Contractions Of Algebras Research Articles

From su(2) Gaudin Models to Integrable Tops

In the present paper we derive two well-known integrable cases of rigid body dynamics (the Lagrange top and the Clebsch system) performing an algebraic contraction on the two-body Lax matrices governing the (classical) su(2) Gaudin models. The procedure preserves the linear r-matrix formulation of the ancestor models. We give the Lax repre- sentation of the resulting integrable systems in terms of su(2) Lax matrices with rational and elliptic dependencies on the spectral parameter. We finally give some results about the many-body extensions of the constructed systems.

Operator Segal algebras in Fourier algebras

Let $G$ be a locally compact group, $\mathrm{A}(G)$ its Fourier algebra and $\mathrm{L}^1(G)$ the space of Haar integrable functions on $G$. We study the Segal algebra ${\mathrm{S}^1\!\mathrm{A}(G)}= {\mathrm{A}(G)}\cap{\rm L}^1(G)$ in ${\mathrm{A}(G)}$.

Contraction of broken symmetries via Kac-Moody formalism

I investigate contractions via Kac-Moody formalism. In particular, I show how the symmetry algebra of the standard two-dimensional Kepler system, which was identified by Daboul and Slodowy as an infinite-dimensional Kac-Moody loop algebra, and was denoted by H2, gets reduced by the symmetry breaking term, defined by the Hamiltonian H(β)=(1∕2m)(p12+p22)−α∕r−βr−1∕2cos((φ−γ)∕2). For this H(β) I define two symmetry loop algebras Li(β), i=1,2, by choosing the “basic generators” differently. These Li(β) can be mapped isomorphically onto subalgebras of H2, of codimension two or three, revealing the reduction of symmetry. Both factor algebras Li(β)∕Ii(E,β), relative to the corresponding energy-dependent ideals Ii(E,β), are isomorphic to so(3) and so(2,1) for E<0 and E>0, respectively, just as for the pure Kepler case. However, they yield two different nonstandard contractions as E→0, namely to the Heisenberg-Weyl algebra h3=w1 or to an Abelian Lie algebra, instead of the Euclidean algebra e(2) for the pure Kepler case. The above-noted example suggests a general procedure for defining generalized contractions, and also illustrates the “deformation contraction hysteresis,” where contraction which involves two contraction parameters can yield different contracted algebras, if the limits are carried out in different order.

THE GENERAL STRUCTURE OF G-GRADED CONTRACTIONS OF LIE ALGEBRAS, II: THE CONTRACTED LIE ALGEBRA

We continue our study of G-graded contractions γ of Lie algebras where G is an arbitrary finite Abelian group. We compare them with contractions, especially with respect to their usefulness in physics. (Note that the unfortunate terminology "graded contraction" is confusing since they are, by definition, not contractions.)We give a complete characterization of continuous G-graded contractions and note that they are equivalent to a proper subset of contractions. We study how the structure of the contracted Lie algebra Lγdepends on γ, and show that, for discrete graded contractions, applications in physics seem unlikely.Finally, with respect to applications to representations and invariants of Lie algebras, a comparison of graded contractions with contractions reveals the insurmountable defects of the graded contraction approach. In summary, our detailed analysis shows that graded contractions are clearly not useful in physics.

Noncommuting contractions of oscillators with constant force

We find two noncommuting contractions of the Lie algebras u(N) and gl(N,� ), realized as the symmetry algebras of N-dimensional isotropic harmonic and repulsive oscillators of spring constant k ∈� , with a constant force of magnitude f . The contraction limit to the symmetry algebra of the Ndimensional free system is (k, f ) → (0, 0). We take two paths in this plane, determined by the order of contraction of the two parameters, and show that they yield two closely related—but distinct—Euclidean-type symmetry algebras for the common contracted system. We also show briefly how the wavefunctions of the one-dimensional harmonic oscillator reduce to plane waves along the above two paths.

Uniformly complete quotient space $UCQ(G)$ and completely isometric representations of $UCQ(G)^*$ on $\mathcal {B}(L_2(G))$

The uniformly complete quotient space UCQ(G) of a locally compact group G is introduced. It is shown that the operator space dual UCQ(G)* is a completely contractive Banach algebra, which contains the completely bounded Fourier multiplier algebra M cb A(G) as a completely contractively complemented Banach subalgebra. A natural completely isometric representation of UCQ(G)* on B(L2(G)) is studied and some equivalent amenability conditions associated with UCQ(G) are proved.

On Conformal Jordan Cells of Finite and Infinite Rank

This work concerns in part the construction of conformal Jordan cells of infinite rank and their reductions to conformal Jordan cells of finite rank. It is also discussed how a procedure similar to Lie algebra contractions may reduce a conformal Jordan cell of finite rank to one of lower rank. A conformal Jordan cell of rank one corresponds to a primary field. This offers a picture in which any finite conformal Jordan cell of a given conformal weight may be obtained from a universal covering cell of the same weight but infinite rank.

Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras

AbstractWe construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. Like all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.

Courant algebroid and Lie bialgebroid contractions

Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, PoissonNijenhuis structures, and triangular Lie bialgebroids as particular examples. MSC 2000: Primary 17B99; Secondary 17B62, 53C15, 53D17.

The structure of a subclass of amenable banach algebras

We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach algebras to be finite‐dimensional and semisimple algebras. Moreover, we show that any contractible (resp., reflexive amenable) Banach algebra in which every maximal left ideal has a Banach space complement is indeed a direct sum of finitely many full matrix algebras. Finally, we characterize Hermitian *‐algebras that are contractible.

Complete contractivity of maps associated with the Aluthge and Duggal transforms

For an arbitrary operator T on Hilbert space, we study the maps ?Φ: f(T) → f(?T) and?Φ: f(T) → f(?T), where?T and?T are the Aluthge and Duggal transforms of T, respectively, and f belongs to the algebra Hol(σ(T)). We show that both maps are (contractive and) completely contractive algebra homomorphisms. As applications we obtain that every spectral set for T is also a spectral set for T and T, and also the inclusion W(f(T?)) - U W(f(T?)) - C W(f(T)) - relating the numerical ranges of f(T), f(T?), and f(T?).

Operator Figà-Talamanca–Herz algebras

Let G be a locally compact group. We use the canonical operator space structure on the spaces $L^p(G)$ for $p \in [1,\infty]$ introduced by G. Pisier to define operator space analogues $OA_p(G)$ of the classical Figa-Talamanca-Herz algebras $A_p(G)$. If $p \in (1,\infty)$ is arbitrary, then $A_p(G) \subset OA_p(G)$ such that the inclusion is a contraction; if p = 2, then $OA_2(G) \cong A(G)$ as Banachspaces spaces, but not necessarily as operator spaces. We show that $OA_p(G)$ is a completely contractive Banach algebra for each $p \in (1,\infty)$, and that $OA_q(G) \subset OA_p(G)$ completely contractively for amenable $G$ if $1 < p \leq q \leq 2$ or $2 \leq q \leq p < \infty$. Finally, we characterize the amenability of G through the existence of a bounded approximate identity in $OA_p(G)$ for one (or equivalently for all) $p \in (1,\infty)$.

Modules of graded contracted Virasoro algebras

We have described graded contractions of the Virasoro algebra. The highest weight representations of the Virasoro algebra have been constructed. The reducibility of representations is analised. In contrast to standard representations the contracted ones are reducible except some special cases. Moreover we have found an exotic module with null-plane on the fifth level.

Contractions: Nijenhuis and Saletan tensors for general algebraic structures

Generalizations in many directions of the contraction procedure for Lie algebras introduced by E.J.Saletan are proposed. Products of arbitrary nature, not necessarily Lie brackets, are considered on sections of finite-dimensional vector bundles. Saletan contractions of such infinite-dimensional algebras are obtained via a generalization of the Nijenhuis tensor approach. In particular, this procedure is applied to Lie algebras, Lie algebroids, and Poisson structures. There are also results on contractions of n-ary products and coproducts.

Contractions of Lie algebras and the separation of variables: interbase expansions

Lie algebra contractions from o(n + 1) to e(n) are used to obtain asymptotic limits of interbase expansions between bases corresponding to different subgroup chains for the group O(n + 1). The contractions lead to interbase expansions for different subgroup chains of the Euclidean group E(n). They provide asymptotic formulae for quantities such as Wigner rotation matrices, Clebsch-Gordan coefficients and Racah coefficients.

Cohomology and the Operator Space Structure of the Fourier Algebra and its Second Dual

Let G be a locally compact group. We introduce the notion of operator weak amenability for a completely contractive Banach algebra. We then study the potential operator weak amenability of the Fourier algebra A(G) of G and various sub-algebras of its second dual. In particular, we will show that A(G) is operator weakly amenable for a large class of groups, which includes all [IN] -groups and all Hermitian groups.

Maximal ideals and the structure of contractible and amenable Banach algebras

Properties of minimal idempotents in contractible and reflexive amenable Banach algebras are exploited to prove that such a kind of Banach algebra is finite demensional if each maximal ideal is contained in a maximal left or a maximal right ideal that is complemented as a Banach subspace. This result covers several known results on this subject.

ALGEBRAIC EQUIVALENCE BETWEEN CERTAIN MODELS FOR SUPERFLUID–INSULATOR TRANSITION

Algebraic contraction is proposed to realize mappings between Hamiltonian models. This transformation contracts the algebra of the degrees of freedom underlying the Hamiltonian. The rigorous mapping between the anisotropic XXZ Heisenberg model, the quantum phase model and the Bose Hubbard model is established as the contractions of the algebra u(2) underlying the dynamics of the XXZ Heisenberg model.

Lie algebra contractions and symmetries of the Toda hierarchy

The Lie algebra L(Δ) of generalized and point symmetries of the equations in the Toda hierarchy is shown to be a semidirect sum of two infinite-dimensional Lie algebras, one perfect, the other Abelian. In the continuous limit the structure of the Lie algebra changes: a contraction occurs with the lattice spacing as the contraction parameter. In particular, for the Toda equation itself, a set of five elements, involving both point symmetries and generalized ones, contracts to the point symmetry algebra of the potential KdV equation.

Hereditary Semigroup Algebras

In “Semigroup Algebras,” Okniński posed the following question: characterize semigroup algebras that are hereditary. In this paper we describe the (prime contracted) semigroup algebras K[S] that are hereditary and Noetherian when S is either a Malcev nilpotent monoid, a cancellative monoid or a monoid extension of a finite non-null Rees matrix semigroup. Furthermore, for the class of monoids which have an ideal series with factors that are non-null Rees matrix semigroups, we obtain an upper bound for the global dimension of its contracted semigroup algebra.