Universal Enveloping Research Articles

The universal envelope of a polarized anti-Jordan triple system of block matrices

We investigate the universal associative envelope [Formula: see text] of the simple polarized anti-Jordan triple system [Formula: see text] of [Formula: see text] [Formula: see text] block matrices: [Formula: see text] with the triple product [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic 0. We prove that [Formula: see text], where [Formula: see text] is the ordinary associative algebra of all [Formula: see text] matrices over [Formula: see text]. From this result we classify up to equivalence all finite-dimensional irreducible representations of the anti-Jordan triple system [Formula: see text]. We conclude the paper with an open problem for further research.

Straightening Identities in the Universal Enveloping Algebra of some Twisted Multiloop Algebras of sl4

We have formulated and proven some straightening identities in the universal enveloping algebras of the twisted multiloop algebras of sl4 with Chevalley involution twist. These formulas could be useful in investigating the representation theory of this algebra.

Geometric tool kit for higher spin gravity (Part II): An introduction to Lie algebroids and their enveloping algebras

These notes provide a self-contained introduction to Lie algebroids, Lie-Rinehart algebras and their universal envelopes. This review is motivated by the speculation that higher-spin gauge symmetries should admit a natural formulation as enveloping algebras of Lie algebroids since rigid higher-spin algebras are enveloping algebras of Lie algebras. Nevertheless, the material covered here may be of general interest to anyone interested in the description of gauge symmetries, connections and covariant derivatives, in terms of Lie algebroids. In order to be self-contained, a concise introduction to the algebraic characterization of vector bundles as projective modules over the algebra of functions on the base manifold is provided.

Globalization of partial group actions on semiprime Lie algebras and unital Jordan algebras

We give necessary and sufficient conditions for the existence of a semiprime globalization for a partial group action on a semiprime Lie algebra L, and with an additional reasonable condition, we show that this semiprime globalization is unique up to isomorphism. Moreover, under the same condition we prove that any globalizable partial group action on L induces a globalizable partial group action on its maximal quotient algebra. For Jordan algebras, we show that a globalizable partial group action on a unital Jordan algebra J induces a globalizable partial group action on the unital special universal envelope for J.

Universal enveloping of a graded Lie algebra

In this paper we construct a graded universal enveloping algebra of a G-graded Lie algebra, where G is not necessarily an abelian group. If the grading group is abelian, then it coincides with the classical construction. We prove the existence and uniqueness of the graded enveloping algebra. As consequences, we prove a graded variant of Witt's Theorem on the universal enveloping algebra of the free Lie algebra, and the graded version of Ado's Theorem, which states that every finite-dimensional Lie algebra admits a faithful finite dimensional representation. Furthermore we investigate if a Lie grading is equivalent to an abelian grading.

Hochschild cohomology of the Weyl conformal algebra with coefficients in finite modules

In this work, we find Hochschild cohomology groups of the Weyl associative conformal algebra with coefficients in all finite modules. The Weyl conformal algebra is the universal associative conformal envelope of the Virasoro Lie conformal algebra relative to the locality N = 2. In order to obtain this result, we adjust the algebraic discrete Morse theory to the case of differential algebras.

Post-Lie algebras in Regularity Structures

Abstract In this work, we construct the deformed Butcher-Connes-Kreimer Hopf algebra coming from the theory of Regularity Structures as the universal envelope of a post-Lie algebra. We show that this can be done using either of the two combinatorial structures that have been proposed in the context of singular SPDEs: decorated trees and multi-indices. Our construction is inspired from multi-indices where the Hopf algebra was obtained as the universal envelope of a Lie algebra, and it has been proved that one can find a basis that is symmetric with respect to certain elements. We show that this Lie algebra comes from an underlying post-Lie structure.

Perfect JC-algebras

Perfect C∗-algebras were introduced by Akeman and Shultz in [Perfect C*-algebras, Mem. Amer. Math. Soc. 55(326) (1985)] and they form a certain subclass of C*-algebras determined by their pure states, and for which the general Stone–Weierstrass conjecture is true. In this paper, we introduce the notion of perfect JC-algebras, and we use the strong relationship between a JC-algebra A and its universal enveloping C∗-algebra [Formula: see text], to establish that if [Formula: see text] is perfect and A is of complex type, then A is perfect. It is also shown that every scattered JC-algebra of complex type is perfect, and the same conclusion holds for every JC-algebra of complex type whose primitive spectrum is Hausdorff.

On the Universality of Microwave Envelope Equations for Narrowband Optoelectronic Oscillators

Narrowband optoelectronic oscillators are time-delayed system that can generate ultrapure microwave signals. In order to study these oscillators, one approach is to derive a time-delayed envelope equation. In this article, we show that this approach is universal and applies to a wide variety of OEO families. By studying different oscillators, it is possible to show that their dynamics is governed by an envelope equation that can be rewritten under a generalized universal form. In this article, this universal microwave envelope equation is proposed and investigated analytically for optoelectronic microwave oscillators, and the bifurcation analysis of the deterministic envelope equation is performed. The universal model is further generalized and written in a stochastic form in order to predict the phase noise performance of the oscillators.

Uniform Envelopes

In the author's PhD thesis (2019) universal envelopes were introduced as a tool for studying the continuously obtainable information on discontinuous functions. To any function $f \colon X \to Y$ between $\operatorname{qcb}_0$-spaces one can assign a so-called universal envelope which, in a well-defined sense, encodes all continuously obtainable information on the function. A universal envelope consists of two continuous functions $F \colon X \to L$ and $\xi_L \colon Y \to L$ with values in a $\Sigma$-split injective space $L$. Any continuous function with values in an injective space whose composition with the original function is again continuous factors through the universal envelope. However, it is not possible in general to uniformly compute this factorisation. In this paper we propose the notion of uniform envelopes. A uniform envelope is additionally endowed with a map $u_L \colon L \to \mathcal{O}^2(Y)$ that is compatible with the multiplication of the double powerspace monad $\mathcal{O}^2$ in a certain sense. This yields for every continuous map with values in an injective space a choice of uniformly computable extension. Under a suitable condition which we call uniform universality, this extension yields a uniformly computable solution for the above factorisation problem. Uniform envelopes can be endowed with a composition operation. We establish criteria that ensure that the composition of two uniformly universal envelopes is again uniformly universal. These criteria admit a partial converse and we provide evidence that they cannot be easily improved in general. Not every function admits a uniformly universal uniform envelope. We can however assign to every function a canonical envelope that is in some sense as close as possible to a uniform envelope. We obtain a composition theorem similar to the uniform case.

First quantization of braided Majorana fermions

A Z2-graded qubit represents an even (bosonic) “vacuum state” and an odd, excited, Majorana fermion state. The multiparticle sectors of N, braided, indistinguishable Majorana fermions are constructed via first quantization. The framework is that of a graded Hopf algebra endowed with a braided tensor product. The Hopf algebra is U(gl(1|1)), the Universal Enveloping Algebra of the gl(1|1) superalgebra. A 4×4 braiding matrix Bt defines the braided tensor product. Bt, which is related to the R-matrix of the Alexander-Conway polynomial, depends on the braiding parameter t belonging to the punctured plane (t∈C⁎); the ordinary antisymmetry property of fermions is recovered for t=1.For each N, the graded dimension m|n of the graded multiparticle Hilbert space is computed. Besides the generic case, truncations occur when t coincides with certain roots of unity which appear as solutions of an ordered set of polynomial equations. The roots of unity are organized into levels which specify the maximal number of allowed braided Majorana fermions in a multiparticle sector.By taking into account that the even/odd sectors in a Z2-graded Hilbert space are superselected, a nontrivial braiding with t≠1 is essential to produce a nontrivial Hilbert space described by qubits, qutrits, etc., since at t=1 the N-particle vacuum and the antisymmetrized excited state encode the same information carried by a classical 1-bit.

Universal Enveloping Algebras of Lie–Rinehart Algebras as a Left Adjoint Functor

We prove how the universal enveloping algebra constructions for Lie-Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie-Rinehart algebra morphisms, of modules over Lie-Rinehart algebras and of the infinitesimal gauge algebra of a module.

A Closed Structure Constant Formula for the Universal Enveloping Algebra of the Lie Algebra sl2

Gourley and Kennedy gave a recursive formula for the structure constants of the universal enveloping algebra of sl2. Using Kostant's formula we give a closed formula for these structure constants.

Derived Intersections and Free dg-Lie Algebroids

We study free dg-Lie algebroids over arbitrary derived schemes, and compute their universal enveloping and jet algebras. We also introduce derived twisted connections, and relate them to lifts on twisted square zero extensions. This construction allows us to provide new conceptual approaches to existing results concerning the derived deformation theory of subschemes.

The Milnor-Moore theorem for L_infty algebras in rational homotopy theory

We give a construction of the universal enveloping A_infty algebra of a given L_infty algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem. This proposes a new A_infty model for simply connected rational homotopy types, and uncovers a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.

Representations of special Jordan triple systems of all symmetric and hermitian n by n matrices

We study the representations of two special Jordan triple systems (with respect to the product xyz + zyx): The first is the Jordan triple system of all symmetric by matrices over a field of characteristic zero, the second is the Jordan triple system of all Hermitian by matrices over the complex numbers . We show that the universal (associative) envelope of is isomorphic to , while the universal (associative) envelope of is isomorphic to . As corollaries, the Jordan triple system has two nontrivial finite-dimensional inequivalent irreducible representations, while the Jordan triple system has four nontrivial inequivalent finite-dimensional irreducible representations.

On the universal envelope of a Jordan triple system of n × n matrices

We study the universal (associative) envelope of the Jordan triple system of all [Formula: see text] [Formula: see text] matrices with the triple product [Formula: see text] over a field of characteristic 0. We use the theory of non-commutative Gröbner–Shirshov bases to obtain the monomial basis and the center of the universal envelope. We also determine the decomposition of the universal envelope and show that there exist only five finite-dimensional inequivalent irreducible representations of the Jordan triple system of all [Formula: see text] matrices.

The universal associative enveloping algebra of a Lie–Jordan algebra with a unit

We describe the universal associative enveloping algebra of a Lie–Jordan algebra with a unit over a field of characteristic 0. We show that if L is a Lie–Jordan algebra with a unit e over a field of characteristic 0, then the (unital) universal associative envelope of L is isomorphic to where The center of the universal associative envelope is also described. We conclude the paper with some open problems for further research.

On the Relationship between Jordan Algebras and Their Universal Enveloping Algebras

The relationship between JW-algebras (resp. JC-algebras) and their universal enveloping von Neumann algebras (resp. C ∗ -algebras) can be described as significant and influential. Examples of numerous relationships have been established. In this article, we established a relationship between the set of split faces of the state space (resp. normal states) of a JC-algebra (resp. a JW-algebra) and the set of split faces of the state space (resp. normal states) of its universal enveloping C ∗ -algebra (resp. von Neumann algebra), and we tied up this relationship with the correspondence between the classes of invariant faces, closed ideals, and central projections of these Jordan algebras and of their universal enveloping algebras.

Universal enveloping of (modified) λ-differential Lie algebras

ABSTRACT We introduce a new operator – a modified λ-differential operator – from a usual differential operator. We construct the free (modified) λ-differential -algebra on a (modified) differential -module, and apply it to erect the universal enveloping (modified) differential algebra of a (modified) λ-differential Lie -algebra. The corresponding Poincaré-Birchoff-Witt theorems are obtained. The Wronskian envelope of a modified λ-differential Lie -algebra is also constructed.