Universal central extensions of linear groups over rings of non-commutative Laurent polynomials, associated $K_1$-groups and $K_2$-groups

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

We give a Matsumoto-type presentation of $K_2$-groups over rings of non-commutative Laurent polynomials, which is a non-commutative version of M. Tomie's result for loop groups. Our main idea is due to U. Rehmann's approach in case of division rings.

Finiteness Properties of Chevalley Groups over the Laurent Polynomial Ring over a Finite Field

We show that if G is a Chevalley group of rank n and F_q[t,t^{-1}] is the ring of Laurent polynomials over a finite field, then G(F_q[t,t^{-1}]) is of type F_{2n-1}. This bound is optimal because it is known -- and we show again -- that the group is not of type F_{2n}.

二元Laurent多项式环上的一个显式非初等矩阵

An explicit non-elementary matrix over a bivariate Laurent polynomial ring

本文针对离散赋值环的分式域F，构造了二元Laurent多项式环上SL₂群中不属于初等子群E₂的显式矩阵，其非初等性证明依赖于特定预印本的技术。

Commutative Schur Rings Over Symmetric Groups II: The Case n=6

We determine the commutative Schur rings over $S_6$ that contain the sum of all the transpositions in $S_6$. There are eight such types (up to conjugacy), of which four have the set of all the transpositions as a principal set of the Schur ring.

Finiteness Properties of Chevalley Groups over the Ring of (Laurent) Polynomials over a Finite Field

In these notes we determine the finiteness length of the groups G(O_S) where G is an F_q-isotropic, connected, noncommutative, almost simple F_q-group and O_S is one of F_q[t], F_q[t^{-1}], and F_q[t,t^{-1}]. That is, k = F_q(t) and S contains one or both of the places s_0 and s_\infty corresponding to the polynomial p(t) = t respectively to the point at infinity. The statement is that the finiteness length of G(O_S) is n-1 if S contains one of the two places and is 2n-1 if it contains both places, where n is the F_q-rank of G. For example, the group SL_3(F_q[t,t^{-1}]) is of type F_3 but not …

Automorphisms of Chevalley groups of different types over commutative rings

In this paper we prove that every automorphism of (elementary) adjoint Chevalley group with root system of rank $>1$ over a commutative ring (with 1/2 for the systems $A_2$, $F_4$, $B_l$, $C_l$; with 1/2 and 1/3 for the system $G_2$) is standard, i.\,e., it is a composition of ring, inner, central and graph automorphisms.