Nice Basis Research Articles

Ad-invariant metrics on nonnice nilpotent Lie algebras

We proved in previous work that all real nilpotent Lie algebras of dimension up to [Formula: see text] carrying an ad-invariant metric are nice, i.e. they admit a nice basis in the sense of Lauret et al. In this paper, we show by constructing explicit examples that nonnice irreducible nilpotent Lie algebras admitting an ad-invariant metric exist for every dimension greater than [Formula: see text] and every nilpotency step greater than [Formula: see text]. In the way of doing so, we introduce a method to construct Lie algebras with ad-invariant metrics called the single extension, as a parallel to the well-known double extension procedure.

Nice Bases of QTAG-Modules

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of universal modules. In this paper, we investigate the class of QTAG-modules having nice basis. It is proved that if H_ω (M) is bounded then M has a bounded nice basis and if H_ω (M) is a direct sum of uniserial modules, then M has a nice basis. We also proved that if M is any QTAG-module, then M⊕D has a nice basis, where D is the h-divisible hull of H_ω (M).

Indefinite Einstein metrics on nice Lie groups

Abstract We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice basis is orthogonal and a more general class associated to order two permutations of the nice basis. We obtain classifications in dimension 8 and, under the assumption that the root matrix is surjective, dimension 9; moreover, we prove that Einstein nilpotent Lie groups of nonzero scalar curvature exist in every dimension ≥ 8 {\geq 8} .

Laurent Positivity of Quantized Canonical Bases for Quantum Cluster Varieties from Surfaces

In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed ${\rm PGL}_2$-local systems on a punctured surface $S$. The moduli space is birational to a cluster $\mathcal{X}$-variety, whose positive real points recover the enhanced Teichm\"uller space of $S$. Their basis is enumerated by integral laminations on $S$, which are collections of closed curves in $S$ with integer weights. Around ten years later, a quantized version of this basis, still enumerated by integral laminations, was constructed by Allegretti and Kim. For each choice of an ideal triangulation of $S$, each quantum basis element is a Laurent polynomial in the exponential of quantum shear coordinates for edges of the triangulation, with coefficients being Laurent polynomials in $q$ with integer coefficients. We show that these coefficients are Laurent polynomials in $q$ with positive integer coefficients. Our result was expected in a positivity conjecture for framed protected spin characters in physics and provides a rigorous proof of it, and may also lead to other positivity results, as well as categorification. A key step in our proof is to solve a purely topological and combinatorial ordering problem about an ideal triangulation and a closed curve on $S$. For this problem we introduce a certain graph on $S$, which is interesting in its own right.

Construction of nice nilpotent Lie groups

We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension n up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for n≤9. On every nilpotent Lie algebra of dimension ≤7, we determine the number of inequivalent nice bases, which can be 0, 1, or 2.We show that any nilpotent Lie algebra of dimension n has at most countably many inequivalent nice bases.

Lattice Bases of Planar Lattices

In this paper we give a characterization of planar lattices, and show that the sublattice [Formula: see text] corresponding to the Hibi ideal of a finite distributive lattice [Formula: see text] has a nice lattice basis if and only if [Formula: see text] is planar. We also consider the Gröbner basis of the lattice basis ideal attached to the planar lattices.

Equivariant quantum cohomology of the Grassmannian via the rim hook rule

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provided a way to compute quantum products of Schubert classes in the Grassmannian of k-planes in complex n-space by doing classical multiplication and then applying a combinatorial rim hook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule then gives an effective algorithm for computing all equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights modulo n, suggesting a direct connection to the Peterson isomorphism relating quantum and affine Schubert calculus.

New construction of quantum error-avoiding codes via group representation of quantum stabilizer codes

In quantum computing, nice error bases as generalization of the Pauli basis were introduced by Knill. These bases are known to be projective representations of finite groups. In this paper, we propose a group representation approach to the study of quantum stabilizer codes. We utilize this approach to define decoherence-free subspaces (DFSs). Unlike previous studies of DFSs, this type of DFSs does not involve any spatial symmetry assumptions on the system-environment interaction. Thus, it can be used to construct quantum error-avoiding codes (QEACs) that are fault tolerant automatically. We also propose a new simple construction of QEACs and subsequently develop several classes of QEACs. Finally, we present numerical simulation results encoding the logical error rate over physical error rate on the fidelity performance of these QEACs. Our study demonstrates that DFSs-based QEACs are capable of providing a generalized and unified framework for error-avoiding methods.

Abstracting definitional interpreters (functional pearl)

In this functional pearl, we examine the use of definitional interpreters as a basis for abstract interpretation of higher-order programming languages. As it turns out, definitional interpreters, especially those written in monadic style, can provide a nice basis for a wide variety of collecting semantics, abstract interpretations, symbolic executions, and their intermixings. But the real insight of this story is a replaying of an insight from Reynold's landmark paper, Definitional Interpreters for Higher-Order Programming Languages , in which he observes definitional interpreters enable the defined-language to inherit properties of the defining-language. We show the same holds true for definitional abstract interpreters. Remarkably, we observe that abstract definitional interpreters can inherit the so-called “pushdown control flow” property, wherein function calls and returns are precisely matched in the abstract semantics, simply by virtue of the function call mechanism of the defining-language. The first approaches to achieve this property for higher-order languages appeared within the last ten years, and have since been the subject of many papers. These approaches start from a state-machine semantics and uniformly involve significant technical engineering to recover the precision of pushdown control flow. In contrast, starting from a definitional interpreter, the pushdown control flow property is inherent in the meta-language and requires no further technical mechanism to achieve.

Soliton Almost Kähler Structures on 6-Dimensional Nilmanifolds for the Symplectic Curvature Flow

The aim of this paper is to study self-similar solutions to the symplectic curvature flow on 6-dimensional nilmanifolds. For this purpose, we focus our attention on the family of symplectic two- and three-step nilpotent Lie algebras admitting a minimal compatible metric and give a complete classification of these algebras together with their respective metric. Such a classification is given by using our generalization of Nikolayevsky’s nice basis criterion, which, for the convenience of the reader, will be repeated here in the context of canonical compatible metrics for geometric structures on nilmanifolds. By computing the Chern–Ricci operator \(\text {P}\) in each case, we show that the above distinguished metrics define a soliton almost Kähler structure. Many illustrative examples are carefully developed.

An equivariant rim hook rule for quantum cohomology of Grassmannians

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the quantum product in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provide a way to compute quantum products of Schubert classes in the Grassmannian of $k$-planes in complex $n$-space by doing classical multiplication and then applying a combinatorial rimhook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule provides an effective algorithm for computing the equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights that is tantalizingly similar to maps in affine Schubert calculus. Une question importante dans la cohomologie quantique des variétés de drapeaux est de trouver des formules positives non récursives pour exprimer le produit quantique dans une base particulièrement bonne, appelée la base de Schubert. Bertram, Ciocan-Fontanine et Fulton donnent une façon de calculer les produits quantiques de classes de Schubert dans la Grassmannienne de $k$-plans dans l’espace complexe de dimension $n$ en faisant la multiplication classique et appliquant une règle combinatoire “rimhook” qui donne le paramètre quantique. Dans cet article, nous donnons une généralisation de ce règle rimhook au contexte où il y a aussi une action du tore complexe. Combiné avec la règle “puzzle” de Knutson et Tao, cela donne une algorithme effective pour calculer les coefficients équivariants de Littlewood-Richard. Il est intéressant d'observer que cette règle demande une spécialisation des poids du tore qui est similaire d’une manière tentante aux applications dans le calcul de Schubert affiné.

Extensions of nice bases on Ulm subgroups of primary Abelian groups with totally projective quotients

SupposeG is an abelian p-group, α is an ordinal andG/pG is totally projective. We show that G has a nice basis if and only if pG has a nice basis. This extends results of Danchev in Bull. Malaysian Math. Sci. Soc. (2010), Bull. Allah. Math. Soc. (2011) and An. St. Univ. Ovidius Constanta (2012), as well as joint work with Keef in Rocky Mount. J. Math. (2011).

On distinguished orbits of reductive representations

Let G be a real reductive Lie group and let τ:G→GL(V) be a real reductive representation of G with (restricted) moment map mg:V∖{0}→g. In this work, we introduce the notion of nice space of a real reductive representation to study the problem of how to determine if a G-orbit is distinguished (i.e. it contains a critical point of the norm squared of mg). We give an elementary proof of the well-known convexity theorem of Atiyah–Guillemin–Sternberg in our particular case and we use it to give an easy-to-check sufficient condition for a G-orbit of an element in a nice space to be distinguished. In the case where G is algebraic and τ is a rational representation, the above condition is also necessary (making heavy use of recent results of Michael Jablonski), obtaining a generalization of Nikolayevskyʼs nice basis criterion. We also provide useful characterizations of nice spaces in terms of the weights of τ. Finally, some applications to ternary forms are presented.

Error bases, group algebra and quantum codes

We find a new approach to study nice error bases and quantum error-correcting codes, namely the group algebra which gives us an algebraic notation for nice error bases and quantum codes. From this algebraic notation we establish a series of linear programming bounds on the most general quantum error-correcting codes.

An Extension of Nice Bases on Ulm Subgroups of Primary Abelian Groups

Abstract Suppose A is an Abelian p-group and is an ordinal such that A/pα A is a direct sum of countable groups. It is shown that A has a nice basis if, and only if, pα A has a nice basis. This strengthens an earlier result of ours in Bull. Allah. Math. Soc. (2011) proved when α< ω2.

S41 Screening for latent TB in HIV: are NICE & BHIVA guidance effective?

HIV infection is the strongest single risk factor for the development of active TB in latently infected individuals. NICE and BHIVA 2011 guidelines both recommend screening for latent TB in...

Nice bases and thickness in primary abelian groups

Nice bases and thickness in primary abelian groups

The nice basis rank of a primary abelian group

An abelian p-group G has a nice basis if it is the ascending union of a sequence of nice subgroups, each of which is a direct sum of cyclic groups. It is shown that if G is any group, then G ⊕ D has a nice basis, where D is the divisible hull of pωG. This leads to a consideration of the nice basis rank of G, i.e., the smallest rank of a divisible group D such that G ⊕ D has a nice basis. This concept is used to show that there exist a reduced group G and a non-reduced group H, both without a nice basis, such that G ⊕ H has a nice basis

Nice elongations of primary abelian groups

Suppose N is a nice subgroup of the primary abelian group G and A = G/N. The paper discusses various contexts in which G satisfying some property implies that A also satisfies the property, or visa versa, especially when N is countable. For example, if n is a positive integer, G has length not exceeding !1 and N is countable, then G is n-summable iff A is n-summable. When A is separable and N is countable, we discuss the condition that any such G decomposes into the direct sum of a countable and a separable group, and we show that it is undecidable in ZFC whether this condition implies that A must be a direct sum of cyclics. We also relate these considerations to the study of nice bases for primary abelian groups.

Complexity of referrals to a specialist psychotherapy service

Consecutive referrals to a specialist psychotherapy service were analysed for complexity. Some 71% had a severe enduring mental illness or personality disorder. Two-thirds of the rest had added psychosocial or behavioural complexity. Only 9% had no severe enduring mental illness, personality disorder, or added complexity; nearly half of these had somatoform disorder. Only 2% of the patients had uncomplicated depression or anxiety disorders. Patients treated in this psychotherapy service are substantially different from those on whom NICE bases its guidance for the treatment of depression and anxiety, and for whom IAPT was conceived, it functions as a de facto complex case service.