A Matsumoto type theorem for linear groups over rings of non-commutative Laurent polynomials

概要

Many researchers have studied the structure of the general linear group and its elementary subgroup over a field F or a commutative ring R. They also have analyzed associated lower K-groups, for example [8] and [16]. Needless to say, the general linear groups are important objects and have many applica- tions in various areas of mathematics, but they particularly have much to do with Lie theory; Lie groups, Lie algebras and their representations. Based on Chevalley’s study on linear groups over any field F , Iwahori–Matsumoto [4] gen- eralized a Bruhat decomposition: for a p-adic Chevalley group G, they showed that there exists a decomposition G =∪w∈Wa BwB with Wa the corresponding affine Weyl group, where B is a generalized Borel subgroup called the “Iwahori subgroup”. Using this notion, Moody–Teo [19], Marcuson [6] and Peterson– Kac [12] discussed a Tits system for Kac–Moody groups (generalized Chevalley groups). As a special case of affine Kac–Moody groups, Morita [9] proved that loop groups have Tits systems with the affine Weyl groups. Linear groups cor- responding to extended affine Lie algebras, which are an affinization of [9], were studied in [11]. There is a direct relationship between the nullity of extended affine Lie algebras and the number of variables of the Laurent polynomials. Morita–Sakaguchi [11] researched groups over a completed quantum torus with two variables. In order to generalize [11] to the case of nullity n ≥ 3, we need a ring Dτ = D[t, t−1] of non-commutative Laurent polynomials over a division ring D (see Section 1).

　Our main object in this paper is the following exact sequence [17]:

　1 → K2(n, Dτ ) → St(n, Dτ ) →ϕ GL(n, Dτ ) → K1(n, Dτ ) → 1.

In Section 1, we reveal the structure of the groups in the above sequence, ex- cept for the presentation of K2(n, Dτ ). We first describe an existence of the Tits system in the elementary subgroup E(n, Dτ ) of the general linear group GL(n, Dτ ) and the associated Steinberg group St(n, Dτ ) in Subsections 1.2 and 1.3, respectively. Using these facts, we show that the above homomorphism ϕ is a central extension of E(n, Dτ ), that is, we confirm that Ker ϕ = K2(n, Dτ ) is a central subgroup of the Steinberg group in Subsection 1.4. It is proved in Subsection 1.5 that ϕ is universal when the center Z(D) of D has at least five elements. Meanwhile, we discuss the structure of the associated K1-group and K2-group in Subsections 1.4 and 1.6. In particular, we check that the K2-group is generated by the Steinberg symbols.

　In Section 2, we give a presentation of K2(n, Dτ ). The structure of linear groups is important object, but the associated lower K-groups are also remark- able ones. Indeed, it is a well-known fact that the K2-group is an invariant which measures the size of central extensions of the group. For the Chevalley group over any field, the presentation of the K2-group has already been given by Matsumoto [7]. The K2-group derived from the loop group was studied in Tomie [19], and Sakagich [15] gave the Matsumoto type presentation for linear groups over the quantum torus in two variables. Moreover, Rehmann [13] [14] has already determined the presentation of the K2-group over a division ring D.

　As mentioned in Matsumoto [7], there exist two types of the K2-group, which are called “symplectic type” when n = 2, and “non-symplectic type” when n ≥ 3. In fact, the K2-group changes its group structure depending on the size of the elementary subgroup: the symplectic type K2-group K2(2, F ) is presented by the symbols c(u, v), u, v ∈ F×, and the following defining relations:

　c(u, v)c(uv, w) = c(u, vw)c(v, w), c(1, 1) = 1,
　c(u, v) = c(u−1, v−1),
　c(u, v) = c(u, (1 − u)v) (1 − u ∈ F×).

On the other hand, the non-symplectic type K2-groups K2(n, F ) is presented by the symbols c(u, v), u, v ∈ F×, and the following defining relations:

　c(uv, w) = c(u, w)c(v, w),
　c(u, vw) = c(u, v)c(u, w),
　c(u, 1 − u) = 1 (1 − u ∈ F×).

In this paper, we determine the presentation of the K2-groups of symplectic type and non-symplectic type, respectiverly (see Section 2), which is a non- commutative version of Tomie’s result for loop groups [19]. Our main idea is due to Rehmann’s approach in the case of division rings [13], [14]. In Subsection 2.1, we give the group presentation of the symplectic K2-group K2(2, Dτ ). Let P be the group presented by generators c(u, v), u, v ∈ D×, with the following defining
relations:

　(P1) c(u, v)c(vu, w) = c(u, vw)c(v, w),
　(P2) c(u, v) = c(uvu, u−1),
　(P3) c(x, y)c(u, v)c(x, y)−1 = c([x, y]u, v)c(v, [x, y]),
　(P4) c(u, v) = c(u, v(1 − u)) (1 − u ∈ D×τ),
　(P5) c(u, v) = c(u, −vu).

Then, there exists a natural homomorphism φ of P onto [D×, D×], whose kernel is isomorphic to K2(2, Dτ ). The presentation of non-symplectic K2(n, Dτ ) is given in Subsection 2.2. Let Q be the group presented by generators c(u, v), u, v ∈ D× with the following defining relations:

　(Q1) c(uv, w) = c(uv,u w)c(u, w),
　(Q2) c(u, vw) = c(u, v)c(vu,v w),
　(Q3) c(u, 1 − u) = 1 (1 − u ∈ D×τ),

where uv = uvu−1. Then, there exists a natural homomorphism φ0 of Q onto [D×τ, D×τ], whose kernel is isomorphic to K2(n, Dτ ).