Quadratic Maps Between Groups

Hartl, Manfred

doi:10.1515/gmj.2009.55

Cited by 4 publications

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“…The operations on the semi-direct product are uniquely determined by the operations on the factors and by the action. All axioms and all other claims follow from direct computation using the notion of quadratic maps from [5] and lemma 1 from [16].…”

Section: Odd Form Algebrasmentioning

confidence: 99%

Centrality of odd unitary $K_2$-functor

Voronetsky

2020

Preprint

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Let (R, ∆) be an odd form algebra. We show that the unitary Steinberg group StU(R, ∆) is a crossed module over the odd unitary group U(R, ∆) in two major cases: if the odd form algebra has a free orthogonal hyperbolic family satisfying local stable rank condition and if the odd form algebra is sufficiently isotropic and quasi-finite. The proof uses only elementary localization techniques and stability results for KU1 and KU2.

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Section: Odd Form Algebrasmentioning

confidence: 99%

Centrality of odd unitary $K_2$-functor

Voronetsky

2020

Preprint

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Quadratic functors on pointed categories

Hartl

Vespa

2011

Advances in Mathematics

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We study polynomial functors of degree 2, called quadratic, with values in the category of abelian groups Ab, and whose source category is an arbitrary category C with null object such that all objects are colimits of copies of a generating object E which is small and regular projective; this includes all pointed algebraic varieties. More specifically, we are interested in such quadratic functors F from C to Ab which preserve filtered colimits and suitable coequalizers; one may take reflexive ones if C is Mal'cev and Barr exact.A functorial equivalence is established between such functors F : C → Ab and certain minimal algebraic data which we call quadratic C-modules: these involve the values on E of the cross-effects of F and certain structure maps generalizing the second Hopf invariant and the Whitehead product.Applying this general result to the case where E is a cogroup these data take a particularly simple form. This application extends results of Baues and Pirashvili obtained for C being the category of groups or of modules over some ring; here quadratic C-modules are equivalent with abelian square groups or quadratic R-

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Quadratic maps between modules

Gaudier

Hartl

2009

Journal of Algebra

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Proposition 1.2. Suppose that R contains an element r such that r and r − 1 are invertible. Then any Rquadratic map f : M → N decomposes uniquely as a sum f = f 1 + f 2 of an R-linear map f 1 and a homoge-This criterion is improved in Example 4.5(2) below. Proof.We can takeby Remark 1.3 below. Uniqueness of f 1 and f 2 follows from the fact that under the hypothesis any map which is R-linear and homogenous R-quadratic is trivial; in fact,Note that the proposition applies whenever R is a field different from F 2 . If R = F 2 any Rquadratic map is homogenous.Finally, we discuss the case R = Z. Passi [15] defines a map f : G → A from a group G to an abelian group A to be (normalized) polynomial of degree. An inductive characterization of this property [9] shows that f is polynomial of degree 2 iff its cross-effect d f is homomorphic in each variable. If G is abelian, this is equivalent towhich is homogenous Z-quadratic; this suffices by Remark 1.3 below.Let us exhibit some elementary properties of R-quadratic maps. First note that if f is R-quadratic,Next for x, y, z in M and r, s in R the first condition in 1.1 can be written as(1.2) Additivity of f [r] then follows from (1.2) with s = r, and its R-linearity can be written as:(1.3) Remark 1.3. Relation (1.3) can be written as f s (rx) = r 2 f s (x), that is f s is a homogeneous R-quadratic map. Thus we see that f is R-quadratic iff its cross-effect is R-bilinear and its cross-actions are homogeneous R-quadratic.Clearly the set R-Quad(M, N) (resp. R-HQuad(M, N)) of the R-quadratic maps (resp. homogeneous R-quadratic maps) from M to N is an R-module, and pre-or postcomposition of an R-quadratic map (resp. homogeneous R-quadratic map) by an R-linear map is an R-quadratic (resp. homogeneous Rquadratic) map.Throughout this paper the tensor product of R-modules M and N is denoted by M ⊗ N instead of M ⊗ R N. Lemma 1.4. For any R-modules M, M , N one has a natural isomorphism R-Quad(M ⊕ M , N) ∼ = R-Quad(M, N) ⊕ R-Quad(M , N) ⊕ R-Hom(M ⊗ M , N).

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Quadratic Maps Between Groups

Cited by 4 publications

References 24 publications

Centrality of odd unitary $K_2$-functor

Centrality of odd unitary $K_2$-functor

Quadratic functors on pointed categories

Quadratic maps between modules

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