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At least 163 records · Page 9Linked to original sources

arXiv1996

Representations of the deformed U(su(2)) and U(osp(1,2)) algebras

The polynomial deformations of the Witten extensions of the U(su(2)) and U(osp(1,2)) algebras are three generator algebras with normal ordering, admitting a two generator subalgebra. The modules and the representations of these algebras are based on the construction of Verma modules, which are quotient modules, generated by ideals of the original algebra. This construction unifies a large number of the known algebras under the same scheme. The finite dimensional representations show new features such as the multiplicity of representations of the same dimensionality, or the existence of finite dimensional representations only for some dimensions.

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arXiv1996

On p-Adic Convergence of Perturbative Invariants of Some Rational Homology Spheres

R.~Lawrence has conjectured that for rational homology spheres, the series of Ohtsuki's invariants converges p-adicly to the SO(3) Witten-Reshetikhin-Turaev invariant. We prove this conjecture for Seifert rational homology spheres. We also derive it for manifolds constructed by a surgery on a knot in S^3. Our derivation is based on a conjecture about the colored Jones polynomial that we have formulated in our previous paper. We also present numerical examples of p-adic convergence for some simple manifolds.

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arXiv1996

Left-Covariant Differential Calculi on SL_q(2) and SL_q(3)

We study $N^2-1$ dimensional left-covariant differential calculi on the quantum group $SL_q(N)$ for which the generators of the quantum Lie algebras annihilate the quantum trace. In this way we obtain one distinguished calculus on $SL_q(2)$ (which corresponds to Woronowicz' 3D-calculus on $SU_q(2)$) and two distinguished calculi on $SL_q(3)$ such that the higher order calculi give the ordinary differential calculus on $SL(2)$ and $SL(3)$, respectively, in the limit $q\rightarrow 1$. Two new differential calculi on $SL_q(3)$ are introduced and developed in detail.

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arXiv1996

Double quantum groups and Iwasawa decomposition

The double quantum groups are the Hopf algebras underlying the complex quantum groups of which the simplest example is the quantum Lorentz group. They are non- standard quantizations of the double group $G \times G$. We construct a corresponding quantized universal enveloping algebra (QUE) and prove that the pairing between a quantum double group and its QUE is nondegenerate. We analyze the representation theory of these double quantum groups, give a detailed version of the Iwasawa decomposition proved by Podles and Woronowicz for the quantum Lorentz group, and show that they are noetherian algebras. Finally we outline a construction of more general non-standard quantum groups using quantum double groups and their generalizations.

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arXiv1996

An Algorithmic Approach to Classify Knots

I briefly discuss a method of obtaining distinct classes of topologically equivalent knots by developing appropriate computer programs.

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arXiv1996

Contraction of Algebraical Structures and Different Couplings of Cayley-Klein and Hopf Structures

Contractions (and graded contractions) of Lie algebra, Lie bialgebra and Hopf algebra are discussed. It is noticed the fundamental role of E.In{ö}n{ü} and E.P.Wigner idea of degenerate transformations. A constructive algorithm for description of contractions of quantum Cayley-Klein algebras $ so_{z}(n+1; {\bf j}) $ with different choice of the set of primitive operators is suggested. For nonsemisimple quantum algebras it gives nonisomorphic Hopf algebras. From physical point of view this algorithm gives the different physical interpretations of primitive operators for mathematically the same nonsemisimple quantum algebra. The case of $ so_{z}(3; {\bf j}) $ is regarded in detail.

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arXiv1996

A note on geometry of Kappa-Minkowski space

The infinitesimal action of Kappa-Poincar'e group on Kappa-Minkowski space is computed both for generators of Kappa-Poincar'e algebra and those of Woronowicz generalized Lie algebra. The notion of invariant operators is introduced and generalized Klein-Gordon equation is written out.

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arXiv1996

Semiinfinite cohomology of associative algebras and bar duality

We describe semiinfinite cohomology of associative algebras in terms of Koszul (or bar) duality. Consider an associative algebra $A$ and two its subalgebras $B$ and $N$ such that $A=B\otimes N$ as a vector space. We prove that the endomorphism algebra of the semiregular $A$-module appears naturally in semiinfinite cohomology theory as a ``two times Koszul dual'' to the algebra $A$. We compare semiinfinite cohomology of universal enveloping algebras with the well-known Lie algebra semiinfinite cohomology. A new description of the critical 2-cocycle is provided. As a consequence we obtain another proof of the fact that additive categories generated by Verma modules over affine Lie algebras on dual levels are antiequivalent.

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arXiv1996

Comparing finite type invariants of knots and integral homology 3-spheres

Using elementary counting methods of weight systems for finite type invariants of knots and integral homology 3-spheres, in the spirit of [B-NG], we answer positively three questions raised in [Ga]. In particular, we exhibit a one-to-one map from the space of finite type invariants of integral homology 3-spheres to the space of finite type invariants of knots in $S^3$.

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arXiv1996

Tensor product of Vertex operator algebras

Let $V_1 \otimes V_2$ be a tensor product of VOAs. Using Zhu theory we discuss the theory of representations of V (associative algebra, modules and fusion rules). We prove that this theory is more or less the same as representation theory of tensor product of the associative algebras.

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arXiv1996

Multivariable Askey-Wilson Polynomials and Quantum Complex Grassmannians

We present a one-parameter family of constant solutions of the reflection equation and define a family of quantum complex Grassmannians endowed with a transitive action of the quantum unitary group. By computing the radial part of a suitable Casimir operator, we identify the zonal spherical functions (i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional irreducible representations) as multivariable Askey-Wilson polynomials containing two continuous and two discrete parameters.

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arXiv1996

On the defining relations of the affine Lie superalgebras and their quantized universal enveloping superalgebras

In this paper, we give defining relations of the affine Lie superalgebras an and defining relations of a super-version of the Drinfeld[D]-Jimbo[J] affine quantized universal enveloping algebras. As a result, we can exactly define the affine quantized universal enveloping superalgebras with generators and relations. Moreover we give a Drinfeld's realizations of $U_h({\hat {sl}}(m|n)^{(1)})$.

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arXiv1996

Homological Reduction of Constrained Poisson Algebras

The ``classical BRST construction'' as developed by Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of the Poisson algebra $P = C^\infty (W)$ of smooth functions on a Poisson manifold $W$ by the ideal $I$ of functions which vanish on a constraint locus. This ideal is called first class if $I$ is closed under the Poisson bracket; geometers refer to the constraint locus as coisotropic. The physicists' model is crucially a differential Poisson algebra extension of a Poisson algebra $P$; its differential contains a piece which reinvented the Koszul complex for the ideal $I$ and a piece which looks like the Cartan-Chevalley-Eilenberg differential. The present paper is concerned purely with the homological (Poisson) algebraic structures, using the notion of ``model'' from rational homotopy theory and the techniques of homological perturbation theory to establish some of the basic results explaining the mathematical existence of the classical BRST-BFV construction. Although the usual treatment of BFV is basis dependent (individual constraints) and nominally finite dimensional, I take care to avoid assumptions of finite dimensionality and work more invariantly in terms of the ideal. In particular, the techniques are applied to the `irregular' case (the ideal is not generated by a regular sequence of constraints), although the geometric interpretation is less complete.

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arXiv1996

Skew Symmetric Bundle Maps on Space-Time

We study the "Lie Algebra" of the group of Gauge Transformations of Space-time. We obtain topological invariants arising from this Lie Algebra. Our methods give us fresh mathematical points of view on Lorentz Transformations, orientation conventions, the Doppler shift, Pauli matrices , Electro-Magnetic Duality Rotation, Poynting vectors, and the Energy Momentum Tensor $T$ .

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arXiv1996

Some Remarks on Quantum and Braided Group Gauge Theory

We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzezinski and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. We study the bosonisation of any braided group provides as a trivial principal bundle in three ways.

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arXiv1996

Bosonic representations of Yangian Double $\cD Y_{\hbar}(\gtg)$ with $\gtg=\gtgl_N,\gtsl_N$

On the basis of `$RTT=TTR$' formalism, we introduce the quantum double of the Yangian $Y_{\hbar}(\gtg)$ for $\gtg=\gtgl_N,\gtsl_N$ with a central extension. The Gauss decomposition of T-matrices gives us the so-called Drinfel'd generators. Using these generators, we present some examples of both finite and infinite dimensional representations that are quite natural deformations of the corresponding affine counterpart.

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arXiv1996

Understanding the Kauffman bracket skein module

The Kauffman bracket skein module $K(M)$ of a 3-manifold $M$ is defined over formal power series in the variable $h$ by letting $A=e^{h/4}$. For a compact oriented surface $F$, it is shown that $K(F \times I)$ is a quantization of the $\g$-characters of the fundamental group of $F$, corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined. Topologically free Kauffman bracket modules are shown to generate finite type invariants. It is shown for compact $M$ that $K(M)$ can be generated as a module by cables on a finite set of knots. Moreover, if $M$ contains no incompressible surfaces, the module is finitely generated.

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arXiv1996

Rings of $SL_2({\mathbb C})$-Characters and the Kauffman Bracket Skein Module

Let $M$ be a compact orientable 3-manifold. The set of characters of $SL_2({\mathbb C})$ representations of the fundamental group of $M$ forms a closed affine algebraic set. We show that its coordinate ring is isomorphic to a specialization of the Kauffman bracket skein module modulo its nilradical. This is accomplished by making the module into a combinatorial analog of the ring, in which tools of skein theory are exploited to illuminate relations among characters. We conclude with an application, proving that a small manifold's specialized module is necessarily finite dimensional.

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