Related papers: A Bloch-Wigner complex for SL_2

We use the properties of the refined Bloch group of a field to prove that H_3 of SL_2 of a global field is never finitely generated, and to calculate - up to some 2-torsion - H_3 of SL_2 of local fields with finite residue field of odd…

For a field F with discrete valuation and residue field $k$ we relate the third homology of SL_2(F) with half-integral coefficients to the third homology of SL_2(k) and a certain refined scissors congruence group of k. As an application, we…

In this paper we investigate the third homology of the projective special linear group ${\rm PSL}_2(A)$. As a result of our investigation we prove a projective refined Bloch-Wigner exact sequence over certain class of rings. The projective…

Let $A$ be a local domain of characteristic $2$ such that its residue field has more than $64$ elements. Then we find an exact relation between the third integral homology of the group $\mathrm{SL}_2(A)$ and Hutchinson's refined Bloch group…

We give a definition of (refined) Bloch groups of general commutative rings which agrees with the standard definition in the case of local rings whose residue field has at least $4$ elements. Under appropriate conditions on a ring $A$,…

In this article we prove a generalization of the Bloch-Wigner exact sequence over commutative rings with many units. When the ring is a domain, we get a generalization of Suslin's Bloch-Wigner exact sequence over infinite fields. Our proof…

For a real closed field $\mathbf{R}$, we use the theory of the refined Bloch group to give a new short proof of the isomorphisms $H_{3}(SL_{2}(\mathbf{R}),\mathbb{Z})\cong K_{3}^{\mathrm{ind}}(\mathbf{R})$ and…

We describe the third homology of SL_2 of local rings over Z[1/2] in terms of a refined Bloch group. We use this to derive a localization sequence for the third homology of SL_2 of certain discrete valuation rings, and to make explicit…

The goal of the paper is to achieve - in the special case of the linear group SL_2 - some understanding of the relation between group homology and its A^1-invariant replacement. We discuss some of the general properties of A^1-invariant…

Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincar\'e…

In this article we study the first, the second and the third homology groups of the elementary group $\textrm{E}_2(A)$, where $A$ is a commutative ring. In particular, we prove a refined Bloch-Wigner type exact sequence over a semilocal…

We study K3 surfaces with complex multiplication following the classical work of Shimura on CM abelian varieties. After we translate the problem in terms of the arithmetic of the CM field and its id\`{e}les, we proceed to study some abelian…

Let $G$ be the simple algebraic group $SL_2$ defined over an algebraically closed field $K$ of characteristic $p>0$. In this paper, we find the second cohomology of all irreducible representations of $G$

We study the field of moduli of singular abelian and K3 surfaces. We discuss both the field of moduli over the CM field and over $\Q$. We also discuss non-finiteness with respect to the degree of the field of moduli. Finally, we provide an…

Starting with a finite-dimensional complex Lie algebra, we extend scalars using suitable commutative topological algebras. We study Birkhoff decompositions for the corresponding loop groups. Some results remain valid for loop groups with…

Let F be a finite field of odd cardinality, and let G= GL2(F). The group G \times G \times G acts on F^2 \otimes F^2 \otimes F^2 via symplectic similitudes, and has a natural Weil representation. Answering a question rasised by V. Drinfeld,…

We prove analogues of the fundamental theorem of algebraic K-theory for the second and third homology of SL_2 over an infinite field k. The statements involve Milnor-Witt K-theory and scissors congruence groups. We use these results to…

To a pair (X,f), X compact ANR and f a continuous angle valued map defined on X, a fixed field and a nonnegative integer one assigns a finite configuration of complex numbers with multiplicities located in the punctured complex plane and a…

Electromagnetic field is considered in the framework of Clifford algebra $\C_2$ over a field of complex numbers. It is shown here that a modulo 2 periodicity of complex Clifford algebras may be connected with electromagnetic field.

All finite-dimensional Leibniz algebra bimodules of a Lie algebra $\mathfrak{sl}_2$ over a field of characteristic zero are described.