Localization and the Weyl algebras
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ELEMENTARY PARTICLES AND FIELDS Theory
Localization and the Weyl Algebras∗ Patrick Moylan** Department of Physics, The Pennsylvania State University, Abington College, USA Received May 18, 2016
Abstract—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 Wm (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. DOI: 10.1134/S106377881703022X
1. INTRODUCTION Localization, or formation of quotients, is a powerful tool in mathematics with many known applications. It is used to relate different algebraic structures which share some common similarities. An important example is the Gelfand–Kirillov conjecture [1]. It is useful for us to give a brief description of it here, since it is very similar to problem studied in this paper, and it also serves to fix some notation used by us. Let K be an algebraically closed field. Given a Lie algebra L over K , denote by U (L) the universal enveloping algebra of L. Since U (L) is a Noetherian domain, it admits a field of fractions which we shall denote by K ) denote the Weyl algebra of index D(L). Let Wn (K n over the field K (it is generated over K by 2n generators p1 , . . . , pn , q1 , . . . , qn subject to the relations [pi , pj ] = [qi , qj ] = 0 and [pi , qj ] = δij for all i, j ≤ n). Given a collection of free variables y1 , . . . , ys we define K ) := Wn (K K ) ⊗ K [y1 , . . . , ys ]. Wn,s (K K) Being a Noetherian domain, the algebra Wn,s (K K ). In [1] also admits a field of fractions denoted Dn,s (K Gelfand and Kirillov put forth the following conjecture: K) = 0 Gelfand–Kirillov conjecture. If char(K and L is the Lie algebra of an algebraic K -group, ∗ **
The text was submitted by the author in English. E-mail: [email protected]
K ) for some n, s depending then D(L) ∼ = Dn,s (K on L. In [1] the conjecture was settled for nilpotent Lie algebras, sl(n) and gl(n). A major breakthrough for the case of L simple occurred in 2010, when Premet [2] proved that the conjecture fails for simple Lie algebras of type Bn (n ≥ 3), Dn (n ≥ 4), E6 , E7 , E8 or F4 . The conjecture also makes sense over fields which are not algebraically closed. Clearly, if D(L) ∼ = F
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