Surjectivity of the Taylor map for complex nilpotent Lie groups

2009 ◽  
Vol 146 (1) ◽  
pp. 177-195 ◽  
Author(s):  
BRUCE K. DRIVER ◽  
LEONARD GROSS ◽  
LAURENT SALOFF-COSTE
Keyword(s):  
Lie Algebra ◽  
Hermitian Form ◽  
Holomorphic Functions ◽  
Companion Paper ◽  
Wigner Transform ◽  
Enveloping Algebra ◽  
Taylor Map ◽  
Hörmander’S Condition ◽  
Simply Connected ◽  
Insight Into

AbstractA Hermitian formqon the dual space,*, of the Lie algebra,, of a simply connected complex Lie group,G, determines a sub-Laplacian, Δ, onG. Assuming Hörmander's condition for hypoellipticity, there is a smooth heat kernel measure, ρt, onGassociated toetΔ/4. In a companion paper [6], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions inL2(G, ρt) ontoJt0(a subspace of) the dual of the universal enveloping algebra of. Here we give a very different proof of the surjectivity of the Taylor map under the assumption thatGis nilpotent. This proof provides further insight into the structure of the Taylor map. In particular we show that the finite rank tensors are dense inJt0when the Lie algebra is graded and the Laplacian is adapted to the gradation. We also show how the Fourier–Wigner transform produces a natural family of holomorphic functions inL2(G, ρt), for appropriatet, whenGis the complex Heisenberg group.

The depth and LS category of a topological space

2018 ◽  
Vol 123 (2) ◽  
pp. 220-238
Author(s):  
Yves Félix ◽  
Steve Halperin
Keyword(s):  
Topological Space ◽  
Hopf Algebra ◽  
Lie Algebra ◽  
Finite Type ◽  
Universal Enveloping Algebra ◽  
Primitive Elements ◽  
Enveloping Algebra ◽  
Cw Complex ◽  
Lusternik Schnirelmann ◽  
Simply Connected

The depth of an augmented ring $\varepsilon \colon A\to k $ is the least $p$, or ∞, such that \begin {equation*} \Ext _A^p(k , A)\neq 0. \end {equation*} When $X$ is a simply connected finite type CW complex, $H_*(\Omega X;\mathbb {Q})$ is a Hopf algebra and the universal enveloping algebra of the Lie algebra $L_X$ of primitive elements. It is known that $\depth H_*(\Omega X;\mathbb {Q}) \leq \cat X$, the Lusternik-Schnirelmann category of $X$. For any connected CW complex we construct a completion $\widehat {H}(\Omega X)$ of $H_*(\Omega X;\mathbb {Q})$ as a complete Hopf algebra with primitive sub Lie algebra $L_X$, and define $\depth X$ to be the least $p$ or ∞ such that \[ \Ext ^p_{UL_X}(\mathbb {Q}, \widehat {H}(\Omega X))\neq 0. \] Theorem: for any connected CW complex, $\depth X\leq \cat X$.

Formally-radical Functions in Elements of a Nilpotent Lie Algebra and Noncommutative Localizations

2010 ◽  
Vol 17 (spec01) ◽  
pp. 749-788 ◽  
Author(s):  
Anar Dosi
Keyword(s):  
Lie Algebra ◽  
Holomorphic Functions ◽  
Universal Enveloping Algebra ◽  
Nilpotent Lie Algebra ◽  
Character Space ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Taylor Spectrum ◽  
Algebra Of Operators

In the present paper, we introduce the sheaf 𝔗𝔤 of germs of non-commutative holomorphic functions in elements of a finite-dimensional nilpotent Lie algebra 𝔤, which is a sheaf of non-commutative Fréchet algebras over the character space of 𝔤. We prove that 𝔗𝔤(D) is a localization over the universal enveloping algebra [Formula: see text] whenever D is a polydisk, which in turn allows to describe the Taylor spectrum of a supernilpotent Lie algebra of operators in terms of the transversality.

scholarly journals CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

2021 ◽  
Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Invariant Theory ◽  
Reductive Group ◽  
Symmetric Tensor ◽  
Free Associative Algebra ◽  
Tensor Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

scholarly journals PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

2021 ◽  
Author(s):  
DMITRI I. PANYUSHEV ◽  
OKSANA S. YAKIMOVA
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Invariant Theory ◽  
Cyclic Permutation ◽  
Poisson Brackets ◽  
Enveloping Algebra ◽  
Order Automorphism ◽  
Finite Dimensional ◽  
Compatible Poisson Brackets ◽  
Commutative Subalgebras

AbstractLet 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).

PBW deformations of quantum symmetric algebras and their group extensions

2016 ◽  
Vol 15 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Piyush Shroff ◽  
Sarah Witherspoon
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Extensions ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Necessary And Sufficient ◽  
Special Case

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

scholarly journals A note on the zeroth products of Frenkel–Jing operators

2017 ◽  
Vol 16 (03) ◽  
pp. 1750053 ◽  
Author(s):  
Slaven Kožić
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Quantum Vertex ◽  
Algebra Theory

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

scholarly journals ELEMENTARY INVARIANTS FOR CENTRALIZERS OF NILPOTENT MATRICES

2009 ◽  
Vol 86 (1) ◽  
pp. 1-15 ◽  
Author(s):  
JONATHAN BROWN ◽  
JONATHAN BRUNDAN
Keyword(s):  
Lie Algebra ◽  
Characteristic Zero ◽  
Universal Enveloping Algebra ◽  
General Linear ◽  
Enveloping Algebra ◽  
Nilpotent Matrix ◽  
Nilpotent Matrices ◽  
Algebra Over A Field

AbstractWe construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.

Symbols in Berezin quantization for representation operators

2021 ◽  
pp. 296-304
Author(s):  
Vladimir F. Molchanov ◽  
Svetlana V. Tsykina
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Grassmann Manifold ◽  
Homogeneous Spaces ◽  
Finite Multiplicity ◽  
Basic Notion ◽  
Berezin Quantization ◽  
Algebraic Version ◽  
Enveloping Algebra ◽  
Algebra Of Operators

The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F^♮ defined on M. These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization — we call it the polynomial quantization — is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G. In this case symbols turn out to be polynomials on the Lie algebra g. In this paper we offer a new theme in the Berezin quantization on G/H: as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G=SL(2;R), H — the subgroup of diagonal matrices, G/H — a hyperboloid of one sheet in R^3; b) G — the pseudoorthogonal group SO_0 (p; q), the subgroup H covers with finite multiplicity the group SO_0 (p-1,q -1)×SO_0 (1;1); the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G.

Sign in / Sign up

Export Citation Format

Share Document