Smooth vectors for highest weight representations

Conformal restriction, highest-weight representations and SLE

We show how to relate Schramm-Loewner Evolutions (SLE) to highestweight representations of infinite-dimensional Lie algebras that are singular at level two, using the conformal restriction properties studied by Lawler, Schramm and Werner in [33]. This confirms the prediction from conformal field theory that two-dimensional critical systems are related to degenerate representations.

Highest weight representations of the Virasoro algebra

We consider representations of the Virasoro algebra, a one-dimensional central extension of the Lie algebra of vectorfields on the unit circle. Positive-energy, highest weight and Verma representations are defined and investigated. The Shapovalov form is introduced, and we study Kac formula for its determinant and some consequences for unitarity and degeneracy of irreducible highest weight repr...

Tau-functions as highest weight vectors for W1+∞ algebra

For each r = (r1, r2, . . . , rN ) ∈ C N we construct a highest weight module Mr of the Lie algebra W1+∞. The highest weight vectors are specific tau-functions of the N -th Gelfand–Dickey hierarchy. We show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors. hep-th/9510211

HIGHEST WEIGHT VECTORS OF IRREDUCIBLE REPRESENTATIONS OF THE QUANTUM SUPERALGEBRA Uq (gl(m,n))

The Iwahori-Hecke algebra Hk(q ) of type A acts on tensor product space V ⊗k of the natural representation of the quantum superalgebra Uq(gl(m, n)). We show this action of Hk(q ) and the action of Uq(gl(m, n)) on the same space determine commuting actions of each other. Together with this result and Gyoja’s q-analogue of the Young symmetrizer, we construct a highest weight vector of each irredu...

HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GLn ON POLYNOMIALS, II