Steinberg Lie Algebras And Their Central Extensions

In this thesis, we will mainly study the universal central extensions of the Steinberg Lie algebras st(n, R) and the Steinberg Lie superalgebras st(m, n, R). It is equivalent to work out the second homology group H₂(st(n, R)) and H₂(st(m, n, R)).Steinberg Lie algebras si(n, R) and/or their universal coverings have been studied by Bloch [B1], Kassel-Loday [KL], Kassel [Ka], Faulkner [F], Allison-Faulkner [AF], Berman-Moody [BM], Gao [G1, 2] and Allison-Gao [AG], and among others. They play an important role in the study of root graded Lie algebras and the additive algebraic K-theory. In most situations, the Steinberg Lie algebra si(n, R) is the universal covering of the Lie algebra sl_n(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(st(n,R)) = 0. It was shown in [Bl] and [KL] that H₂(st(n,R)) = 0 for n≥5.In the Chapter 3 of this thesis, we shall work out H₂(st(n, R)) explicitly for n = 3,4, which is not necessarily equal to 0 when the base commutative ring K is of small characteristic. Our first main result of this thesis is the following.Theorem 1: let K be a unital commutative ring and R be a unital associative K-algebra. Assume that R has a K-basis containing the identity element. Thenwhere R_m(m,∈N∩{0}) is defined in the Section 2 of Chapter 3.Similarly with the theory of the Steinberg Lie algebras st(n, R), 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 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))₀ (?) (0)1. Here we would like to emphasize the Z₂-gradation of the kernel. We shall work out H₂(st(m, n, R)) explicitly for m + n = 3,4 in Chapter 4. We will treat for m = 2, n = 1, m = 3, n = 1 and m = 2, n = 2 case by case, and obtain the second main result of this thesis,Theorem 2: let K be a unital commutative ring and R be a unital associative K-algebra. Assume that R has a K-basis containing the identity element. Thenwhich are Z₂-graded spaces.