Lie identities on symmetric elements of restricted enveloping algebras

Let L be a restricted Lie algebra over a field of characteristic p > 2 and denote by u(L) its restricted enveloping algebra. We determine the conditions under which the set of symmetric elements of u(L) with respect to the principal involution is Lie solvable, Lie nilpotent, or bounded Lie Engel.     

On the Lie algebra of skew-symmetric elements of an enveloping algebra

Let L be a restricted Lie algebra over a field of characteristic p>2 and denote by u(L) its restricted enveloping algebra. We establish when the Lie algebra of skew-symmetric elements of u(L) under the principal involution is solvable, nilpotent, or satisfies an Engel condition.     

Sur les semi-caracteres des groupes de Lie resolubles connexes

Let G be a connected solvable Lie group, π a normal factor representation of G and ψ a nonzero trace on the factor generated by G. We denote by $\text{D}$(G) the space of C^∞ functions on G which are compactly supported. We show that there exists an element u of the enveloping algebra U$\text{G}$_$\text{c}$ of the complexification of the Lie algebra of G for which the linear form $\text{? ψ(π(u 1 ?))}$ on $\text{D}$(G) is a nonzero semiinvariant distribution on G. The proof uses results about characters for connected solvable Lie groups and results about the space of primitive ideals of the enveloping algebra U$\text{G}$_$\text{c}$.     

Séparation des orbites coadjointes d'un groupe exponentiel par leur enveloppe convexe

Looking to the separation of irreducible unitary representations of an exponential Lie group G through the image of their moment map, we propose here a new way: instead to extend the moment map to the universal enveloping algebra of G, we define a non linear mapping Φ from the dual of the Lie algebra g of G to the dual g^+∗ of a larger solvable group G⁺, and we extend the representation from G to G⁺, in such a manner that the corresponding coadjoint orbits in g^+∗ have distinct closed convex hull. This allows us to separate the irreducible unitary representations of G.     

Lie metabelian restricted universal enveloping algebras

We characterize the restricted Lie algebras L whose restricted universal enveloping algebra u(L) is Lie metabelian. Moreover, we show that the last condition is equivalent to u(L) being strongly Lie metabelian.Received: 1 October 2004     

Restricted Enveloping Algebras with Minimal Lie Derived Length

Let L be a non-abelian restricted Lie algebra over a field of characteristic p > 0 and let u(L) denote its restricted enveloping algebra. In Siciliano (Publ Math (Debr) 68:503–513, 2006) it was proved that if u(L) is Lie solvable then the Lie derived length of u(L) is at least ⌈log₂(p + 1)⌉. In the present paper we characterize the restricted enveloping algebras whose Lie derived length coincides with this lower bound.     

On Auslander-Reiten Quivers of Enveloping Algebras of Restricted Lie Algebras

Let (L, [p]) be a finite dimensional restricted Lie algebra over an algebraically closed field F of characteristic p ≥ 3, X ∈ L* a linear form. In this article we study the Auslander-Reiten quivers of certain blocks of the reduced enveloping algebra u(L,x). In particular, it is shown that the enveloping algebras of supersolvable Lie algebras do not possess AR-components of Euclidean type.     

Co-Poisson structures on polynomial Hopf algebras

The Hopf dual H° of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized. Some correspondences between co-Poisson and Poisson structures are also established.     

Quantum differential operators on the quantum plane

The universal enveloping algebra of a Lie algebra acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl₂.     

A subelliptic Taylor isomorphism on infinite-dimensional Heisenberg groups

Let G denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on G that are square integrable with respect to a heat kernel measure which is formally subelliptic, in the sense that all appropriate finite-dimensional projections are smooth measures. We prove a unitary equivalence between a subclass of these square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Cameron–Martin” Lie subalgebra. The isomorphism defining the equivalence is given as a composition of restriction and Taylor maps.     

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure μ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Lie algebra” of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the L ²(ν)-closure of holomorphic polynomials by their values on the Cameron–Martin subgroup.     

Special Lie superalgebras with maximality condition for subalgebras

We consider finitely generated Lie superalgebras over a field of characteristic zero satisfying Capelli identities. We prove that any such an algebra with the maximality condition for abelian subalgebras is finite dimensional. In particular, any special Lie superalgebra with the maximality condition for its subalgebras has a finite dimension. We also prove that the universal enveloping algebra U(L) of special Lie superalgebra L is Noetherian if and only if $\dim L<\infty$ .     

The multiplicative Weyl functional calculus

The Weyl calculus discussed in the author's previous papers starts with a fixed set of n noncommuting self-adjoint operators and associates an operator to a real function of n variables. The calculus is not multiplicative with respect to point-wise multiplication of functions. However, if the n self-adjoint operators generate a unitary Lie group representation, a “skew product” of functions can be defined which yields multiplicativity. This skew product depends only on the Lie group, not on the particular representation. In the case of the Heisenberg group, this skew product makes it possible to write the Schrödinger equation as an integro-differential equation on the phase plane. Strong convergence of the dynamical group, as Planck's constant goes to zero, to the classical Hamiltonian flow is proved under various conditions on the Hamiltonian.     

Enveloping algebras that are principal ideal rings

Let L be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of L is a principal ideal ring if and only if L is an extension of a finite-dimensional torus by a cyclic restricted Lie algebra.     

Filtered Multiplicative Bases of Restricted Enveloping Algebras

We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie algebra over a field of positive characteristic p.     

Description of bilateral ideals in a class of noncommutative rings. II

For generalized Weyl algebras containing the universal enveloping algebra Usl (2,K) of the Lie algebra sl (2) over a field with characteristic zero, bilateral ideals are classified. We show that a product of ideals is commutative and any proper ideal can be uniquely decomposed into a product of primary ideals.     

A sort-Jacobi algorithm for semisimple lie algebras

A structure preserving sort-Jacobi algorithm for computing eigenvalues or singular values is presented. It applies to an arbitrary semisimple Lie algebra on its (−1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well-known normal form problems such as e.g. the symmetric, Hermitian, skew-symmetric, symmetric and skew-symmetric R-Hamiltonian eigenvalue problem and the singular value decomposition.     

Reduction of the adjoint representation of sl(2,1): generators of primitive ideals

We present a reduction of the adjoint representation of the Lie superalge-bra,sl(2,1) and a study of the quotient algebra B(^c,k)= u/u(C?c)+u(D?kc), where c,k are two complex numbers. Under some additional conditions, we prove that every irreducible infinite dimensional representation of B(^c,k) is faithful, and that B(^C,K) is a primitive algebra. We give explicitly a set of generators of primitive degenerate ideal of infinite codimension. Essentially we prove that any minimal primitive ideal of u(sl(2,1)) is generated, as a 2-sided ideal, by its intersection with the algebra of gg-iuvariants.     

Convexity and unitary representations of nilpotent Lie groups

Summary The moment map of symplectic geometry is extended to associate to any unitary representation of a nilpotent Lie group aG-invariant subset of the dual of the Lie algebra. We prove that this subset is the closed conex hull of the Kirillov orbit of the representation.Supported by NSERC research grant no. A7918