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Journal of Generalized Lie Theory and Applications
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Matrix Bosonic realizations of a Lie colour algebra with three generators and ve relations of Heisenberg Lie type

Gunnar SIGURDSSON a and Sergei D. SILVESTROV b

aDepartment of Theoretical Physics, School of Engineering Sciences, Royal Institute of Technology (KTH) { AlbaNova University Center, SE-106 91 Stockholm, Sweden, E-mail: [email protected]

bCentre for Mathematical Sciences, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden, E-mail:[email protected]

Received Date: January 15, 2009; Revised Date: August 10, 2009

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Abstract

We describe realizations of a Lie colour algebra with three generators and ve relations by matrices of power series in noncommuting indeterminates satisfying Heisenberg's canonical commutation relation of quantum mechanics. The obtained formulas are used to construct new operator representations of this Lie colour algebra using canonical representation of the Heisenberg commutation relation and creation and annihilation operators of the quantum mechanical harmonic oscillator.

Introduction

The main object studied in this paper is the unital associative algebra with three generators A1, A2, and A3 satisfying de ning commutation relations

Equation

The main goal is to show how A1, A2, and A3 can be expressed, using elements A and B, obeying Heisenberg's canonical commutation relation

Equation

The canonical representation of the commutation relation (1.6) is given by choosing A as the usual di erentiation operator and B as multiplication by x acting on di erentiable functions of one real variable x, on polynomials in one variable, or on some other suitable linear space of functions invariant under these operators. In quantum mechanics, these operators, when considered on the Hilbert space of square integrable functions, are essentially the same as the canonical Heisenberg-Schrodinger observables of momentum and coordinate, di ering just by a complex scaling factor. The Heisenberg canonical commutation relation (1.6) is also satis ed by the annihilation and creation operators in a quantum harmonic oscillator.

Since the 1970s, generalized (colour) Lie algebras have been an object of constant interest in both mathematics and physics [1,3-5,8-17,19-21,23,25-27]. Description of representations of these algebras is an important and interesting general problem. It is well known that representations of three-dimensional Lie algebras play an important role in the representation theory of general Lie algebras and groups, both as test examples and building blocks. Similarly, one would expect the same to be true for three-dimensional Lie colour algebras and superalgebras with respect to general Lie colour algebras and superalgebras. The representations of nonisomorphic algebras have di erent structure. In [26,27], three-dimensional Lie colour algebras are classi ed in terms of their structure constants, that is, in terms of commutation relations between generators. In [11,17,23], quadratic central elements and involutions on these algebras are calculated. In [16,25], Hilbert space *-representations are described for the graded analogues of the Lie algebra Equation and of the Lie algebra of the group of plane motions, two of the nontrivial algebras from the classi cation. The classi cation of *-representations in [16,25] is achieved, using the method of dynamical systems based on generalized Mackey imprimitivity systems.

The colour Heisenberg Lie algebra is another important nontrivial algebra in the classi- cation of three-dimensional Lie colour algebras obtained in [26,27]. In the paper [24] we approached representations of this algebra in a totally di erent way than it was done in [16,25]. Namely, we studied those representations which can be obtained as power series in operator representations of Heisenberg's canonical commutation relations by rst obtaining in general realizations of the colour Heisenberg Lie algebra generators in terms of power series in elements of an associative algebra obeying the Heisenberg's canonical commutation relations and then combining these realizations with canonical representations of Heisenberg's canonical commutation relations.

In this paper we extend these investigations of realizations via Heisenberg's canonical commutation relations to another colour Lie algebra with three generators and ve relations. This algebra can be considered as another colour analogue of the Heisenberg Lie algebra. However, we show that a structure of this algebra is quite di erent from that for the algebra considered in [24] as far as realization via Heisenberg's canonical commutation relations is concerned. In Section 2 we show that, with a natural choice for A1 as the rst generator of the Heisenberg algebra corresponding to di erentiation, there are no nonzero power series in Heisenberg generators which can be taken as A2 and A3 so that the three relations (1.1)-(1.3) are satis ed. In Lemma 2.2, we describe all such formal power series solutions A2 and A3 for the rst two relations (1.1)-(1.2) as in [24]. In Theorem 2.4, we present all such formal power series solutions for A2 and A3 satisfying the three relations (1.1)-(1.3) showing that A3 = 0 is the only possibility. Using this result we get in Corollary 2.7 that A2 = A3 = 0 must hold for such realizations of the ve relations (1.1)-(1.5). However, by considering 2 × 2 matrices with entries chosen as formal power series in the noncommuting indeterminates A and B satisfying Heisenberg's canonical commutation relations, we demonstrate how it is possible to construct nontrivial realizations of (1.1)-(1.5). We also construct concrete operator representations by applying this construction to the canonical representation of Heisenberg canonical commutation relations and to the simple quantum mechanical harmonic oscillator.

Matrix power series realizations

Throughout this article Equation denotes the eld of complex numbers and Equation the set of nonnegative integers. By Equation and Equation we mean the ring of polynomials and formal power series over Equation, respectively.

Consider a set {A1;A2;A3} in some associative algebra over Equation with unit element I satisfying commutation relations (1.1)-(1.5). From relation (1.1) we obtain Equation and hence by (1.2) it follows that

Equation

Using only (1.2) and (1.3) we may conclude that Equation commutes with both A1 and A2, that is, Equation. If merely (1.1) and (1.2) are satis ed and if Equation for some nonzero Equation, then by (2.1) we have

Equation

Applying the famous Wintner-Wielandt theorem [18,28,29], we have that no elements in any unital normed algebra can satisfy the Heisenberg canonical commutation relation

Equation

So, we obtain from (2.2) the following result.

Proposition 2.1. The commutation relations

Equation

together with Equation cannot be satis ed by bounded operators on a Hilbert space or even generally by elements in any unital normed algebra.

When computing with power series in noncommuting elements A and B of an associative unital algebra Equation, we use the usual addition and multiplication rules of the Magnus algebra of noncommutative formal power series in two indeterminates (see [2]). However, we assume that A and B are not free, but satisfy at least the Heisenberg commutation relation as elements in Equation. We denote the algebra that we are working with byEquation. In addition to the subalgebra of Equation generated by A and B, consisting of noncommutative polynomials in A and B, the algebra Equation may contain other elements which are in nite noncommutative power series in A and B not belonging to Equation. The problem of equality of two elements in Equation is a very complex matter in itself, deeply connected both to the properties of noncommutative power series and Heisenberg's relation and to the structure of the algebra Equation, and properties of A and B in Equation. We say that an element of Equation is in the (B, A)- normal form (resp., (A, B)-normal form) if it is a noncommutative power series built of only ordered monomials Equation. In order to be able to enjoy the equality properties in a similar way with formal power series as in the polynomial case, we assume throughout this article that two formal power series in A and B, written in the (B, A)-normal form (resp., in the (A, B)-normal form), are equal if and only if their coecients are the same and in particular such a series is zero if and only if all coecients are zero. This important equality assumption is actually an assumption on Equation, on the algebra Equation as well as on A and B as elements in Equation. In the particular case of polynomials in A and B, that is, for the subalgebra of Equation generated by A and B, the assumption yields the same property as in Equation, namely thatEquation are linearly independent as subsets of Equation.

With the assumption above we may claim the equality of two elements of Equation if they are equal to the same element in (B, A)-normal form (resp., in (A, B)-normal form). However, it is important to observe that Equation may well contain elements which cannot be represented on (B, A)-normal form, or (A, B)-normal form or even on either of them. In most of the statements in this article we will adhere to the (B, A)-normal forms and the corresponding equality assumptions. But, we will also comment and at some instances will formulate the corresponding results when instead the (A, B)-normal forms and corresponding equality of series is used. Which of these assumptions is used will be clear from the context. We refer to [6,7] for further discussion on power series extensions of the Heisenberg algebra, Diamond lemma, and normal forms.

In the paper [24] we have proved the following useful result.

Lemma 2.2. Let A1=A and assume that A2 and A3 are elements of the algebra Equation written in the (B, A)-normal form, that is,

Equation

Then A1, A2, and A3 satisfy the commutation relations

Equation

if and only if

Equation

Remark 2.3. It follows by this lemma that the commutation relations

Equation

can be satis ed by polynomial A2 and A3 ( nite sums) only if A2 = A3 = 0:

Theorem 2.4. Suppose that A1, A2, and A3 are elements of the algebra Equation such that A1 = A and A2, A3 are formal power series in the (B, A)-normal form given as

Equation

Then A1, A2, and A3 will satisfy the commutation relations

Equation

if and only if

Equation

where Equation

Proof. By Lemma 2.2 we have, considering only the two anticommutation relations, a general solution of the form

Equation

Applying the rules of Lemma 4 in [24] yields

Equation

and hence

Equation

which is a functional-di erential equation for V and W. Thus A2A3A3A2 = 0 implies 2BW(–A)W(A) = 0, which for a formal power series W(A) yields W(–A)W(A) = 0. The set of complex formal power series Equation is an integral domain and hence W(A) = 0. This means that A2 = T(B, A)V (A) and A3 = 0.

Remark 2.5. Note that in Theorem 2.4, A1A2 + A2A1 = A3 = 0 is the only nontrivial relation left. The other two relations are trivially satis ed when A3 = 0, independently of A1 and A2. So, under conditions of Theorem 2.4, one can say that

Equation

is equivalent to

Equation

Remark 2.6. In Lemma 2.2, the formal series A2 and A3 are expressed in the (B, A)-normal form. This is a natural ordering when we think of A as the usual di erentiation operator and B as a multiplication operator M acting on di erentiable functions on the real line, given by Equation. In other situations, it may be more appropriate to consider the reversed order.

Taking A2 and A3 in Lemma 2.2 to be in the (A, B)-normal form, but keeping A1 = A, the general solution will be changed to the following form:

Equation

where

Equation

If we consider a solution in the (A;B)-normal form satisfying the three relations in Theorem 2.4, a completely similar proof shows that also in this case A2 = V (A)U(A, B) and A3 = 0.

Corollary 2.7. Suppose that A1, A2 and A3 are elements of Equation such that A1 = A, and A2 and A3 are formal power series in the (B, A)-normal form given as

Equation

Then A1, A2 and A3 satisfy

Equation

if and only if A2 = A3 = 0.

Proof. By Theorem 2.4, we know that A2 = T(B, A)V (A) and A3 = 0. Since

Equation

the relation Equation holds only if V (–A)V (A) = 0, which, for a power series V (A), yields V (A) = 0 and A2 = 0.

Now let A1, A2 and A3 be 2 × 2 matrices on the form

Equation

where A11, A12, A21, A22, L, and M are elements of some associative algebra or ring Equation. Then

Equation

Therefore, A1A2 + A2A1 = A3 is equivalent to the conditions

Equation

and similarly we have A1A3 + A3A1 = 0 if and only if

Equation

Remark 2.8. Observe that for any elements L and M in Equation the relations

Equation

are always satis ed, since moreover we have A2A3 = A3A2 = 0. Thus, under the conditions (2.3) and (2.4), elements A1, A2, and A3 satisfy commutation relations (1.1)-(1.5). This gives a method of construction for realizations of (1.1)-(1.5).

Let us assume that either L or M are left or right invertible. Then A21 = 0, and hence A1 is upper triangular, that is,

Equation

So, we have the following useful statement.

Lemma 2.9. Let L and M be elements of an associative algebra or ring Equation, such that at least one of them is left or right invertible in Equation. Then the elements

Equation

of the algebra (ring) Equation of 2 × 2 matrices over Equation satisfy the commutation relations

Equation

if and only if

Equation

together with

Equation

Given A1 and A2 of the form in (2.7) and (2.5), respectively, there exists an M so that A3 of the form in (2.5) satisfies (2.6) if and only if

Equation

Proof. The rst part was proved before the statement. The last relation follows by eliminating M using (2.8). On the other hand, if (2.9) holds, then M = A11L + LA22 satis es (2.8).

Lemma 2.10. Let the matrices

Equation

be elements of Equation, the algebra (ring) of 2 × 2 matrices over an associative algebra (ring) Equation, and assume Equation Then

Equation

are satis ed if and only if B1, B2, and B3 satisfy

Equation

Moreover, Equation and A2A3 =A3A2 =A2A3 νA3A2 =0, independently onEquation

Proof. The proof follows from the following easily checked equalities:

Equation

By Lemma 2.2 we know that if ABBA = I for A and B in some unital associative algebra Equation, then

Equation

satisfy commutation relations

Equation

and also if B1 = A and B2 and B3 are (B, A)-normally ordered power series in A and B, then B1, B2, and B3 satisfy (2.11) if and only if B2 and B3 are of the form (2.10). A similar result is obtained if we choose to write B2 and B3 in the (A, B)-normal form.

Note also that we can apply Lemma 2.2 and its version in the (A, B)-normal form on a new set of generators Equation. Observing that T(–A, B) = U(A, B) and U(B,–A) = T(B, A), the resulting general solutions are

Equation

Combining this with Lemma 2.10, we can construct a realization of (1.1)-(1.5) using A and B satisfying the Heisenberg commutation relation ABBA = I.

Theorem 2.11. Assume A and B are two elements of an associative unital algebra or a ring (with identity) Equation, satisfying the Heisenberg commutation relation AB – BA = I. Let Equation and define the following 2 × 2 matrices:

Equation

whereEquation. Then A1, A2, and A3 satisfy the commutation relations

Equation

Moreover, A2A3 = A3A2 = 0.

When the (A, B)-normal forms and the corresponding equality assumption in Equation is used, the following analogue of Theorem 2.11 holds.

Theorem 2.12. Assume A and B are two elements of an associative unital algebra or a ring (with identity) Equation, satisfying the Heisenberg commutation relation AB – BA = I. Let Equation, and de ne the following 2 × 2 matrices:

Equation

where Equation. Then A1, A2, and A3 satisfy the commutation relations

Equation

Moreover, A2A3 = A3A2 = 0.

Example 2.13. Introduce the operators

Equation

acting on Equation. These operators satisfy AB – BA = I. Note also that U(A, B) = U(M, –) = T(M, ). Let Equation be any operator. Then A1, A2, and A3 defined in Theorem 2.12 are realized on Equation by the operators

Equation

acting on Equation. By the same theorem they satisfy the ve relations for any power series V (M) and W(M) in M. Note that here Equation is the parity operator (see [24]). For example, if V (t) = 1 and W(t) = t2, then

Equation

Example 2.14. Referring to [22] and [24, Example 5], we consider a sequence of functions Equation defined by

Equation

where a* is the linear operator

Equation

defined for any positive real constant x0. It follows that

Equation

for n = 0, 1, 2, ..., where Hn are the Hermite polynomials. Defining the linear operator a as

Equation

it can be shown that Equation Moreover,

Equation

for any Equation The sequence of functionsEquation describes the energy eigenstates of the simple quantum mechanical harmonic oscillator. Consider now the two di erential operators a* ("creation" operator) and a ("annihilation" operator) de ned on the linear space Ω = Equation consisting of all complex linear combinations of functions from the set of eigenfunctions Equation. Since now aa* – a*a = 1, we can de ne A and B in Theorem 2.11 as A = a and B = a* acting on Ω. Suppose Equation is any operator. Then A1, A2, and A3 introduced in Theorem 2.11 are realized as the operators

Equation

acting on Equation.

Recall that Equation. Hence

Equation

Moreover, for Equation,

Equation

Now let Equation andEquation, whereEquation. Since

Equation

it readily follows that

Equation

keeping in mind that Equation.

Similarly, using the fact that Equation, one obtains

Equation

This means that

Equation

In the first expression, we have to assume thatEquation. Introduce a basisEquation given by

Equation

For Equation, it follows that

Equation

And, for Equation,

Equation

Moreover, for all Equation,

Equation

Acknowledgements

This work was supported by the Swedish Research Council, The Crafoord Foundation, The Royal Swedish Academy of Sciences, The Royal Physiographic Society in Lund, and The Swedish Foundation for International Cooperation in Research and Higher Education (STINT). The rst author is grateful to the Centre for Mathematical Sciences, Lund University for hospitality and support during his visits in Lund.

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