56 research outputs found
Integral formulas for wave functions of quantum many-body problems and representations of gl(n)
We derive explicit integral formulas for eigenfunctions of quantum integrals
of the Calogero-Sutherland-Moser operator with trigonometric interaction
potential. In particular, we derive explicit formulas for Jack's symmetric
functions. To obtain such formulas, we use the representation of these
eigenfunctions by means of traces of intertwining operators between certain
modules over the Lie algebra , and the realization of these modules
on functions of many variables.Comment: 6 pages. One reference ([FF]) has been corrected. New references and
an introduction have been adde
Quantum integrable systems and representations of Lie algebras
In this paper the quantum integrals of the Hamiltonian of the quantum
many-body problem with the interaction potential K/sinh^2(x) (Sutherland
operator) are constructed as images of higher Casimirs of the Lie algebra gl(N)
under a certain homomorphism from the center of U(gl(N)) to the algebra of
differential operators in N variables. A similar construction applied to the
affine gl(N) at the critical level k=-N defines a correspondence between higher
Sugawara operators and quantum integrals of the Hamiltonian of the quantum
many-body problem with the potential equal to constant times the Weierstrass
function. This allows one to give a new proof of the Olshanetsky-Perelomov
theorem stating that this Hamiltonian defines a completely integrable quantum
system. We also give a new expression for eigenfunctions of the quantum
integrals of the Sutherland operator as traces of intertwining operators
between certain representations of gl(N).Comment: 17 pages, no figure
Double Affine Hecke Algebras and Their Applications
I am very excited to have been asked to deliver an invited address at the Fall 2017 meeting of the AMS Western Section (UC Riverside). I will talk about double affine Hecke algebras and their applications
Exact sequences of tensor categories with respect to a module category
We generalize the definition of an exact sequence of tensor categories due to Bruguières and Natale, and introduce a new notion of an exact sequence of (finite) tensor categories with respect to a module category. We give three definitions of this notion and show their equivalence. In particular, the Deligne tensor product of tensor categories gives rise to an exact sequence in our sense. We also show that the dual to an exact sequence in our sense is again an exact sequence. This generalizes the corresponding statement for exact sequences of Hopf algebras. Finally, we show that the middle term of an exact sequence is semisimple if so are the other two terms. Keywords: Tensor categories; Module categorie
On Cohen–Macaulayness of Algebras Generated by Generalized Power Sums
Generalized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen–Macaulay. It turns out that the Cohen–Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation theoretic results and deformation theory, we establish Cohen–Macaulayness of the algebra of q, t-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, proving a conjecture of Brookner, Corwin, Etingof, and Sam. We also apply representation-theoretic techniques to studying m-quasi-invariants of deformed Calogero–Moser systems. In an appendix to this paper, M. Feigin uses representation theory of Cherednik algebras to compute Hilbert series for such quasi-invariants, and show that in the case of one light particle, the ring of quasi-invariants is Gorenstein.National Science Foundation (U.S.) (grant DMS-1000113
Finite dimensional Hopf actions on algebraic quantizations
Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra, we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z 1 ,… z s ] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s = 0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal enveloping algebras of finite dimensional Lie algebras, spherical symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras (in particular, quantized quiver varieties), finite W-algebras and their central reductions, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, and Sklyanin algebras. The generalization in the last three cases uses a result from algebraic number theory due to A. Perucca.National Science Foundation (U.S.) (Grant CHE-1464804)National Science Foundation (U.S.) (Grant DMS-1550306
On properties of the lower central series of associative algebras
We give an accessible introduction into the theory of lower central series of associative algebras, exhibiting the interplay between algebra, geometry and representation theory that is characteristic for this subject, and discuss some open questions. In particular, we provide shorter and clearer proofs of the main results of this theory. We also discuss some new theoretical and computational results and conjectures on the lower central series of the free algebra in two generators modulo a generic homogeneous relation. Keywords: Lower central series; associative algebra; free algebr
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