The Experts below are selected from a list of 12075 Experts worldwide ranked by ideXlab platform
Ian M Musson - One of the best experts on this subject based on the ideXlab platform.
-
on the goldie quotient ring of the Enveloping Algebra of a classical simple lie superAlgebra
Journal of Algebra, 2001Co-Authors: Ian M MussonAbstract:Abstract If g is a classical simple Lie superAlgebra ( g ≠ P(n)), the Enveloping Algebra U( g ) is a prime ring and hence has a simple artinian ring of quotients Q(U( g )) by Goldie's Theorem. We show that if g has Type I then Q(U( g )) is a matrix ring over Q(U( g 0)). On the other hand, if g = osp(1, 2r) then by extending the center of U( g ) we obtain a prime ring whose Goldie quotient ring is a matrix ring over the quotient division ring of a Weyl Algebra. This is an analog of a result of Gelfand and Kirillov.
-
the Enveloping Algebra of the lie superAlgebra 1 2
Representation Theory of The American Mathematical Society, 1997Co-Authors: Ian M MussonAbstract:Let g be the Lie superAlgebra osp(1, 2r) and U(g) the Enveloping Algebra of g. In this paper we obtain a description of the set of primitive ideals PrimU(g) as an ordered set. We also obtain the multiplicities of composition factors of Verma modules over U(g), and of simple highest weight modules forU(g) when regarded as a U(g0)-module by restriction. 0.1. Let g = g0 ⊕ g1 be a finite dimensional complex classical simple Lie superAlgebra. In [M1] we showed that any primitive ideal in U(g) is the annihilator of a simple highest weight module L(λ), (see 0.2–0.5 for notation). To complete the description of the set of primitive ideals PrimU(g), it is necessary to say when two modules L(λ), L(μ) have the same annihilator. For Lie superAlgebras of Type I, this was done in [L2] using a bijection between PrimU(g0) and PrimU(g). However this bijection does not preserve inclusions. In this paper we study the case where g = osp(1, 2r) and obtain a description of Prim U(g) as an ordered set. We also obtain the multiplicities of composition factor of Verma modules over U(g), and of L(λ) when regarded as a U(g0)-module by restriction. The orthosymplectic Lie superAlgebra osp(V, β) may be defined as the Lie superAlgebra of all linear operators on a Z2-graded vector space V preserving a nondegenerate even bilinear supersymmetric form β. We refer to [K1, 2.1.2] or [Sch, II. 4.3.A, page 129] for more details. In [M3] we give an alternative construction for g = osp(1, 2r) using the rth Weyl Algebra. This leads to a construction of an analog of the Joseph ideal in U(g). There are several related reasons why we might expect U(g) to be structurally similar to U(g0) when g = osp(1, 2r). For example the Harish-Chandra map yields an isomorphism Z(g) ' S(h) , all weights in h∗ are typical, and all finite dimensional modules are completely reducible. The results of this paper tend to confirm this expectation. Some of the proofs in this paper work for Lie superAlgebras other than osp(1, 2r). For example most results hold for typical representations of s`(r, 1). In order to state our results in greater detail, we introduce some notation. Received by the editors January 27, 1997 and, in revised form, July 25, 1997. 1991 Mathematics Subject Classification. Primary 17B35. Research partially supported by National Science Foundation grant DMS 9500486. c ©1997 American Mathematical Society
-
on the center of the Enveloping Algebra of a classical simple lie superAlgebra
Journal of Algebra, 1997Co-Authors: Ian M MussonAbstract:0.1. Let g be a semisimple Lie Algebra with Cartan subAlgebra h , Weyl Ž . group W, and Enveloping Algebra U g . Several results illustrate the Ž . Ž . important role played by the center Z g of U g . First there is the Ž . Harish-Chandra homomorphism which shows that Spec Z g ( h*rW, Ž . and describes the action of Z g on highest weight modules. Next we w x recall the separation of variables Theorem of Kostant Ko; D, 8.2.4 which Ž . states that there is an ad g-invariant subspace K of U g such that the Ž . Ž . multiplication map K m Z g a U g is an isomorphism of ad g-modules. w Kostant’s theorem is a key ingredient in the proof of a result of Duflo D, x 8.4.3 stating that the annihilator of a Verma module is generated by its Ž . intersection with Z g . A version of the Harish-Chandra homomorphism has been given by Kac for basic classical simple Lie superAlgebras. We recall the details in Subsection 1.1. The proof of Kostant’s theorem depends heavily on the fact that every finite dimensional g-module is completely reducible. By contrast if g is a simple Lie superAlgebra, such that every finite dimensional g-module is completely reducible, then either g is a simple Lie Algebra or g is an Ž . w orthosymplectic Lie superAlgebra g s osp 1, 2 r for some r G 1 Sch, x Theorem 1, p. 239 . Ž . Ž . In this paper we show that for g s osp 1, 2 r we again have U g ( Ž . K m Z g for K ad-invariant. In addition if E is any finite dimensional
-
a classification of primitive ideals in the Enveloping Algebra of a classical simple lie superAlgebra
Advances in Mathematics, 1992Co-Authors: Ian M MussonAbstract:For a Lie superAlgebra 9 we denote the even and odd parts of 9 by go and g,, respectively. The simple Lie superAlgebra 9 is called classical if f0 is reductive. For 9 classical simple we study primitive ideals in the Enveloping Algebra U(g). Our main result is that any graded primitive ideal is the annihilator of a graded simple quotient of a Verma module. This is an analogue of the well-known theorem of Duflo [D] on primitive ideals in the Enveloping Algebra of a semisimple Lie Algebra. The proof is based on Duflo’s theorem and some work of E. Letzter [Ll, L2] on primitive ideals in finite ring extensions. The definition of a Verma module depends on the existence of a triangular decomposition in 9. This is dicussed in Section 1. A more precise statement of the main theorem is given Section 2. In Section 3 we discuss some corollaries, for example we show that if JZ # Q(H) then graded prime ideals are prime (Corollary 3.1), and if f # P(n), then any factor ring of U(g) has the same left and right Krull dimension (Corollary 3.3). Classical simple Lie superAlgebras which are not Lie Algebras have been classified by Kac [Kl, Theorem 2, p. 441 (see also [Sch, Theorem 1, p. 1401). In the notation of Kac these Algebras are as follows. Scheunert’s notation, if different is given in parentheses. A(m,n)=sl(m+l,n+l), m#n,m,n~O(spl(m+1,n+1)) A(n, n) = son + 1, n + l)/(Zzn+z>, n>O(spl(n+1,n+1)/@rz,+2)
Carlos Tamarit - One of the best experts on this subject based on the ideXlab platform.
-
the noncommutative u 1 higgs kibble model in the Enveloping Algebra formalism and its renormalizability
Journal of High Energy Physics, 2007Co-Authors: C P Martin, Domingo Sanchezruiz, Carlos TamaritAbstract:We discuss the renormalizability of the noncommutative U(1) Higgs-Kibble model formulated within the Enveloping-Algebra approach. We consider both the phase of the model with unbroken gauge symmetry and the phase with spontaneously broken gauge symmetry. We show that against all odds the gauge sector of the model is always one-loop renormalizable at first order in θμν, perhaps, hinting at the existence of a new symmetry of the gauge sector of the model. However, we also show that the matter sector of the model is non-renormalizable whatever the phase.
-
the noncommutative u 1 higgs kibble model in the Enveloping Algebra formalism and its renormalizability
arXiv: High Energy Physics - Theory, 2006Co-Authors: C P Martin, Domingo Sanchezruiz, Carlos TamaritAbstract:We discuss the renormalizability of the noncommutative U(1)Higgs-Kibble model formulated within the Enveloping-Algebra approach. We consider both the phase of the model with unbroken gauge symmetry and the phase with spontaneously broken gauge symmetry. We show that against all odds the gauge sector of the model is always one-loop renormalizable at first order in theta^{mu nu}, perhaps, hinting at the existence of a new symmetry of the gauge sector of the model. However, we also show that the matter sector of the model is non-renormalizable whatever the phase.
Chelsea Walton - One of the best experts on this subject based on the ideXlab platform.
-
maps from the Enveloping Algebra of the positive witt Algebra to regular Algebras
Pacific Journal of Mathematics, 2016Co-Authors: Susan J Sierra, Chelsea WaltonAbstract:We construct homomorphisms from the universal Enveloping Algebra of the positive (part of the) Witt Algebra to several different Artin-Schelter regular Algebras, and determine their kernels and images. As a result, we produce elementary proofs that the universal Enveloping Algebras of the Virasoro Algebra, the Witt Algebra, and the positive Witt Algebra are neither left nor right noetherian.
-
the universal Enveloping Algebra of the witt Algebra is not noetherian
Advances in Mathematics, 2014Co-Authors: Susan J Sierra, Chelsea WaltonAbstract:Abstract This work is prompted by the long standing question of whether it is possible for the universal Enveloping Algebra of an infinite dimensional Lie Algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal Enveloping Algebra of the Witt (or centerless Virasoro) Algebra is not noetherian. To show this, we prove our main result: the universal Enveloping Algebra of the positive part of the Witt Algebra is not noetherian. We employ algebro-geometric techniques from the first author's classification of (noncommutative) birationally commutative projective surfaces. As a consequence of our main result, we also show that the Enveloping Algebras of many other (infinite dimensional) Lie Algebras are not noetherian. These Lie Algebras include the Virasoro Algebra and all infinite dimensional Z -graded simple Lie Algebras of polynomial growth.
Susan J Sierra - One of the best experts on this subject based on the ideXlab platform.
-
ideals in the Enveloping Algebra of the positive witt Algebra
Algebras and Representation Theory, 2020Co-Authors: Alexey V Petukhov, Susan J SierraAbstract:Let W+ be the positive Witt Algebra, which has a $\mathcal {C}$ -basis $\{e_{n}: n \in \mathcal {Z}_{\geq 1}\}$ , with Lie bracket [ei,ej] = (j − i)ei+j. We study the two-sided ideal structure of the universal Enveloping Algebra U(W+) of W+. We show that if I is a (two-sided) ideal of U(W+) generated by quadratic expressions in the ei, then U(W+)/I has finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending chain condition. We conjecture that analogous facts hold for arbitrary ideals of U(W+), and verify a version of these conjectures for radical Poisson ideals of the symmetric Algebra S(W+).
-
maps from the Enveloping Algebra of the positive witt Algebra to regular Algebras
Pacific Journal of Mathematics, 2016Co-Authors: Susan J Sierra, Chelsea WaltonAbstract:We construct homomorphisms from the universal Enveloping Algebra of the positive (part of the) Witt Algebra to several different Artin-Schelter regular Algebras, and determine their kernels and images. As a result, we produce elementary proofs that the universal Enveloping Algebras of the Virasoro Algebra, the Witt Algebra, and the positive Witt Algebra are neither left nor right noetherian.
-
the universal Enveloping Algebra of the witt Algebra is not noetherian
Advances in Mathematics, 2014Co-Authors: Susan J Sierra, Chelsea WaltonAbstract:Abstract This work is prompted by the long standing question of whether it is possible for the universal Enveloping Algebra of an infinite dimensional Lie Algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal Enveloping Algebra of the Witt (or centerless Virasoro) Algebra is not noetherian. To show this, we prove our main result: the universal Enveloping Algebra of the positive part of the Witt Algebra is not noetherian. We employ algebro-geometric techniques from the first author's classification of (noncommutative) birationally commutative projective surfaces. As a consequence of our main result, we also show that the Enveloping Algebras of many other (infinite dimensional) Lie Algebras are not noetherian. These Lie Algebras include the Virasoro Algebra and all infinite dimensional Z -graded simple Lie Algebras of polynomial growth.
Branislav Jurco - One of the best experts on this subject based on the ideXlab platform.
-
Enveloping Algebra valued gauge transformations for non abelian gauge groups on non commutative spaces
European Physical Journal C, 2000Co-Authors: Branislav Jurco, Stefan Schraml, Peter Schupp, J WessAbstract:An Enveloping Algebra-valued gauge field is constructed, its components are functions of the Lie Algebra-valued gauge field and can be constructed with the Seiberg-Witten map. This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces.
-
differential calculus on quantized simple lie groups
Letters in Mathematical Physics, 1991Co-Authors: Branislav JurcoAbstract:Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's Enveloping Algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal Enveloping Algebra, is obtained for our differential calculi. Real forms for q ∈ ℝ are also discussed.