Factorisation of Macdonald polynomials
We discuss the problem of factorisation of the symmetric Macdonald polynomials and present the obtained results for the cases of 2 and 3 variables.
q-alg↗
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Quantization of the Algebra of Chord Diagrams
In this paper we define an algebra structure on the vector space $L(Σ)$ generated by links in the manifold $Σ\times [0,1]$ where $Σ$ is an oriented surface. This algebra has a filtration and the associated graded algebra $L_{Gr}(Σ)$ is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra of chord diagrams $ch(Σ)$ on $Σ$ to $L_{Gr}(Σ)$. We show that multiplication in $L(Σ)$ provides a geometric way to define a deformation quantization of the algebra of chord diagrams, provided there is a universal Vassiliev invariant for links in $Σ\times [0,1]$. The quantization descends to a quantization of the moduli space of flat connections on $Σ$ and it is universal with respect to group homomorphisms. If $Σ$ is compact with free fundamental group we construct a universal Vassiliev invariant.
q-alg↗
BV-Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of (W-) Strings
Given a simple, simply laced, complex Lie algebra $\bfg$ corresponding to the Lie group $G$, let $\bfnp$ be the subalgebra generated by the positive roots. In this paper we construct a BV-algebra $\fA[\bfg]$ whose underlying graded commutative algebra is given by the cohomology, with respect to $\bfnp$, of the algebra of regular functions on $G$ with values in $\mywedge (\bfnp\backslash\bfg)$. We conjecture that $\fA[\bfg]$ describes the algebra of {\it all} physical (i.e., BRST invariant) operators of the noncritical $\cW[\bfg]$ string. The conjecture is verified in the two explicitly known cases, $\bfg=\sltw$ (the Virasoro string) and $\bfg=\slth$ (the $\cW_3$ string).
hep-th↗
Separation of variables and integral relations for special functions
We show that the method of separation of variables gives a natural generalisation of integral relations for classical special functions of one variable. The approach is illustrated by giving a new proof of the ``quadratic'' integral relations for the continuous q-ultraspherical polynomials. The separating integral operator M expressed in terms of the Askey-Wilson operator is studied in detail: apart from writing down the characteristic (``separation'') equations it satisfies, we find its spectrum, eigenfunctions, inversion, invariants (invariant q-difference operators), and give its interpretation as a fractional q-integration operator. We also give expansions of the A1 Macdonald polynomials into the eigenfunctions of the separating operator M and vice versa.
q-alg↗
New Quantum Groups by Double-Bosonisation
We obtain new family of quasitriangular Hopf algebras $C^{0|n}_q\lcross \widetilde{U_q(su_n)}\rcross C^{0|n}_q$ via the author's recent double-bosonisation construction for new quantum groups. They are versions of $U_q(su_{n+1})$ with a fermionic rather than bosonic quantum plane of roots adjoined to $U_q(su_n)$. We give the $n=2$ case in detail. We also consider the anyonic-double of an anyonic ($Z/nZ$-graded) braided group and the double-bosonisation of the free braided group in $n$ variables.
q-alg↗
Quantum Deformations of Space-Time Symmetries and Interactions
We discuss quantum deformations of Lie algebra as described by the noncoassociative modification of its coalgebra structure. We consider for simplicity the quantum $D=1$ Galilei algebra with four generators: energy $H$, boost $B$, momentum $P$ and central generator $M$ (mass generator). We describe the nonprimitive coproducts for $H$ and $B$ and show that their noncocommutative and noncoassociative structure is determined by the two-body interaction terms. Further we consider the case of physical Galilei symmetry in three dimensions. Finally we discuss the noninteraction theorem for manifestly covariant two-body systems in the framework of quantum deformations of $D=4$ \poin algebra and a possible way out.
hep-th↗
Gauge transformations and quasitriangularity
Natural conditions on a Poisson/quantum group G to implement Poisson/quantum gauge transformations on the lattice are investigated. In addition to our previous result that transformations on one lattice link require G to be coboundary, it is shown that for a sequence of links one needs a quasitriangular G.
q-alg↗
Anyonic Realization of the Quantum Affine Lie Algebras
We give a realization of the quantum affine Lie algebras $\uqa$ and $\uqc$ in terms of anyons defined on a one-dimensional chain (or on a two-dimensional lattice), the deformation parameter $q$ being related to the statistical parameter $ν$ of the anyons by $q = e^{iπν}$. In the limit of the deformation parameter going to one we recover the Feingold-Frenkel fermionic construction of undeformed affine Lie algebras.
q-alg↗
On the existence of deformed Lie-Poisson structures for quantized groups
The geometrical description of deformation quantization based on quantum duality principle makes it possible to introduce deformed Lie-Poisson structure. It serves as a natural analogue of classical Lie bialgebra for the case when the initial object is a quantized group. The explicit realization of the deformed Lie-Poisson structure is a difficult problem. We study the special class of such constructions characterized by quite a simple form of tanjent vector fields. It is proved that in such a case it is sufficient to find four Lie compositions that form two deformations of the first order and four Lie bialgebras. This garantees the existence of two families of deformed Lie-Poisson structures due to the intrinsic symmetry of the initial compositions. The explicit example is presented.
q-alg↗
A Note on First Order Differential Calculus on Quantum Principal Bundles
The relationship between the exactness of a first order differential calculus on a comodule algebra $P$ and the Galois property of $P$ is investigated.
q-alg↗
Higher Order Differential Calculus on $SL_q(N)$
Let $Γ$ be an $N^2$-dimensional bicovariant first order differential calculus on a Hopf algebra $SL_q(N)$. There are three possibilities to construct a differential Z-graded Hopf algebra $Γ^\wedge$ which contains $Γ$ as its first order part. Let $q$ be a transcendental complex number. For $N>2$ these three Z-graded Hopf algebras coincide. For Woronowicz' external algebra we calculate the dimensions of the spaces of left-invariant and bi-invariant $k$-forms. In this case each bi-invariant form is closed. In case of $4D_\pm$ calculi on $SL_q(2)$ the universal calculus is strictly larger than the other two calculi. In particular, the bi-invariant 1-form is not closed.
q-alg↗
Deformation quantization of Poisson manifolds, I
I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.
q-alg↗
Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras
This is a continuation of our "Lecture on Kac--Moody Lie algebras of the arithmetic type" \cite{25}. We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice), reflection group $W\subset W(S)$, fundamental polyhedron $\Cal M$ of $W$ and an acceptable (corresponding to twisting coefficients) set $P({\Cal M})\subset M$ of vectors orthogonal to faces of $\Cal M$ (simple roots). One can construct the corresponding Lorentzian Kac--Moody Lie algebra ${\goth g}={\goth g}^{\prime\prime}(A(S,W,P({\Cal M})))$ which is graded by $M$. We show that $\goth g$ has good behavior of imaginary roots, its denominator formula is defined in a natural domain and has good automorphic properties if and only if $\goth g$ has so called {\it restricted arithmetic type}. We show that every finitely generated (i.e. $P({\Cal M})$ is finite) algebra ${\goth g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1)))$ may be embedded to ${\goth g}^{\prime\prime}(A(S,W,P({\Cal M})))$ of the restricted arithmetic type. Thus, Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a natural class to study. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the best automorphic properties for the denominator function if they have {\it a lattice Weyl vector $ρ$}. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type with generalized lattice Weyl vector $ρ$ are called {\it elliptic}
alg-geom↗
The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
We find automorphic corrections for the Lorentzian Kac--Moody algebras with the simplest generalized Cartan matrices of rank 3: A_{1,0} = 2 0 -1 0 2 -2 -1 -2 2 and A_{1,I} = 2 -2 -1 -2 2 -1 -1 -1 2 For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form χ_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30 with respect to a 2-congruence subgroup. We find infinite product or sum expansions for these forms. Our method of construction of χ_{35} leads to the direct construction of Siegel modular forms by infinite product expansions, whose divisors are the Humbert surfaces with fixed discriminants. Existence of these forms was proved by van der Geer in 1982 using some geometrical consideration. We announce a list of all hyperbolic symmetric generalized Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.
alg-geom↗
On Quantum Groups in the Hubbard Model with Phonons
The correct Hamiltonian for an extended Hubbard model with quantum group symmetry as introduced by A. Montorsi and M. Rasetti is derived for a D-dimensional lattice. It is shown that the superconducting SUq(2) holds as a true quantum symmetry only for D = 1 and that terms of higher order in the fermionic operators in addition to phonons are required for a quantum symmetric hamiltonian. The condition for quantum symmetry is "half filling" and there is no local electron-phonon coupling. A discussion of Quantum symmetries in general is given in a formalism that should be readily accessible to non Hopf-algebraists.
cond-mat↗
Normal Ordering in the Theory of Correlation Functions of Exactly Solvable Models
We study models of quantum statistical mechanics which can be solved by the algebraic Bethe ansatz. The general method of calculation of correlation functions is based on the method of determinant representations. The auxiliary Fock space and auxiliary Bose fields are introduced in order to remove the two body scattering and represent correlation functions as a mean value of a determinant of a Fredholm integral operator; the representation has a simple form for large space and time separations. In this paper we explain how to calculate the mean value in the auxiliary Fock space of asymptotic expression of the Fredholm determinant. It is necessary for the evaluation of the asymptotic form of the physical correlation functions.
hep-th↗
Exact perturbative solution of the Kondo problem
We explicitly evaluate the infinite series of integrals that appears in the "Anderson-Yuval" reformulation of the anisotropic Kondo problem in terms of a one-dimensional Coulomb gas. We do this by developing a general approach relating the anisotropic Kondo problem of arbitrary spin with the boundary sine-Gordon model, which describes impurity tunneling in a Luttinger liquid and in the fractional quantum Hall effect. The Kondo solution then follows from the exact perturbative solution of the latter model in terms of Jack polynomials.
cond-mat↗
Linear Connections on the Two Parameter Quantum Plane
We apply a recently proposed definition of a linear connection in non commutative geometry based on the natural bimodule structure of the algebra of differential forms to the case of the two-parameter quantum plane. We find that there exists a non trivial family of linear connections only when the two parameters obeys a specific relation.
q-alg↗