283 research outputs found

    Cohomology and Deformation of Leibniz Pairs

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    Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebra AA together with a Lie algebra LL mapped into the derivations of AA. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential.Comment: 15 page

    Empowering the individual within a productive franchise relationship.

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    1. The rapidly changing economic and social environment, in our ever shrinking global Community, calls for a drastic review and reappraisal of the 'way we do business'. A projected doubling of the world population in the next 50 - 70 years, all of it within the already struggling developing nations, must be taken as a 'wake up call' for the world's 'Community of Professionals'. 2. The National Centrę for Work Based Learning Partnerships has embarked upon a mission designed to assist Professionals to update their own knowledge and skills portfolios whilst at the same time providing a 'product' containing a distilled, reflected and thought out treasure trove of experience. Such practical experience will help build up the 'body of knowledge' of the greater Community in which they operate i.e. other Professionals. 3. The candidate himself has embarked on a personal journey which, when started, was structured to achieve the following: • To capitalise upon previous professional learning, taking advantage of a unique opportunity to reflect upon the merging of formal and informal study, practical experience and new insights gained over the years to produce a brand new composition of far greater quality and scope than previously achieved. • To capture the essence of own Franchising experience, knowledge and feel for the benefit of Franchisees, Franchisors, Government and Academia, i.e. to the benefit of the greater Community. • To think 'outside of the box' and offer new ideas and suggestions for consideration i.e. how a revitalised and courageous Franchising can contribute to the well being of the global and local communities. • To gain the pleasure of achieving the Doctorate in Professional Studies degree. • To be recognised 'in the public domain' by other Professionals in the economic, social and Franchising communities. 4. As far back as 1943 Sir Winston Churchill recognised and acknowledged that skills and expertise, as well as, the wide distribution of same throughout society are the key to success, when he said: "The empires of the future are the empires of the mind". (From a speech given at Harvard University 06.09.1943

    A nested sequence of projectors and corresponding braid matrices R^(θ)\hat R(\theta): (1) Odd dimensions

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    A basis of N2N^2 projectors, each an N2×N2{N^2}\times{N^2} matrix with constant elements, is implemented to construct a class of braid matrices R^(θ)\hat{R}(\theta), θ\theta being the spectral parameter. Only odd values of NN are considered here. Our ansatz for the projectors PαP_{\alpha} appearing in the spectral decomposition of R^(θ)\hat{R}(\theta) leads to exponentials exp(mαθ)exp(m_{\alpha}\theta) as the coefficient of PαP_{\alpha}. The sums and differences of such exponentials on the diagonal and the antidiagonal respectively provide the (2N2−1)(2N^2 -1) nonzero elements of R^(θ)\hat{R}(\theta). One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves 1/2(N+3)(N−1){1/2}(N+3)(N-1) free parameters mαm_{\alpha}. The diagonalizer of R^(θ)\hat{R}(\theta) is presented for all NN. Transfer matrices t(θ)t(\theta) and L(θ)L(\theta) operators corresponding to our R^(θ)\hat{R}(\theta) are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of N2N^2 constant N×NN\times N matrices. The θ\theta-dependence factors out for such combinations. R^(θ)\hat R(\theta) is developed in a power series in θ\theta. The basic difference arising for even dimensions is made explicit. Some special features of our R^(θ)\hat{R}(\theta) are discussed in a concluding section.Comment: latex file, 32 page

    Deformations Associated with Rigid Algebras

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    The deformations of an infinite dimensional algebra may be controlled not just by its own cohomology but by that of an associated diagram of algebras, since an infinite dimensional algebra may be absolutely rigid in the classical deformation theory for single algebras while depending essentially on some parameters. Two examples studied here, the function field of a sphere with four marked points and the first Weyl algebra, show, however, that the existence of these parameters may be made evident by the cohomology of a diagram (presheaf) of algebras constructed from the original. The Cohomology Comparison Theorem asserts, on the other hand, that the cohomology and deformation theory of a diagram of algebras is always the same as that of a single, but generally rather large, algebra constructed from the diagram

    A Group-Theoretic Consequence of the Donald-Flanigan Conjecture

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    AbstractThe Donald-Flanigan conjecture asserts that for any finite group G and prime p dividing its order #G, the group algebra FpG can be deformed into a semisimple, and hence rigid, algebra. We show that this implies that there is some element g ∈ G whose centralizer CG(g) has a normal subgroup of index p. The method is to observe that the Donald-Flanigan deformation must be a jump, whence, from the deformation theory, H1(FpG, FpG) ≠ 0. Using a standard result linking Hochschild and group cohomology one sees that some H1(CG(g), Fp) must be non-zero, giving the result. (Our corollary to the D-F conjecture has recently been verified by Fleischmann, Janiszczak, and Lempken using the classification of finite simple groups.

    Quantization on Curves

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    Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. Essential deformations are classified by the Harrison component of Hochschild cohomology, that vanishes on smooth manifolds and reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian ∗*-products. This paper begins a study of abelian quantization on plane curves over \Crm, being algebraic varieties of the form R2/I where I is a polynomial in two variables; that is, abelian deformations of the coordinate algebra C[x,y]/(I). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co-)homology and its Barr-Gerstenhaber-Schack decomposition. Homology is the same for all plane curves C[x,y]/(I), but the cohomology depends on the local algebra of the singularity of I at the origin.Comment: 21 pages, LaTex format. To appear in Letters Mathematical Physic

    Global Geometric Deformations of the Virasoro algebra, current and affine algebras by Krichever-Novikov type algebra

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    In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever-Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac-Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory.Comment: 17 page

    A Cohomological Perspective on Algebraic Quantum Field Theory

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    Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model. This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited. To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory

    Cremmer-Gervais r-matrices and the Cherednik Algebras of type GL2

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    We give an intepretation of the Cremmer-Gervais r-matrices for sl(n) in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL2 on certain subspaces of their polynomial representations. This is used to compute the nilpotency index of the Jordanian r-matrices, thus answering a question of Gerstenhaber and Giaquinto. We also give an interpretation of the Cremmer-Gervais quantization in terms of the corresponding double affine Hecke algebra.Comment: 6 page

    Deformation of dual Leibniz algebra morphisms

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    An algebraic deformation theory of morphisms of dual Leibniz algebras is obtained.Comment: 10 pages. To appear in Communications in Algebr
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