Search

In this short note we study the center of the universal enveloping algebra of the strange Lie superalgebra Q(N). We obtain an analogue of the well known Perelomov-Popov formula (1968) for central elements of this algebra - an expression of the central characters through the highest weight parameters., Comment: 5 pages, thanks added

We consider the Lie algebra $\mathfrak{g} = \mathfrak{p}_n $ of (n + 1) × (n + 1) matrices with zeros in the last row. This algebra has received the name of mirabolic; it has many remarkable properties and plays an important role in representation theory. In this paper we study open coadjoint orbits for the corresponding Lie group P n .

The Yangian double DY(A(m, n)) of the Lie superalgebra A(m, n) is described in terms of generators and defining relations. Normally ordered bases in the Yangian and its dual in the quantum double are introduced. We calculate the pairing between the elements of these bases and obtain a formula for the universal R-matrix of the Yangian double as well as a formula for the universal R-matrix (introduced by Drinfeld) of the Yangian.

The five simple exceptional complex Lie superalgbras of vector fields are described. One of them is new; the other four are explicitely described for the first time. All of the exceptional Lie superalgebras are obtained with the help of the Cartan prolongation or a generalized prolongation. The description of several of the exceptional Lie superalgebras is associated with the Lie superalgebra AS - the nontrivial central extension of the supertraceless subalgebra SPE(4) of the periplectic Lie superalgebra PE(4) that preserves the nondegenerate odd bilinear form on the (4|4)-dimensional superspace. (A nontrivial central extension of SPE(n) only exists for n=4.)

1. S. D. Fisher and C. Micchelli, Duke Math. 3., 47, No. 4, 789-801 (1980). 2. S. D. Fisher and M. I. Stessin, Preprint, Northwestern University, Evanston (1990). 3. W. K. Hayman and P. L. Kennedy, Subharmonic Functions, Academic Press, New York (1976). 4. H. Widom, a. Approx. Theory, 5, 343-361 (1972). 5. S. D. Fisher and C. Micchelli, Amer. J. Math., 166, No. 3, 593-609 (1984), Math. Ann., 273, 131-147 (1985). 6. Yu. A. Farkov, in: Function Theory and Approximation, Proc. III Saratov Winter School, Publ. Saratov University, Part. 3 (1988), pp. 72-74. 7. H. Shapiro and A. Shields, Amer. J. Math., 83, Part. 3, 513-532 (1961). 8. O. G. Parfenov, Mat. Sb., 131 (173), No. 4 (12), 501-518 (1986).