Central extensions of Steinberg Lie superalgebras of small rank

It was shown by A.V.Mikhalev and I.A.Pinchuk in [MP] that the second homology group H2(st(m,n,R)) of the Steinberg Lie superalgebra st(m,n,R) is trivial for m+n ≥ 5. In this paper, we will work out H2(st(m,n,R)) explicitly for m + n = 3, 4. Introduction Steinberg Lie algebras stn(R) play an important role in (additive) algebraic K-theory. They have been studied by many people (see [L] and [GS], and the references therein). The point is that for any unital associative algebra R over a field the Steinberg Lie algebra stn(R) is the universal central extension of sln(R) with the kernel isomorphic to the first cyclic homology group HC1(R) except when both n and the characteristic of the field are small. As seen in [GS], if n = 3, 4, H2(stn(R)) is not necessarily equal to 0. Recently, A.V.Mikhalev and I.A.Pinchuk [MP] studied the Steinberg Lie superalgebras st(m,n,R) which are central extensions of Lie superalgebras sl(m,n,R). They further showed that when m + n ≥ 5, st(m,n,R) is the universal central extension of sl(m,n,R) whose kernel is isomorphic to (HC1(R))0̄ ⊕ (0)1̄, here we would like to emphasize the Z2gradation of the kernel. In this paper, we shall work out H2(st(m,n,R)) explicitly for m+ n = 3, 4 without any assumption on charK by adopting the definition for Lie superalgebras (including char K = 2 case) introduced by Neher [N]. It is equivalent to work on the Steinberg Lie superalgebras st(m,n,R) for small m + n. This completes the determination of the universal central extensions of the Lie superalgebras st(m,n,R) and sl(m,n,R) as well. 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.