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At least 19 recordsLinked to original sources
arXiv2010

A Lie-theoretic construction of representations of the degenerate affine and double affine Hecke algebras of type $BC_n$

Let G=GL(N), K=GL(p)xGL(q), where p+q=N, and n be a positive integer. We construct a functor from the category of Harish-Chandra modules for the pair (G,K) to the category of representations of the degenerate affine Hecke algebra of type B_n, and a functor from the category of K-monodromic twisted D-modules on G/K to the category of representations of the degenerate double affine Hecke algebra of type BC_n; the second functor is an extension of the first one. These functors are generalizations of the type A functors from q-alg/9710037 and math/0702670, respectively.

math.RT↗
arXiv2017

A Formality quasi-isomorphism for Hochschild cochains over rationals can be constructed recursively

It is believed arXiv:0808.2762, arXiv:math/9904055 that, among the coefficients entering Kontsevich's formality quasi-isomorphism arXiv:q-alg/9709040, there are irrational (possibly even transcendental) numbers. In this paper, we prove that a formality quasi-isomorphism for Hochschild cochains of a polynomial algebra over rationals can be constructed recursively. The proof that the proposed recursive algorithm works, is based on the existence of formality quasi-isomorphism over reals. However, the algorithm requires no explicit knowledge of the coefficients entering Kontsevich's construction. Although this algorithm completely bypasses Tamarkin's approach arXiv:math/0003052, arXiv:math/9803025, the construction is inspired by Proposition 5.8 from the classical paper (Algebra i Analiz, 1990) by V. Drinfeld.

math.KT↗
arXiv2021

Type $C$ Webs

We define a $\mathbb{C}(q)$-linear pivotal category $\mathbf{Web}(\mathfrak{sp}_{2n})$ and prove that it is equivalent to the full subcategory of finite-dimensional representations of $U_q(\mathfrak{sp}_{2n})$ tensor-generated by the fundamental representations. This answers the type $C$ case of the main open problem from Kuperberg's 1996 paper "Spiders for rank 2 Lie algebras" (arXiv:q-alg/9712003).

math.RT↗
arXiv2003

Two applications of elementary knot theory to Lie algebras and Vassiliev invariants

Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [Bar-Natan, Garoufalidis, Rozansky and Thurston, arXiv:q-alg/9703025] and [Deligne, letter to Bar-Natan, January 1996, http://www.ma.huji.ac.il/~drorbn/Deligne/], which give, respectively, the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams. It turns out that the Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on a wire), and its intertwining property is analogous to the computation of 1+1=2 on an abacus. The Wheels conjecture is proved from the fact that the k-fold connected cover of the unknot is the unknot for all k. Along the way, we find a formula for the invariant of the general (k,l) cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the Duflo-Kirillov map S(g)-->U(g) for metrized Lie (super-)algebras g.

math.QA↗
arXiv2013

On top Chern classes of universal bundles on moduli spaces of rank two coherent sheaves on the projective plane, or How to calculate the correlation function in SYM N=2 Nf=4 quantum field theory on complex projective plane

We explain how to calculate the correlation function for SYM N=2 SO(3)-gauge QFT with 4 flavors in terms of top Chern classes of the universal bundles over the moduli spaces of rank 2 stable torsion free coherent sheaves with det=-1 on the complex projective plane. We give a direct geometrical description of these moduli spaces, study a 2-dimensional torus action on them, and calculate several initial coefficients of the correlation function via Bott residue formula.

alg-geom↗
arXiv1995

Cohomology of Drinfel'd algebras: A General Nonsense Approach

In our paper [Markl, Shnider: Drinfel'd Algebra Deformations and the Associahedra, IMRN 1994, no. 4, 169-176] we announced a construction of a cohomology controlling deformations of quasi-coassociative (or Drinfel'd) bialgebras. The full version of the paper will appear as [Markl, Shnider: Drinfel'd Algebra Deformations, Homotopy Comodules and the Associahedra] in Trans. Amer. Math. Soc. The construction in the paper was based on very explicit arguments using deep combinatorial properties of the associahedra. The present paper gives an alternative, general nonsense approach to the construction. So, we just prove the existence of such a cohomology without explicitly constructing it. This should be compared with the two approaches to the cohomology of associative algebras: we either describe explicitly the Hochschild complex and say "Behold! this is the cohomology" or we prove the existence of a projective resolution and define the cohomology as the derived functor.

q-alg↗
arXiv1995

On Hopf algebras and the elimination theorem for free Lie algebras

The elimination theorem for free Lie algebras, a general principle which describes the structure of a free Lie algebra in terms of free Lie subalgebras, has been recently used by E. Jurisich to prove that R. Borcherds' ``Monster Lie algebra'' has certain large free Lie subalgebras, illuminating part of Borcherds' proof that the moonshine module vertex operator algebra obeys the Conway-Norton conjectures. In the present expository note, we explain how the elimination theorem has a very simple and natural generalization to, and formulation in terms of, Hopf algebras. This fact already follows from general results contained in unpublished 1972 work, unknown to us when we wrote this note, of R. Block and P. Leroux.

q-alg↗
arXiv1996

Topics in hidden symmetries. IV

This note being devoted to some aspects of the inverse problem of representation theory explicates the links between researches on the Sklyanin algebras and the author's (based on the noncommutative geometry) approach to the setting free of hidden symmetries in terms of "the quantization of constants". Namely, the Racah-Wigner algebra for the Sklyanin algebra is constructed. It may be considered as a result of the quantization of constants in the Racah-Wigner algebra for the Lie algebra $sl(2,C)$. The Racah-Wigner algebra for the Sklyanin algebra is an example of the noncommutative weighted shift operator algebras (NWSO-algebras), which generalize the mho-algebras introduced by the author earlier. If the Sklyanin algebra is interpreted as an algebra of anomalous spins then the Racah-Wigner algebra for it may be regarded as an enlargement of the Sklyanin algebra by operators of the anomalous spin-spin interaction (of tensor type).

q-alg↗
arXiv2019

Idempotents of Hecke algebras of type A

We use a skein-theoretic version of the Hecke algebras of type A to present three-dimensional diagrammatic views of Gyoja's idempotent elements, based closely on the corresponding Young diagram. In this context we give straightforward calculations for the eigenvalues of two natural central elements in the Hecke algebras, namely the full curl and the sum of the Murphy operators. We discuss their calculation also in terms of the framing factor associated to the appropriate irreducible representation of the quantum group SU(N,q).

q-alg↗
arXiv1997

On BC type basic hypergeometric orthogonal polynomials

The five parameter family of multivariable Askey-Wilson polynomials is studied with four parameters generically complex. The multivariable Askey-Wilson polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partially discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partially discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable q-Racah polynomials and multivariable big and little q-Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the multivariable Askey-Wilson polynomials. In particular new proofs of several well known q-analogues of the Selberg integral are obtained.

q-alg↗
arXiv1996

Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations

Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. The standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. In particular, elliptic solutions of Hirota's equation give complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's $Q$-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to Bethe ansatz are studied. The nested Bethe ansatz equations for $A_{k-1}$-type models appear as discrete time equations of motions for zeros of classical $τ$-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation and a new determinant formula for eigenvalues of the quantum transfer matrices are obtained.

hep-th↗
arXiv1997

The sh Lie structure of Poisson brackets in field theory

A general construction of an sh Lie algebra from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket.

hep-th↗
arXiv1997

The hidden symmetry algebras of a class of quasi-exactly solvable multi dimensional operators

Let $P(N,V)$ denote the vector space of polynomials of maximal degree less than or equal to $N$ in $V$ independent variables. This space is preserved by the enveloping algebra generated by a set of linear, differential operators representing the Lie algebra $gl(V+1)$. We establish the counterpart of this property for the vector space $P(M,V) \oplus P(N,V)$ for any values of the integers $M,N,V$. We show that the operators preserving $P(M,V) \oplus P(N,V)$ generate an abstract superalgebra (non linear if $Δ=\mid M-N\mid\geq 2$). A family of algebras is also constructed, extending this particular algebra by $Δ-1$ arbitrary complex parameters.

q-alg↗
arXiv1997

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↗
arXiv1997

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↗
arXiv1995

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↗