Contractions Of Algebras Research Articles

LAPLACE EQUATIONS, CONFORMAL SUPERINTEGRABILITY AND BÔCHER CONTRACTIONS

Quantum superintegrable systems are solvable eigenvalue problems. Their solvability is due to symmetry, but the symmetry is often ``hidden''.<p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;">The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems and their algebras are related by geometric limits, induced by generalized Inönü-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. The systems can be best understood by transforming them to Laplace conformally superintegrable systems and using ideas introduced in the 1894 thesis of Bôcher to study separable solutions of the wave equation. The contractions can be subsumed into contractions of the conformal algebra <em>so</em>(4,C) to itself. Here we announce main findings, with detailed classifications in papers under preparation.</p>

Contractions of 3-folds: Deformations and invariants

This paper discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper [Noncommutative deformation and flops, Duke Math. J. 165(8) (2016) 1397–1414], the author and Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It characterizes their geometric and homological properties in a unified manner, using the theory of noncommutative deformations. Toda has now clarified the enumerative significance of the contraction algebra for flopping curves, calculating its dimension in terms of Gopakumar-Vafa invariants [Noncommutative width and Gopakumar–Vafa invariants, Manuscripta Math. 148(3–4) (2015) 521–533]. Before reviewing these results, and others, the author gives a brief introduction to the rich geometry of flopping curves on 3-folds, starting from the resolutions of Kleinian surface singularities. This is based on a talk given at VBAC 2014 in Berlin.

Bôcher Contractions of Conformally Superintegrable Laplace Equations

The explicit solvability of quantum superintegrable systems is due to symmetry, but the symmetry is often "hidden". The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems on constant curvature spaces are related by geometric limits, induced by generalized In\"on\"u-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. However, the limits have no satisfactory Lie algebra contraction interpretations for underlying spaces with 1- or 0-dimensional Lie algebras. We show that these systems can be best understood by transforming them to Laplace conformally superintegrable systems, with flat space conformal symmetry group ${\rm SO}(4,{\mathbb C})$, and using ideas introduced in the 1894 thesis of B\^ocher to study separable solutions of the wave equation in terms of roots of quadratic forms. We show that B\^ocher's prescription for coalescing roots of these forms induces contractions of the conformal algebra $\mathfrak{so}(4,{\mathbb C})$ to itself and yields a mechanism for classifying all Helmholtz superintegrable systems and their limits. In the paper [Acta Polytechnica, to appear, arXiv:1510.09067], we announced our main findings. This paper provides the proofs and more details.

The quantum-to-classical transition: contraction of associative products

The quantum-to-classical transition is considered from the point of view of contractions of associative algebras. Various methods and ideas to deal with contractions of associative algebras are discussed that account for a large family of examples. As an instance of them, the commutative algebra of functions in phase space, corresponding to classical physical observables, is obtained as a contraction of the Moyal star-product which characterizes the quantum case. Contractions of associative algebras associated to Lie algebras are discussed, in particular the Weyl–Heisenberg and SU(2) groups are considered.

Algebraic contraction rate for distance between entropy solutions of scalar conservation laws

We establish an algebraic contraction rate in a modified Wasserstein distance for solutions of scalar conservation laws with uniformly convex flux. We also show that our estimate is optimal w.r.t. scaling in time and discuss why it gives non-trivial information in relation to the stability of the rarefaction wave.

Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems

Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by In\"on\"u-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as ${\hbar}\to 0$ and nonrelativistic phenomena from special relativistic as $c\to \infty$, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classified contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras $e(2,{\mathbb C})$ in flat space and $o(3,{\mathbb C})$ on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of In\"on\"u-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by In\"on\"u-Wigner contractions. We present tables of the contraction results.

Wilson polynomials/functions and intertwining operators for the generic quantum superintegrable system on the 2-sphere

The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, HΨ = EΨ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L2 of H in terms of an eigenbasis of another symmetry operator L1, but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2,C). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions.

(Quasi)-exact-solvability on the sphere Sn

An Exactly Solvable (ES) potential on the sphere Sn is reviewed and a related Quasi-Exactly Solvable (QES) potential is found and studied. After mapping the sphere to a simplex, it is found that the metric (of constant curvature) is in polynomial form, and both the ES and the QES potentials are rational functions. Their hidden algebra is gln, realized in a finite-dimensional representation by first order differential operators acting on RPn. It is shown that variables in the Schrödinger eigenvalue equation can be separated in polyspherical coordinates and there is always complete integrability. The QES system is completely integrable for n = 2 and non-maximally superintegrable for n ≥ 3. There is no separable coordinate system in which it is exactly solvable. We point out that by taking contractions of superintegrable systems, such as induced by Inönü-Wigner Lie algebra contractions, we can find other QES superintegrable systems, and we illustrate this by contracting our Sn system to a QES non-maximal superintegrable system on Euclidean space En, an extension of the Smorodinsky-Winternitz potential.

Relativity symmetries and Lie algebra contractions

We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n) symmetry as an isometry on an m+n dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n) preserving a symmetry of the same type at dimension m+n−1, e.g. a G(m,n−1), together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4), which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3) may be relevant to real physics.

Quadratic algebra contractions and second-order superintegrable systems

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quantum system on the 2-sphere (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.

Zero products preserving maps from the Fourier algebra of amenable groups

Let $G$ be a locally compact amenable group. The goal of this paper is to investigate the problem of surjective zero products preserving maps from the Fourier algebra of $G$ into a completely contractive Banach algebra. We show that if $B$ is a completely contractive Banach algebra which is faithful and factors weakly, then every surjective completely bounded linear map from $A(G)$ into $B$ which preserves zero products is a weighted homomorphism. Moreover an equivalent condition is given for such a map to be a homomorphism. In particular, this result implies that if $B$ is a commutative C$^*$-algebra or a matrix space and $T:A(G)\rightarrow B$ is a continuous surjective linear map which preserves zero products, then $T$ is a weighted homomorphism and there is an equivalent condition for $T$ to be a homomorphism.

Asymptotic representations and q-oscillator solutions of the graded Yang–Baxter equation related to Baxter Q-operators

We consider a class of asymptotic representations of the Borel subalgebra of the quantum affine superalgebra Uq(glˆ(M|N)). This is characterized by Drinfeld rational fractions. In particular, we consider contractions of Uq(gl(M|N)) in the FRT formulation and obtain explicit solutions of the graded Yang–Baxter equation in terms of q-oscillator superalgebras. These solutions correspond to L-operators for Baxter Q-operators. We also discuss an extension of these representations to the ones for contracted algebras of Uq(glˆ(M|N)) by considering the action of renormalized generators of the other side of the Borel subalgebra. We define model independent universal Q-operators as the supertrace of the universal R-matrix and write universal T-operators in terms of these Q-operators based on shift operators on the supercharacters. These include our previous work on Uq(slˆ(2|1)) case [1] in part, and also give a cue for the operator realization of our Wronskian-like formulas on T- and Q-functions in [2,3].

Preordered forests, packed words and contraction algebras

We introduce the notions of preordered and heap-preordered forests, generalizing the construction of ordered and heap-ordered forests. We prove that the algebras of preordered and heap-preordered forests are Hopf for the cut coproduct, and we construct a Hopf morphism to the Hopf algebra of packed words. Moreover, we define another coproduct on the preordered forests given by the contraction of edges. Finally, we give a combinatorial description of morphism defined on Hopf algebras of forests with values in the Hopf algebras of shuffles or quasi-shuffles.

The theory of contractions of 2D 2nd order quantum superintegrable systems and its relation to the Askey scheme for hypergeometric orthogonal polynomials

We describe a contraction theory for 2nd order superintegrable systems, showing that all such systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. Analogously, all of the quadratic symmetry algebras of these systems can be obtained by a sequence of contractions starting from S9. By contracting function space realizations of irreducible representations of the S9 algebra (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.This relates the scheme directly to explicitly solvable quantum mechanical systems. Amazingly, all of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2, C). The present paper concentrates on describing this intimate link between Lie algebra and superintegrable system contractions, with the detailed calculations presented elsewhere. Joint work with E. Kalnins, S. Post, E. Subag and R. Heinonen.

A note on p-completely bounded homomorphisms of the Figà-Talamanca–Herz algebras

For a locally compact group G and 1<p<∞ let Ap(G) be the Figà-Talamanca–Herz algebras, which include in particular the Fourier algebra of G, A(G) (p=2). It is shown that for any amenable group H, a proper affine map α:Y⊂H→G induces a p-completely contractive algebra homomorphism ϕα:Ap(G)→Ap(H) by setting ϕα(u)=u∘α on Y and ϕα(u)=0 off of Y. Moreover, we show that if both G and H are amenable then any p-completely contractive algebra homomorphism ϕ:Ap(G)→Ap(H) is of this form. These results are the analogs in the context of the Figà-Talamanca–Herz algebras of the ones in the Fourier algebra setting (p=2) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.

One-parameter contractions of Lie-Poisson brackets

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra {\mathcal A}=\mathbb K[\mathbb A^n] is said to be of Kostant type, if its centre Z({\mathcal A}) is freely generated by homogeneous polynomials F_1,\ldots,F_r such that they give Kostant's regularity criterion on \mathbb A^n ( d_xF_i are linear independent if and only if the Poisson tensor has the maximal rank at x ). If the initial Poisson algebra is of Kostant type and F_i satisfy a certain degree-equality, then the contraction is also of Kostant type. The general result is illustrated by two examples. Both are contractions of a simple Lie algebra \gt g corresponding to a decomposition \gt g=\gt h \oplus V , where \gt h is a subalgebra. Here {\mathcal A}={\mathcal S}(\gt g)=\mathbb K[\gt g^*] , Z({\mathcal A})={\mathcal S}(\gt g)^\gt g , and the contracted Lie algebra is a semidirect product of \gt h and an Abelian ideal isomorphic to \gt g/\gt h as an \gt h -module. In the first example, \gt h is a symmetric subalgebra and in the second, it is a Borel subalgebra and V is the nilpotent radical of an opposite Borel.

Contractions of Certain Lie Algebras in the Context of the DLF-Theory

Contractions of the Lie algebras d = u(2), f = u(1 ,1) to the oscillator Lie algebra l are realized via the adjoint action of SU(2,2) when d, l, f are viewed as subalgebras of su(2,2). Here D, L, F are the corresponding (four-dimensional) real Lie groups endowed with bi-invariant metrics of Lorentzian signature. Similar contractions of (seven-dimensional) isometry Lie algebras iso(D), iso(F) to iso(L) are determined. The group SU(2,2) acts on each of the D, L, F by conformal transformation which is a core feature of the DLF-theory. Also, d and f are contracted to T, S-abelian subalgebras, generating parallel translations, T, and proper conformal transformations, S (from the decomposition of su(2,2) as a graded algebra T + Ω + S, where Ω is the extended Lorentz Lie algebra of dimension 7).

Invariant Cones in Lie Algebras and Positive Energy Representations and Contractions of Conformal Algebras

We recall some important results, due to Kostant and others, about invariant convex cones in Lie algebras and positive energy representations. We apply these results to a study of positive energy representation of the conformal groups in n dimensions, and we present a proof of the converse of a theorem attributed to I.E. Segal, which relates positive energy representations to positivity of the action of the generator of time translations for representations of the n-dimensional conformal group.We also discuss related notions of deformation and contractions of Lie algebras and describe a deformation of the Poincaré subalgebra of the conformal algebra which generalizes the usual treatment. We consider the positive energy representations of the anti-deSitter subalgebras in the physically important four dimensional case, and apply this generalization to argue that the singelton representations cannot have nontrivial contractions to representations of the Poincaré algebra. We believe that our results represent a sharpening of the meaning of "kinematical confinement", introduced by Flato, Fronsdal and their coworkers.

Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials

We show explicitly that all 2nd order superintegrable systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. We extend the Wigner-In\"on\"u method of Lie algebra contractions to contractions of quadratic algebras and show that all of the quadratic symmetry algebras of these systems are contractions of that of S9. Amazingly, all of the relevant contractions of these superintegrable systems on flat space and the sphere are uniquely induced by the well known Lie algebra contractions of e(2) and so(3). By contracting function space realizations of irreducible representations of the S9 algebra (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems, and using Wigner's idea of "saving" a representation, we obtain the full Askey scheme of hypergeometric orthogonal polynomials. This relationship directly ties the polynomials and their structure equations to physical phenomena. It is more general because it applies to all special functions that arise from these systems via separation of variables, not just those of hypergeometric type, and it extends to higher dimensions.

Graded contractions of the Gell-Mann graded $sl(3,\mathbb {C})$sl(3,C)

The Gell-Mann grading, one of the four gradings of \documentclass[12pt]{minimal}\begin{document}$\operatorname{sl}(3,\mathbb {C})$\end{document}sl(3,C) that cannot be further refined, is considered as the initial grading for the graded contraction procedure. Using the symmetries of the Gell-Mann grading, the system of contraction equations is reduced and solved. Each non-trivial solution of this system determines a Lie algebra which is not isomorphic to the original algebra \documentclass[12pt]{minimal}\begin{document}$\operatorname{sl}(3,\mathbb {C})$\end{document}sl(3,C). The resulting 53 contracted algebras are divided into two classes — the first is represented by the algebras which are also continuous Inönü–Wigner contractions, the second is formed by the discrete graded contractions.