Quantum Algebra And Topology

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.

Deformed $\W$--algebra $\W_{q,t}(\g)$ associated to an arbitrary simple Lie algebra $\g$ is defined together with its free field realizations and the screening operators. Explicit formulas are given for generators of $\W_{q,t}(\g)$ when $\g$ is of classical type. These formulas exhibit a deep connection between $\W_{q,t}(\g)$ and the analytic Bethe Ansatz in integrable models associated to quantum affine algebras $U_q(\G)$ and $U_t(\GL)$. The scaling limit of $\W_{q,t}(\g)$ is closely related to affine Toda field theories.

We give explicit formulas for the generators of q -deformed W-algebras associated to Lie algebras D n , E 6 and G 2 , and compute the Poisson brackets between the generators.

Deformed Harmonic Oscillator Algebras are generated by four operators, two mutually adjoint a and a † , and two self-adjoint N and the unity 1 such as: [a,N]=a,[ a † ,N]=− a † , a † a=ψ(N) and a a † =ψ(N+1) . The Bargmann Hilbert space is defined as a space of functions, holomorphic in a ring of the complex plane, equipped with a scalar product involving a true integral. In a Bargmann representation, the operators of a Deformed Harmonic Oscillator Algebra act on a Bargmann Hilbert space and the creation (or the annihilation operator) is the multiplication by z . We discuss the conditions of existence of Deformed Harmonic Oscillator Algebras assumed to admit a given Bargmann representation.

Twisted Hopf algebra s l ξ (2) gives rise to a deformation of the Yangian Y(sl(2)) . The corresponding deformations of the integrable XXX-spin chain and the Gaudin model are discussed.

Generalizing the case of the usual harmonic oscillator, we look for Bargmann representations corresponding to deformed harmonic oscillators. Deformed harmonic oscillator algebras are generated by four operators a, a † ,N and the unity 1 such as [a,N]=a,[ a † ,N]=− a † , a † a=ψ(N) and a a † =ψ(N+1) . We discuss the conditions of existence of a scalar product expressed with a true integral on the space spanned by the eigenstates of a (or a † ). We give various examples, in particular we consider functions ψ that are linear combinations of q N , q −N and unity and that correspond to q-oscillators with Fock-representations or with non-Fock-representations.

We study deformations of the standard embedding of the Lie algebra $\Vect(S^1)$ of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle T ∗ S 1 (with respect to the Poisson bracket). We consider two analogous but different problems: (a) formal deformations of the standard embedding of $\Vect(S^1)$ into the Lie algebra of functions on $\dot T^*S^1:=T^*S^1\setminusS^1$ which are Laurent polynomials on fibers, and (b) polynomial deformations of the $\Vect(S^1)$ subalgebra inside the Lie algebra of formal Laurent series on T ˙ ∗ S 1 .

We introduce a class of induced representations of the degenerate double affine Hecke algebra of gl_N and analyze their structure mainly by means of intertwiners. We also construct them from modules of the affine Lie algebra using Knizhnik-Zamolodchikov connections in the conformal field theory. This construction provides a natural quotient of induced modules, which turns out to be the unique irreducible one under a certain condition. Some cunjectual formulas are presented for the symmetric part of these quotients.

We give a criterion for the Demazure crystal B w (λ) defined by Kashiwara to have a tensor product structure. We study the $\sln$ symmetric tensor case, and see some Demazure characters are expressed using Kostka-Foulkes polynomials.

We discuss the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so. On the other hand, finite group invariants, such as the set of homomorphisms from the knot group to M_11, can detect knot invertibility. For many natural classes of knot invariants, including Vassiliev invariants and quantum Lie group invariants, we can conclude that the invariants either distinguish all oriented knots, or there exist prime, unoriented knots which they do not distinguish.