Positivity for \(K_1\) and \(K_2\)
Let \(R\) be a commutative ring. The authors introduce the notion of a ring \(R\) with positive cone \(R_{+}\), this is a subset that we may regard as the subset of positive elements of \(R\). Using this, they define the ``totally positive matrices'' as the matrices where all of their minors are elements of \(R_{+}\), and of ``strictly totally positive'' matrices where all of their minors are elements of \(R_{+}\{ 0 \}\). Using these concepts they develop a positive version \(K_1(R,R_{+})\) of \(K_1(R)\), a corresponding \(K_{2}\), and a corresponding positive Steinberg group. They provide specific calculations in various interesting cases. These inlude specific generators for both the positive Steinberg group \(St(F,F_{+})\) and for \(K_{2}(F,F_{+})\) when \(F\) is a field with a particular positive cone. They also prove, among other facts, the (group) isomorphism \(K_{2}(F)\sim \mathbb Z/2\oplus \mathbb Z/2\oplus K_{2}(F,F_{+})\), when \(F=\mathbb Q(\sqrt{d})\), \(d\) a square free positive rational integer, providing a specific Matsumoto type presentation in this case. Another important example is when \(SL(n,R)=E(n,R)\) (for example if \(R\) is Euclidean), in this case they prove the (semigroup) isomorphism \(K_{1}(R,R_+)\sim R^\times_+\) (the positive invertible elements of \(R\)). From this they obtain in the case of polynomial rings, the (group) isomorphisms \(K_{1}(\mathbb R[t,t^{-1}], \mathbb R[t,t^{-1}]_+)\sim\mathbb Z\oplus K_{1}(\mathbb R,\mathbb R_{+})\), \(K_{1}(\mathbb R[t],\mathbb R[t]_+)\sim K_{1}(\mathbb R,\mathbb R_{+})\), and a similar statement for \(K_{2}\).
scientific article; zbMATH DE number 1496001
scientific article