [X3, X1] = X2     E(2)  [Miller]

        [X1, X2] = X3  [X2, X3] = 1  [X3, 1] = X2        so(3) su(2)

        [X1, X2] = -X3  [X2, X3] = X1  [X3, X1] = 2       so(2, 1) su(1, 1)

   Hammermesh claims there are only 4 independent structures for a real
   Lie ALGEBRAS with r=3.  I believe it, if the algebras are actually
   complex, and he has misspoken.

   For real forms Petrov shows 9 real and 7 complex GROUP structures.

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Created: November 24, 2002
Last Updated: August 19, 2003