Polynomial maps on groups

Passi, Inder Bir S.

doi:10.1016/0021-8693(68)90017-3

Cited by 61 publications

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“…Polynomial groups P n,R (G) = I R (G)/I n+1 R (G) were introduced by Passi in [20] (see also [21]), along with a notion of polynomial maps from groups to R-modules such that the map p n,R :…”

Section: Fox Polynomial Groupsmentioning

confidence: 99%

On Fox Quotients of Arbitrary Group Algebras

Hartl

2010

Int. J. Algebra Comput.

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Certain subquotients of group algebras are determined as a basis for subsequent computations of relative Fox and dimension subgroups. More precisely, for a group G and N-series G of G let I n R,G (G), n ≥ 0, denote the filtration of the group algebra R(G) induced by G , and I R (G) its augmentation ideal. For subgroups H of G, left ideals J of R(H) and right H -submodules M of I Z Z (G) the quotients I R (G)J/M J are studied by homological methods, notably for M = I R (G

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Section: Fox Polynomial Groupsmentioning

confidence: 99%

On Fox Quotients of Arbitrary Group Algebras

Hartl

2010

Int. J. Algebra Comput.

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“…When we restrict ourselves to finite Abelian ^-groups, then the simplest case, of course, is that of elementary Abelian p-groups. Both P n (G) and Q,,(G) have been computed in [11] in this case. The reader is referred to this paper for details.…”

Section: (Z) = Zngx Mmentioning

confidence: 99%

“…If M is a cyclic group, then 9 is an isomorphism [11]. It follows from Theorem 7.3. that if M is a finite Abelian group and 9 :…”

Section: Then 6 N Is An Isomorphism If and Only Ifn^ P + R(p-l)mentioning

confidence: 99%

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The associated graded ring of a group ring

Passi¹

1979

Group Rings and Their Augmentation Ideals

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“…So this definition of polynomial maps is very satisfactory when R is a field; for general rings R, however, it is too restrictive: for R = M = N = Z, the map assigning n 2 to n should certainly be considered as being quadratic, but does not split as a sum of a linear and a homogenous quadratic map. This example actually comes from group theory where a notion of polynomial maps from groups to abelian groups was introduced by Passi [15] in the context of dimension subgroups, but later on turned out to admit many other applications in nilpotent group theory, too [8,10,11]. A more general notion of quadratic maps between arbitrary groups [12] arises in the new field of "quadratic algebra" which furnishes an appropriate algebraic framework for dealing with various quadratic phenomena arising in homotopy theory, such as metastable homotopy, secondary homotopy groups and operations, 3-types, quadratic homology etc.…”

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confidence: 99%

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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Polynomial maps on groups

Cited by 61 publications

References 1 publication

On Fox Quotients of Arbitrary Group Algebras

On Fox Quotients of Arbitrary Group Algebras

The associated graded ring of a group ring

Quadratic maps between modules

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