A New Lie Bialgebra Structure

We describe the Lie bialgebra structure on the Lie superalgebra sl(2, 1) related to an r−matrix that cannot be obtained by a Belavin-Drinfeld type construction. This structure makes sl(2, 1) into the Drinfeld double of a four-dimensional subalgebra. It is well-known that non-degenerate r−matrices (describing quasitriangular Lie bialgebra structures) on simple Lie algebras are classified by Belavin-Drinfeld triples, (the original references are [BD1, BD2], more pedagogical presentations providing ample background can be found in [CP, ES]). A similar construction using Belavin-Drinfeld type triples is possible for simple Lie superalgebras with nondegenerate Killing form. Surprisingly, though, in the super setting, there are certain non-degenerate r−matrices that do not fit such a description, see [K]. The purpose of this note is to study the super Lie bialgebra structure associated to such an r−matrix on the simple Lie superalgebra sl(2, 1). We start in Section 1 with some background on super Lie bialgebra structures and some basic constructions related to them. In Section 2, we explicitly describe the r-matrix that we will be interested in. Section 3 describes in detail the associated super Lie bialgebra structure on sl(2, 1); this structure makes sl(2, 1) into the Drinfeld double of a four-dimensional subalgebra. A comparison with the standard super Lie bialgebra structure is also provided in this section. We end in Section 4 with a brief discussion of the results and further directions for investigation. Acknowledgments. The author would like to thank N. Reshetikhin, V. Serganova and M. Yakimov for their comments and suggestions. 1. Lie Bialgebra Structures on Lie Superalgebras 1.1. Cohomology of Lie superalgebras. The cohomology theory of Lie superalgebras is more complicated than that of Lie algebras. Even for simple Lie superalgebras and for low dimensions, it is not yet completed. Here we summarize certain basic facts that we will use. For more on the cohomology theory of Lie superalgebras one can look at [F, SZ]. Recall that if g is a Lie algebra, then an n−cochain taking values in a g−module M is an alternating n−linear map f(x1, x2, · · · , xn) of n variables in g. We can view 2000 Mathematics Subject Classification. Primary 17B62, 17B20.