We describe representations of the simply connected covering group of the Poincaré group, {˜ {B}}... more We describe representations of the simply connected covering group of the Poincaré group, {˜ {B}}, which are associated with certain positive energy representations of SO0(2, 3), the simply connected cover of SO0(2, 3). The translation generators of these representations of {˜ {B}} can be viewed as solutions of certain algebraic equations with coefficients from a commutative algebraic extension of the
Mathematical Proceedings of the Cambridge Philosophical Society, 1988
Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and suf... more Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q + 1, q + 1)/S[U(q + 1, q) × U(1)], and SU(q + 1, q + 1)/SL(n, ℂ) ⋊ H(n) (p, q ≥ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal fi...
Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vi... more Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vice versa, the contribution of those two transactions to the checkbook’s final balance is the same. By the commutative property, order does not matter for the algebraic addition of a finite number of terms. However, for a super banker who conducts an infinite number of transactions, order may matter. If a series (sum of all transactions/terms) is convergent and the order of term does not matter, then the series is absolutely convergent. If a series is convergent but the order of terms does matter, then it is conditionally convergent. Georg Bernhard Riemann proved the disturbing result that the final sum of a conditionally convergent series could be any number at all or divergent. In two, three and higher dimensions, the matter is even worse, and such series with double and triple sums are not even well-defined without first giving sum interpretation to the (standard) order in which the se...
We study solutions of the wave equation with circular Dirichlet boundary conditions on a flat two... more We study solutions of the wave equation with circular Dirichlet boundary conditions on a flat two-dimensional Euclidean space, and we also study the analogous problem on a certain curved space which is a Lorentzian variant of the 3-sphere. The curved space goes over into the usual flat space-time as the radius R of the curved space goes to infinity. We show, at least in some cases, that solutions of certain Dirichlet boundary value problems are obtained much more simply in the curved space than in the flat space. Since the flat space is the limit R → ∞ of the curved space, this gives an alternative method of obtaining solutions of a corresponding problem in Euclidean space. I.
We have previously described an embedding of the Poincaré Lie algebra into an extension of the Li... more We have previously described an embedding of the Poincaré Lie algebra into an extension of the Lie field of the group SO0(1,4), and we used this embedding to construct irreducible representations of the Poincaré group out of representations of SO0(1,4). Some q generalizations of these results have been obtained by us i.e. we embed classical structures into quantum structures. Here we report on analogous findings for SO0(2,3): we express the basis elements of the Lie algebra of the Poincaré Lie group as irrational functions of the generators of the anti de Sitter group SO0(2,3), and thus obtain an embedding of the Poincaré Lie algebra into an extension of the Lie field of SO0(2,3). We have also obtained some generalizations to higher dimensions. We apply our resulting formulae to certain unitary, discrete series representations of SO0(2,3) associated with functions on SO0(2,3)/SO0(1,3). From the embedding theorem we obtain representations of the Poincaré Lie algebra which are associa...
The closely related fundamental theorems of projective and affine geometry are keys to understand... more The closely related fundamental theorems of projective and affine geometry are keys to understanding some important but seemingly unrelated topics in mathematical physics. Specifically, they can be used in proofs of Wigner’s theorem on ray correspondences in quantum mechanics and also to establish the scale extended Poincare group as the basic causality preserving symmetry group of special relativity. We describe these two theorems and show their roles in establishing the just mentioned results.
Let \( {W}_{n} (\mathbb{R})\) be the Weyl algebra of index n over \( \mathbb{R} \) and let \( \ti... more Let \( {W}_{n} (\mathbb{R})\) be the Weyl algebra of index n over \( \mathbb{R} \) and let \( \tilde{\mathfrak{D}} \) (so(2,1)) be a certain extension of the skew field of fractions of U(so(2;1)), the universal enveloping algebra of \( \mathfrak{U} \)so(2,1). In a previous work we have established a skew field isomorphism between \( \tilde{\mathfrak{D}} \) (so(2,1)) and \( \mathfrak{D}_{(1,1)} (\mathbb{R}) \) where\( \mathfrak{D}_{(1,1)} (\mathbb{R}) \) is the fraction field of\( {W}_{1,1} (\mathbb{R})\simeq {W}_{1} (\mathbb{R})\bigotimes(\mathbb{R})_{(y)}\) with \( (\mathbb{R})_{(y)}\) being the ring of polynomials over \( \mathbb{R} \) in the indeterminate . Using this isomorphism, we were able to construct, out of unitary and irreducible representations of the universal covering group of SO0(2,1), representations of \( {W}_{1} (\mathbb{R})\) with all of the desired properties required by physics, including hermicity of the momentum and position operators. Thus, we have obtained t...
In the present paper we start consideration of anti de Sitter holography in the general case of t... more In the present paper we start consideration of anti de Sitter holography in the general case of the \((q+1)\)-dimensional anti de Sitter bulk with boundary q-dimensional Minkowski space-time. We present the group-theoretic foundations that are necessary in our approach. Comparing what is done for \(q=3\) the new element in the present paper is the presentation of the bulk space as the homogeneous space \(G/H = SO(q,2)/SO(q,1)\), which homogeneous space was studied by Sekiguchi.
We describe an embedding of the Poincare Lie algebra into an extension of the Lie field of SO(2,3... more We describe an embedding of the Poincare Lie algebra into an extension of the Lie field of SO(2,3) (the anti-de Sitter group). We also describe higher dimensional analogs of this embedding, which connect SO(p,q) groups to their associated motion groups (higher dimensional Poincare groups). Further some results on q generalizations of these results are given. We apply our results to certain unitary, continuous series representations of SO(2,3) associated with functions on SO(2,3)/SO(1,3). From the embedding theorem we obtain representations of the Poincare Lie algebra which are associated with unitary, tachyonic representations of the Poincare group, provided these representations are integrable to group representations.
Let \(\mathbf{W}_n(\mathbb {R})\) be the Weyl algebra of index n. We have shown that by using ext... more Let \(\mathbf{W}_n(\mathbb {R})\) be the Weyl algebra of index n. We have shown that by using extension and localization, it is possible to construct homomorphisms of \(\mathbf{W}_n(\mathbb {R})\) onto its image in a localization, or a quotient thereof, of \({\mathfrak {U}}(\mathfrak {so}(2,q))\), the universal enveloping algebra of \(\mathfrak {so}(2,q)\), for n depending upon q [1]. Here we treat the \(\mathfrak {so}(2,1)\) case in complete detail. We establish an isomorphism of skew fields, specifically, \(\widetilde{\mathfrak {D}(\mathfrak {so}}(2,1))\simeq \mathfrak {D}_{(1,1)}(\mathbb {R})\) where \(\mathfrak {D}_{1,1}(\mathbb {R})\) is the fraction field of \(\mathbf{W}_{1.1}(\mathbb {R}) \simeq \mathbf{W}_1(\mathbb {R})\otimes \mathbb {R}(y)\) with \(\mathbb {R}(y)\) being the ring of polynomials in the indeterminate y and \(\widetilde{\mathfrak {D}(\mathfrak {so}}(2,1))\) is a certain extension of the skew field of fractions of \(\mathfrak {U}(\mathfrak {so}(2,1))\), which ...
It is well-known that Einstein’s first attempt to explain E = mc2 which was published in Annalen ... more It is well-known that Einstein’s first attempt to explain E = mc2 which was published in Annalen der Physik in 1905, has been criticized as problematic. In particular, it has been shown by Ives and reiterated by Jammer that it suffers from the error of circular reasoning. Attempts have been made in the scientific literature to discount the circular reasoning objection of Ives, Jammer, Arzelies and others. Fritz Rohrlich in 1990 gave a remarkably simple and concise derivation of E = mc2 along lines similar to Einstein’s but based on both momentum and energy conservation, in contrast to Einstein’s which uses only energy considerations. Rohrlich’s approach using momentum conservation is an alternative to Einstein’s, which is free from objection in logical error, and we make it quite clear on the importance of the implicit assumption of momentum conservation in any attempt to refute the circular reasoning error in Einstein’s paper. It is our contention that this point is overlooked or a...
We consider generalizations to \({SO}_0(2,q)\) of results in [13, 15, 16, 21] dealing with parall... more We consider generalizations to \({SO}_0(2,q)\) of results in [13, 15, 16, 21] dealing with parallelizations of associated induced bundles for \(SO_0(2,4) = SU(2,2)/{\mathbb {Z}}_2\) representations which are induced from (non-unitary) finite dimensional representations of a maximal parabolic subgroup.
The known radiative decay rates of the ''old'' vector mesons are ... more The known radiative decay rates of the ''old'' vector mesons are used to ; determine the suppression factor for the spectrum generating SU(4) group. From ; the suppression factor the decay rate for J(psi) $Yields$ chi(2.75) + $gamma$ and ; for the radiative decays of other vector mesons. (auth)
American Journal of Undergraduate Research, Mar 17, 2006
We give the general solutions to the wave equation with certain Dirichlet boundary conditions on ... more We give the general solutions to the wave equation with certain Dirichlet boundary conditions on two different three dimensional space times. The results of this paper should be useful for the calculation of the Casmir energy for a circular boundary in the plane, and to problems dealing with cylindrical wave guides.
In this paper, we give an explicit construction of the unitary irreducible representations of the... more In this paper, we give an explicit construction of the unitary irreducible representations of the Poincare groups in 2, 3, and 4 space-time dimensions on Hilbert spaces associated with the Schrodinger representation of the Weyl algebra for n = 1, 2, and 3, respectively. Our method of constructing the representations uses extension and localization of the enveloping algebras associated with these Weyl algebras and the Poincare algebras.
In a recent paper [1] we described a novel treatment of the unitary irreducible representations o... more In a recent paper [1] we described a novel treatment of the unitary irreducible representations of the Poincare groups in 2, 3 and 4 space-time dimensions as unitary operators on the representation spaces of the Schrodinger representation of the Heisenberg-Weyl algebra of index r = 1, 2, and 3, respectively. Here we relate this approach to the usual method of describing the representations of these Poincare groups, i.e. the Wigner-Mackey construction.
Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vi... more Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vice versa, the contribution of those two transactions to the checkbook’s final balance is the same. By the commutative property, order does not matter for the algebraic addition of a finite number of terms. However, for a super banker who conducts an infinite number of transactions, order may matter. If a series (sum of all transactions/terms) is convergent and the order of term does not matter, then the series is absolutely convergent. If a series is convergent but the order of terms does matter, then it is conditionally convergent. Georg Bernhard Riemann proved the disturbing result that the final sum of a conditionally convergent series could be any number at all or divergent. In two, three and higher dimensions, the matter is even worse, and such series with double and triple sums are not even well-defined without first giving sum interpretation to the (standard) order in which the se...
Let Wn(ℝ) be the Weyl algebra of index n. It is well known that so(p, q) Lie algebras can be view... more Let Wn(ℝ) be the Weyl algebra of index n. It is well known that so(p, q) Lie algebras can be viewed as quadratic polynomial (Lie) algebras in Wn(ℝ) for p + q = n with the Lie algebra multiplication being given by the bracket [a, b] = ab − ba for a, b quadratic polynomials in Wn(ℝ). What does not seem to be so well known is that the converse statement is, in a certain sense, also true, namely, that, by using extension and localization, it is possible, at least in some cases, to construct homomorphisms of Wn(ℝ) onto its image in a localization of U(so(p + 2, q)), the universal enveloping algebra of so(p + 2, q), and m = p + q. Since Weyl algebras are simple, these homomorphisms must either be trivial or isomorphisms onto their images. We illustrate this remark for the so(2, q) case and construct a mappping from Wq(ℝ) onto its image in a localization of U(so(2, q)). We prove that this mapping is a homomorphism when q = 1 or q = 2. Some specific results about representations for the lowest dimensional case of W1(ℝ) and U(so(2, 1)) are given.
We describe representations of the simply connected covering group of the Poincaré group, {˜ {B}}... more We describe representations of the simply connected covering group of the Poincaré group, {˜ {B}}, which are associated with certain positive energy representations of SO0(2, 3), the simply connected cover of SO0(2, 3). The translation generators of these representations of {˜ {B}} can be viewed as solutions of certain algebraic equations with coefficients from a commutative algebraic extension of the
Mathematical Proceedings of the Cambridge Philosophical Society, 1988
Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and suf... more Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q + 1, q + 1)/S[U(q + 1, q) × U(1)], and SU(q + 1, q + 1)/SL(n, ℂ) ⋊ H(n) (p, q ≥ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal fi...
Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vi... more Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vice versa, the contribution of those two transactions to the checkbook’s final balance is the same. By the commutative property, order does not matter for the algebraic addition of a finite number of terms. However, for a super banker who conducts an infinite number of transactions, order may matter. If a series (sum of all transactions/terms) is convergent and the order of term does not matter, then the series is absolutely convergent. If a series is convergent but the order of terms does matter, then it is conditionally convergent. Georg Bernhard Riemann proved the disturbing result that the final sum of a conditionally convergent series could be any number at all or divergent. In two, three and higher dimensions, the matter is even worse, and such series with double and triple sums are not even well-defined without first giving sum interpretation to the (standard) order in which the se...
We study solutions of the wave equation with circular Dirichlet boundary conditions on a flat two... more We study solutions of the wave equation with circular Dirichlet boundary conditions on a flat two-dimensional Euclidean space, and we also study the analogous problem on a certain curved space which is a Lorentzian variant of the 3-sphere. The curved space goes over into the usual flat space-time as the radius R of the curved space goes to infinity. We show, at least in some cases, that solutions of certain Dirichlet boundary value problems are obtained much more simply in the curved space than in the flat space. Since the flat space is the limit R → ∞ of the curved space, this gives an alternative method of obtaining solutions of a corresponding problem in Euclidean space. I.
We have previously described an embedding of the Poincaré Lie algebra into an extension of the Li... more We have previously described an embedding of the Poincaré Lie algebra into an extension of the Lie field of the group SO0(1,4), and we used this embedding to construct irreducible representations of the Poincaré group out of representations of SO0(1,4). Some q generalizations of these results have been obtained by us i.e. we embed classical structures into quantum structures. Here we report on analogous findings for SO0(2,3): we express the basis elements of the Lie algebra of the Poincaré Lie group as irrational functions of the generators of the anti de Sitter group SO0(2,3), and thus obtain an embedding of the Poincaré Lie algebra into an extension of the Lie field of SO0(2,3). We have also obtained some generalizations to higher dimensions. We apply our resulting formulae to certain unitary, discrete series representations of SO0(2,3) associated with functions on SO0(2,3)/SO0(1,3). From the embedding theorem we obtain representations of the Poincaré Lie algebra which are associa...
The closely related fundamental theorems of projective and affine geometry are keys to understand... more The closely related fundamental theorems of projective and affine geometry are keys to understanding some important but seemingly unrelated topics in mathematical physics. Specifically, they can be used in proofs of Wigner’s theorem on ray correspondences in quantum mechanics and also to establish the scale extended Poincare group as the basic causality preserving symmetry group of special relativity. We describe these two theorems and show their roles in establishing the just mentioned results.
Let \( {W}_{n} (\mathbb{R})\) be the Weyl algebra of index n over \( \mathbb{R} \) and let \( \ti... more Let \( {W}_{n} (\mathbb{R})\) be the Weyl algebra of index n over \( \mathbb{R} \) and let \( \tilde{\mathfrak{D}} \) (so(2,1)) be a certain extension of the skew field of fractions of U(so(2;1)), the universal enveloping algebra of \( \mathfrak{U} \)so(2,1). In a previous work we have established a skew field isomorphism between \( \tilde{\mathfrak{D}} \) (so(2,1)) and \( \mathfrak{D}_{(1,1)} (\mathbb{R}) \) where\( \mathfrak{D}_{(1,1)} (\mathbb{R}) \) is the fraction field of\( {W}_{1,1} (\mathbb{R})\simeq {W}_{1} (\mathbb{R})\bigotimes(\mathbb{R})_{(y)}\) with \( (\mathbb{R})_{(y)}\) being the ring of polynomials over \( \mathbb{R} \) in the indeterminate . Using this isomorphism, we were able to construct, out of unitary and irreducible representations of the universal covering group of SO0(2,1), representations of \( {W}_{1} (\mathbb{R})\) with all of the desired properties required by physics, including hermicity of the momentum and position operators. Thus, we have obtained t...
In the present paper we start consideration of anti de Sitter holography in the general case of t... more In the present paper we start consideration of anti de Sitter holography in the general case of the \((q+1)\)-dimensional anti de Sitter bulk with boundary q-dimensional Minkowski space-time. We present the group-theoretic foundations that are necessary in our approach. Comparing what is done for \(q=3\) the new element in the present paper is the presentation of the bulk space as the homogeneous space \(G/H = SO(q,2)/SO(q,1)\), which homogeneous space was studied by Sekiguchi.
We describe an embedding of the Poincare Lie algebra into an extension of the Lie field of SO(2,3... more We describe an embedding of the Poincare Lie algebra into an extension of the Lie field of SO(2,3) (the anti-de Sitter group). We also describe higher dimensional analogs of this embedding, which connect SO(p,q) groups to their associated motion groups (higher dimensional Poincare groups). Further some results on q generalizations of these results are given. We apply our results to certain unitary, continuous series representations of SO(2,3) associated with functions on SO(2,3)/SO(1,3). From the embedding theorem we obtain representations of the Poincare Lie algebra which are associated with unitary, tachyonic representations of the Poincare group, provided these representations are integrable to group representations.
Let \(\mathbf{W}_n(\mathbb {R})\) be the Weyl algebra of index n. We have shown that by using ext... more Let \(\mathbf{W}_n(\mathbb {R})\) be the Weyl algebra of index n. We have shown that by using extension and localization, it is possible to construct homomorphisms of \(\mathbf{W}_n(\mathbb {R})\) onto its image in a localization, or a quotient thereof, of \({\mathfrak {U}}(\mathfrak {so}(2,q))\), the universal enveloping algebra of \(\mathfrak {so}(2,q)\), for n depending upon q [1]. Here we treat the \(\mathfrak {so}(2,1)\) case in complete detail. We establish an isomorphism of skew fields, specifically, \(\widetilde{\mathfrak {D}(\mathfrak {so}}(2,1))\simeq \mathfrak {D}_{(1,1)}(\mathbb {R})\) where \(\mathfrak {D}_{1,1}(\mathbb {R})\) is the fraction field of \(\mathbf{W}_{1.1}(\mathbb {R}) \simeq \mathbf{W}_1(\mathbb {R})\otimes \mathbb {R}(y)\) with \(\mathbb {R}(y)\) being the ring of polynomials in the indeterminate y and \(\widetilde{\mathfrak {D}(\mathfrak {so}}(2,1))\) is a certain extension of the skew field of fractions of \(\mathfrak {U}(\mathfrak {so}(2,1))\), which ...
It is well-known that Einstein’s first attempt to explain E = mc2 which was published in Annalen ... more It is well-known that Einstein’s first attempt to explain E = mc2 which was published in Annalen der Physik in 1905, has been criticized as problematic. In particular, it has been shown by Ives and reiterated by Jammer that it suffers from the error of circular reasoning. Attempts have been made in the scientific literature to discount the circular reasoning objection of Ives, Jammer, Arzelies and others. Fritz Rohrlich in 1990 gave a remarkably simple and concise derivation of E = mc2 along lines similar to Einstein’s but based on both momentum and energy conservation, in contrast to Einstein’s which uses only energy considerations. Rohrlich’s approach using momentum conservation is an alternative to Einstein’s, which is free from objection in logical error, and we make it quite clear on the importance of the implicit assumption of momentum conservation in any attempt to refute the circular reasoning error in Einstein’s paper. It is our contention that this point is overlooked or a...
We consider generalizations to \({SO}_0(2,q)\) of results in [13, 15, 16, 21] dealing with parall... more We consider generalizations to \({SO}_0(2,q)\) of results in [13, 15, 16, 21] dealing with parallelizations of associated induced bundles for \(SO_0(2,4) = SU(2,2)/{\mathbb {Z}}_2\) representations which are induced from (non-unitary) finite dimensional representations of a maximal parabolic subgroup.
The known radiative decay rates of the ''old'' vector mesons are ... more The known radiative decay rates of the ''old'' vector mesons are used to ; determine the suppression factor for the spectrum generating SU(4) group. From ; the suppression factor the decay rate for J(psi) $Yields$ chi(2.75) + $gamma$ and ; for the radiative decays of other vector mesons. (auth)
American Journal of Undergraduate Research, Mar 17, 2006
We give the general solutions to the wave equation with certain Dirichlet boundary conditions on ... more We give the general solutions to the wave equation with certain Dirichlet boundary conditions on two different three dimensional space times. The results of this paper should be useful for the calculation of the Casmir energy for a circular boundary in the plane, and to problems dealing with cylindrical wave guides.
In this paper, we give an explicit construction of the unitary irreducible representations of the... more In this paper, we give an explicit construction of the unitary irreducible representations of the Poincare groups in 2, 3, and 4 space-time dimensions on Hilbert spaces associated with the Schrodinger representation of the Weyl algebra for n = 1, 2, and 3, respectively. Our method of constructing the representations uses extension and localization of the enveloping algebras associated with these Weyl algebras and the Poincare algebras.
In a recent paper [1] we described a novel treatment of the unitary irreducible representations o... more In a recent paper [1] we described a novel treatment of the unitary irreducible representations of the Poincare groups in 2, 3 and 4 space-time dimensions as unitary operators on the representation spaces of the Schrodinger representation of the Heisenberg-Weyl algebra of index r = 1, 2, and 3, respectively. Here we relate this approach to the usual method of describing the representations of these Poincare groups, i.e. the Wigner-Mackey construction.
Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vi... more Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vice versa, the contribution of those two transactions to the checkbook’s final balance is the same. By the commutative property, order does not matter for the algebraic addition of a finite number of terms. However, for a super banker who conducts an infinite number of transactions, order may matter. If a series (sum of all transactions/terms) is convergent and the order of term does not matter, then the series is absolutely convergent. If a series is convergent but the order of terms does matter, then it is conditionally convergent. Georg Bernhard Riemann proved the disturbing result that the final sum of a conditionally convergent series could be any number at all or divergent. In two, three and higher dimensions, the matter is even worse, and such series with double and triple sums are not even well-defined without first giving sum interpretation to the (standard) order in which the se...
Let Wn(ℝ) be the Weyl algebra of index n. It is well known that so(p, q) Lie algebras can be view... more Let Wn(ℝ) be the Weyl algebra of index n. It is well known that so(p, q) Lie algebras can be viewed as quadratic polynomial (Lie) algebras in Wn(ℝ) for p + q = n with the Lie algebra multiplication being given by the bracket [a, b] = ab − ba for a, b quadratic polynomials in Wn(ℝ). What does not seem to be so well known is that the converse statement is, in a certain sense, also true, namely, that, by using extension and localization, it is possible, at least in some cases, to construct homomorphisms of Wn(ℝ) onto its image in a localization of U(so(p + 2, q)), the universal enveloping algebra of so(p + 2, q), and m = p + q. Since Weyl algebras are simple, these homomorphisms must either be trivial or isomorphisms onto their images. We illustrate this remark for the so(2, q) case and construct a mappping from Wq(ℝ) onto its image in a localization of U(so(2, q)). We prove that this mapping is a homomorphism when q = 1 or q = 2. Some specific results about representations for the lowest dimensional case of W1(ℝ) and U(so(2, 1)) are given.
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