Classical Radicals of Invariants of Algebraic Skew Derivations

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

Proton-neutron symplectic model description of $^{20}$Ne

A microscopic description of the low-lying positive-parity rotational bands in $^{20}$Ne is given within the framework of the symplectic-based proton-neutron shell-model approach provided by the proton-neutron symplectic model (PNSM). For this purpose a model Hamiltonian is used which includes an algebraic interaction, lying in the enveloping algebra of the $Sp(12,R)$ dynamical group of the PNSM, that introduces both horizontal and vertical mixings of different $SU(3)$ irreducible representations within the $Sp(12,R)$ irreducible collective space of $^{20}$Ne. A good overall description is obtained for the excitation energies of the ground and first two excited $\beta$ bands, as well as for the ground state intraband $B(E2)$ quadrupole collectivity and the known interband $B(E2)$ transition probabilities between the low-lying collective states without the use of an effective charge.

Algebras, traces, and boundary correlators in $$ \mathcal{N} $$ = 4 SYM

We study supersymmetric sectors at half-BPS boundaries and interfaces in the 4d $$ \mathcal{N} $$ N = 4 super Yang-Mills with the gauge group G, which are described by associative algebras equipped with twisted traces. Such data are in one-to-one correspondence with an infinite set of defect correlation functions. We identify algebras and traces for known boundary conditions. Ward identities expressing the (twisted) periodicity of the trace highly constrain its structure, in many cases allowing for the complete solution. Our main examples in this paper are: the universal enveloping algebra $$ U\left(\mathfrak{g}\right) $$ U g with the trace describing the Dirichlet boundary conditions; and the finite W-algebra $$ \mathcal{W}\left(\mathfrak{g},{t}_{+}\right) $$ W g t + with the trace describing the Nahm pole boundary conditions.

Filtrations on block subalgebras of reduced enveloping algebras

We study the interaction between the block decompositions of reduced enveloping algebras in positive characteristic, the Poincaré-Birkhoff-Witt (PBW) filtration, and the nilpotent cone. We provide two natural versions of the PBW filtration on the block subalgebra [Formula: see text] of the restricted universal enveloping algebra [Formula: see text] and show these are dual to each other. We also consider a shifted PBW filtration for which we relate the associated graded algebra to the algebra of functions on the Frobenius neighborhood of [Formula: see text] in the nilpotent cone and the coinvariants algebra corresponding to [Formula: see text]. In the case of [Formula: see text] in characteristic [Formula: see text] we determine the associated graded algebras of these filtrations on block subalgebras of [Formula: see text]. We also apply this to determine the structure of the adjoint representation of [Formula: see text].

Strongly elliptic operators and exponentiation of operator Lie algebras

An intriguing feature which is often present in theorems regardingthe exponentiation of Lie algebras of unbounded linear operators onBanach spaces is the assumption of hypotheses on the Laplacianoperator associated with a basis of the operator Lie algebra.The main objective of this work is to show that one can substitutethe Laplacian by an arbitrary operator in the enveloping algebra andstill obtain exponentiation, as long as its closure generates astrongly continuous one-parameter semigroup satisfying certain normestimates, which are typical in the theory of strongly ellipticoperators.

Affinoid Dixmier Modules and the Deformed Dixmier-Moeglin Equivalence

The affinoid enveloping algebra $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K of a free, finitely generated $\mathbb {Z}_{p}$ ℤ p -Lie algebra ${\mathscr{L}}$ L has proven to be useful within the representation theory of compact p-adic Lie groups, and we aim to further understand its algebraic structure. To this end, we define the notion of a Dixmier module over $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K , a generalisation of the Verma module, and we prove that when ${\mathscr{L}}$ L is nilpotent, all primitive ideals of $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K can be described in terms of annihilator ideals of Dixmier modules. Using this, we take steps towards proving that this algebra satisfies a version of the classical Dixmier-Moeglin equivalence.

LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART

For a finite-dimensional Lie algebra $\mathfrak {L}$ over $\mathbb {C}$ with a fixed Levi decomposition $\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$ , where $\mathfrak {g}$ is semisimple, we investigate $\mathfrak {L}$ -modules which decompose, as $\mathfrak {g}$ -modules, into a direct sum of simple finite-dimensional $\mathfrak {g}$ -modules with finite multiplicities. We call such modules $\mathfrak {g}$ -Harish-Chandra modules. We give a complete classification of simple $\mathfrak {g}$ -Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak {g} = \mathfrak {sl}_2$ , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak {sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules.

PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

Let 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).