EXACT SOLVABILITY OF THE CALOGERO AND SUTHERLAND MODELS

Translationally invariant symmetric polynomials as coordinates for N-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland N-body Hamiltonians, after appropriate gauge transformations, can be presented as a quadratic polynomial in the generators of the algebra sl N in finitedimensional degenerate representation. The exact solvability of these models follows from the existence of the infinite flag of such representation spaces, preserved by the above Hamiltonians. A connection with Jack polynomials is discussed.

THE STRUCTURE OF REPRESENTATION FOR THE W(3) MINIMAL MODEL

Some exact results in the representation theory of the W(3) algebra are presented. The embedding structure of the completely degenerate representation is studied in detail. The character formula is obtained by the Feigen-Fuchs-Rocha-Caridi method. It is manifestly irreducible and coincides with the branching coefficients of diagonal embedding [Formula: see text] in the unitary case. The W(n) character for general n can also be obtained completely in parallel. Four-point functions and fusion rules are calculated explicitly for the Z3 Potts model as a W(3) minimal theory, which agree with the Verlinde formula.

The ~ -representations of symmetric homogeneous algebras

In 1947 I. E. Segal proved that to each non-degenerate ~ -representation R of L1 (= L1 (G) for a compact group G) with representation space , there corresponds a continuous unitary representation W of G, also with representation space , which satisfiesfor each fL1 and hk . This was extended to Lp,1 p < , in 1970 by E. Hewitt and K. A. Ross. We now generalize this result to any symmetric homogeneous convolution Banach alebra of pseudomeasures on G. Further we prove that the correspondence preserves irreduibility.

SU(1, 3) as a Dynamical Group; Analysis of All the Discrete Representations

A method using the most degenerate representation of SU(1, 3) as a dynamical group for the strong interactions of the spin 1/2+ baryons is extended to consider all the discrete representations. It is shown that physical conditions place very severe restrictions on the parameters characterizing such a representation.