un 2 00 6 Universal coverings of Steinberg Lie superalgebras st ( m , n , R ) when m + n = 3 , 4

The second homology group H2(st(m,n,R)) of the Steinberg Lie superalgebra st(m,n,R)(m + n ≥ 5), which is trivial, has been studied by A.V.Mikhalev and I.A.Pinchuk in [MP]. In this paper, we will work out H2(st(m,n,R)) explicitly for m + n = 3, 4 which is also trivial when m + n = 3, but not necessarily trivial when m + n = 4. Introduction Steinberg Lie algebras stn(R) and/or their universal coverings have been studied by Bloch [Bl], Kassel-Loday [KL], Kassel [Ka], Faulkner [F], Allison-Faulkner [AF], Berman-Moody [BM], [G1, 2] and [AG], and among others, which are Lie algebras graded by finite root systems of type Al with l ≥ 2. When n ≥ 5, the Steinberg Lie algebra stn(R) is the universal covering of the Lie algebra sln(R) whose kernel is isomorphic to the first cyclic homology group HC1(R) of the associative algebra R and the second Lie algebra homology group H2(stn(R)) = 0. But when n = 3, 4, H2(stn(R)) is not necessarily equal to 0, which have been worked out in [GS]. Similarly, we can consider the Steinberg Lie superalgebras st(m,n,R), which are Lie superalgebras graded by root system of type A(m,n). When m + n ≥ 5, the Steinberg Lie superalgebra st(m,n,R) is the universal covering of the Lie superalgebra sl(m,n,R) whose kernel is isomorphic to (HC1(R))0̄ ⊕ (0)1̄, where HC1(R) is the first cyclic homology group of the associative algebra R and the second Lie algebra homology group H2(st(m,n,R)) = 0, The corresponding author. Research of the second author was partially supported by NSERC of Canada and Chinese Academy of Science. 2000 Mathematics Subject Classification: 17B55, 17B60.