An Atavistic Lie Algebra

An infinite-dimensional Lie Algebra is proposed which includes, in its subalgebras and limits, most Lie Algebras routinely utilized in physics. It relies on the finite oscillator Lie group, and appears applicable to twisted noncommutative QFT and CFT.


Introduction
Consider the familiar forced oscillator driven by an external time-dependent source.
Since H = exp(−λ(t)α † + λ * (t)α) α † α − |λ(t)| 2 exp(λ(t)α † − λ * (t)α), the integration of the model relies on the coherent states [1] underlain by the noncompact oscillator group G (or H 4 ) [2]. This is the solvable, rank 2, dimension 4, Lie group generated by the oscillator creation and annihilation operators, their Heisenberg commutator (which is central), and the occupation number operator. It turns out that this group, G, as well as related Vertex algebras [3], control the structure of an infinite dimensional Lie Algebra brought into play in this letter. This algebra, remarkably, contains as subalgebras or limits of it, most of the Lie algebras routinely encountered in physics.
The infinite dimensional Lie algebra we introduce in this letter and explore below is [J a m 1 ,m 2 , J b n 1 ,n 2 ] = e is(m 1 e −a n 2 −m 2 e a n 1 ) J a+b m 1 +e a n 1 , m 2 +e −a n 2 − e is(n 1 e −b m 2 −n 2 e b m 1 ) J a+b n 1 +e b m 1 , n 2 +e −b m 2 , (2) where the upper indices, a, b, the lower ones, m 1 , m 2 , .. and the parameter s are arbitrary, unless restricted by some further expediency.
It satisfies the Jacobi identity, which is evident from its merely amounting to the antisymmetrization of the associative group product, J a m 1 ,m 2 J b n 1 ,n 2 = e is(m 1 e −a n 2 −m 2 e a n 1 ) J a+b m 1 +e a n 1 , m 2 +e −a n 2 .
(Associativity amounts to (J a m 1 ,m 2 J b n 1 ,n 2 )J c k 1 ,k 2 = J a m 1 ,m 2 (J b n 1 ,n 2 J c k 1 ,k 2 ).) Naturally, the symmetrization of this product into an anticommutator further yields a consistent graded extension of the Lie algebra.
The algebra (2) contains, as a subalgebra specified by vanishing second subscripts, m 2 = n 2 = ... = 0, the Vertex (coherent-state) Algebra [4], Another, more familiar and general, subalgebra is the infinite Lie algebra specified by vanishing of all superscripts a = b = ... = 0, and by integer subscripts: it is the Sine Algebra [5], This one, in turn, for s = − /2, is recognized as the Moyal Bracket algebra on the basis of the Fourier modes exp(im 1 x + im 2 p) of a toroidal phase space with unit radii [5]. Up to a normalization, the Moyal Brackets represent the antisymmetrization of Groenewold's celebrated star product [6], Alternatively, for s = 2π/N, and integer N, the sine algebra is seen to represent GL(N), and to thus include all classical Lie algebras, (A N , B N , C N , D N ) [7]. Furthermore [5], for s → 0 (or N → ∞), this algebra goes to the algebra of Poisson Brackets, also realized on a toroidal phase space, SDiff(T 2 ), cf [8]; and this one contains the Virasoro algebra as a subalgebra, through judicious summation over the first subscripts m 1 , n 1 , ..., [8].
This, then, is the basis of our remark that the Lie Algebra introduced encompasses and frames quite a range of the Lie algebras normally utilized in physics.
In this letter, we explore basic features of this Lie algebra, eqn (2), its applicability, and generalization. We provide a few explicit useful realizations and representations of restrictions thereof, which illuminate its structure.
Throughout, we stress the somewhat untypical correspondence mentioned, which applies both to the algebra (2) itself, as well as the subalgebras we sketched: namely, a finite-dimensional Lie group product, which yields the infinite Lie algebras when antisymmetrized-and graded (anticommutator) extensions, when symmetrized.

The Atavistic Algebra, realizations, and representations
The Atavistic Algebra (2) product (3) might be rewritten more symmetrically, through the redefinition so that the product (3) amounts to Nevertheless, we stick to the original form, to minimize superfluous indices.
The semidirect product nature of eqn (3) and thus of the Atavistic Algebra (2) is readily recognizable by recalling the ⋆-product underlying the sine algebra (5). We first revert to the version of the star product (6) parameterized for our purpose, where x, p range from 0 to 1. The most widely used property of this product is its associativity. Thus, strings of operators of the form f (x, p)⋆, for any functions f (x, p) on the unit T 2 , may be equivalently evaluated indifferently to the grouping of multiplication chosen. Consequently, choosing a Fourier mode basis for integers m 1 , m 2 , and defining the standard product law follows [5], underlying eqn (5) when antisymmetrized. The finite Lie group corresponding to this product is well-known to be the (dimension 3) Heisenberg group [9].
Now consider a phase-space-area-preserving dilation operator D(a), which braids associatively as A standard realization of this operator is Moreover, this operator formally commutes with the above star product, as is plain in integral kernel representations of the star product [6]. More directly, this also follows from braiding D(a) past the mode as above, eqn (11). Braiding past the simple right-hand side must equal stepwise braiding past each of the factors on the ⋆-product on the left, so it is manifest that the ⋆-product remains invariant. Of course, the reason for the invariance is the area-preservation feature of the dilation operator chosen, since the phase proportioned by s is a two-dimensional cross product, amounting to a phase-spacearea element.
Furthermore, evidently, Consequently, it follows directly that the Atavistic Algebra elements J a m 1 ,m 2 may be constructed out of M m 1 ,m 2 D(a) = J 0 m 1 ,m 2 D(a), i.e., J a m 1 ,m 2 = exp(i(m 1 x + m 2 p)) ⋆ D(a).
Writing this out explicitly, the reader may recognize that the product (3) and thus the Atavistic Algebra (2) are, indeed, satisfied.
Thus, operation by the J a m 1 ,m 2 on a function f (x, p) on this phase space consists of sequential rescalings and shifts of its variables and multiplication by a phase, The form (7) corresponds to V a m 1 ,m 2 = D(a)J a m 1 ,m 2 .

Reduction of variables
Now note that the torus variables x, p commute with each other, so that, effectively, the above realization is a direct product of the same type of operator J a m 1 ,m 2 ≡ e ism 1 m 2 /2 e im 1 x e sm 2 ∂x e ax∂x , acting on variables x, and, disjointly, acting on variables p, which is to say In turn, the product of these operators is J a m 1 ,m 2 J b n 1 ,n 2 = e −is(m 1 e −a n 2 −m 2 e a n 1 )/2 J a+b m 1 +e a n 1 , m 2 +e −a n 2 , i.e., the same as eqn (3), for minus half the value of the parameter s. It is plain that, in general (irrespective of realization), This halving of representations and phases is already apparent in the first of refs [5]: a realization introduced by Hoppe there effectively represents the star product on half the variables of phase space. Plainly, this recursive phase-addition phenomenon extends to arbitrarily long direct product strings of Js, as a reflection of the simple coproduct structure of their logarithms 1 .
One would not expect finite dimensional representations for the generic Atavistic Algebra. If the expression eqn (24) looks familiar, it is because it is. It is, in fact, the parameterization of the oscillator Lie group G investigated by Miller [2]. He provides, in gratifying detail, the representations of this Lie group on finite vector spaces (by matrices), unfaithfully; and, faithfully, on infinitedimensional Bargmann space, by differential operators, eqn (19)-for instance, since J a m 1 ,m 2 f (x) = e im 1 (x−sm 2 /2) f (e a (x+ sm 2 )), representations on associated Laguerre polynomials are specified by this simple action on the generating functions of these polynomials. Representations on Hermite Hilbert space then follow [2].
The four generators of the oscillator group G are abstractions of the operators α, α † , 1 1, augmented by the occupation number operator, normally α † α. These are now abstracted to Faithful representations of this need be infinite dimensional; but unfaithful ones could be finite dimensional, as long as the central generator E is not the identity. This can be arranged by nilpotent matrices.
For example [2], one might consider 3×3 nonhermitean matrices, and satisfies, instead, J a m 1 ,m 2 J b n 1 ,n 2 = exp −is 2 (m 1 e −a n 2 − m 2 e a n 1 )E J a+b m 1 +e a n 1 , m 2 +e −a n 2 .
However, exp − is 2 (m 1 e −a n 2 − m 2 e a n 1 )E = 1 1 − is 2 (m 1 e −a n 2 − m 2 e a n 1 )E is not a pure phase, as the (traceless) center E is not the identity matrix.
The inadequacy of this construction in representing the pure phase appears generic to all finite dimensional representations sought. Nevertheless, we illustrate explicitly in the next section that finite dimensional matrix representations can be available for suitable restrictions of the Atavistic Algebra.
The product now reads This product (and the corresponding antisymmetrization Lie Algebra) can be represented by Sylvester's celebrated p × p matrix basis for GL(p) groups [13]. His standard clock and shift unitary unimodular matrices, are for indices r, t defined mod p, r = 0, 1, 2, ..., p − 1. Consequently, the characteristic braiding identity [9,5,7].
The complete set of p 2 unitary unimodular p × p matrices where and TrM (m 1 ,m 2 ) = pδ m 1 ,0 δ m 2 ,0 , suffice to span the group algebra of GL(p). Since Now, in addition, consider the discrete scaling (doubling) matrix [14], again for indices r, t defined mod p, r = 0, 1, 2, ..., p − 1. The cyclic structure holds by virtue of the identity 2 p−1 = 1 mod p. (Note that a smaller power of 2 may return to 1 before the power p − 1, e.g. 2 3 = 1 mod 7, but this does not compromise the present construction.) The action of the doubling matrix is Consequently, and hence the unitary matrices satisfy eqn (30). Thus, these J s provide a p-dimensional representation, and hence a representation of the restricted Atavistic Lie Algebra. Through the direct product recursive procedure of the preceding section, they can be augmented to selected higher dimensional ones.
However, since Sylvester's basis is complete, R is representable in terms of the above p 2 Ms-in fact, it is the phased sum of all p × p matrices M, normalized by p, since, ∀m 1 , m 2 , Thus, since, e.g., is represented trivially, the representations displayed are not faithful.
It is not hard to generalize to scaling (squeezing) matrices [14] with analogous effects. For instance, "Fermat's Little Theorem" [15] dictates n p−1 = 1 mod p, for any odd prime p and, e.g., 0 < n < p. Thus, instead of R, a matrix T scaling by n, T rt ≡ δ nr,t , T p−1 = 1 1, acts through log base n, i.e. a =ã ln n, yielding analogous representations for the restricted Atavistic Lie Algebra. The question as to whether suitable tensor products of such spaces could yield faithful representations, instead, remains open.

Generalizations to linear canonical transforms and discussion
It turns out that D(a) in Section 2 need not be so restricted, if an associative product of the above type is sought. Indeed, any Sp(2) linear symplectic transformation will do, when it comes to leaving