Crossed Product Algebra

In mathematics, a factor system (sometimes called factor set) is a fundamental tool of Otto Schreier’s classical theory for group extension problem. It consists of a set of automorphisms and a binary function on a group satisfying certain condition (so-called ''cocycle condition''). In fact, a factor system constitutes a realisation of the cocycles in the second cohomology group in group cohomology.

Introduction

Suppose is a group and is an abelian group. For a group extension
: $1 \to A \to X \to G \to 1,$
there exists a factor system which consists of a function and homomorphism such that it makes the cartesian product a group as
: $(g,a)*(h,b) := (gh, f(g,h)a^b).$
So must be a "group 2-cocycle" (symbolically, ). In fact, does not have to be abelian, but the situation is more complicated for non-abelian groups

If is trivial and gives inner automorphisms, then that group extension is split, so become semidirect product of with .

If a group algebra is given, then a factor system ''f'' modifies that algebra to a skew-group algebra by modifying the group operation to .

Application: for Abelian field extensions

Let ''G'' be a group and ''L'' a field on which ''G'' acts as automorphisms. A ''cocycle'' or ''(Noether) factor system'' is a map ''c'': ''G'' × ''G'' → ''L''^* satisfying

:$c(h,k)^g c(hk,g) = c(h,kg) c(k,g) .$

Cocycles are ''equivalent'' if there exists some system of elements ''a'' : ''G'' → ''L''^* with

:$c'(g,h) = c(g,h) (a_g^h a_h a_^) .$

Cocycles of the form

:$c(g,h) = a_g^h a_h a_^$

are called ''split''. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group H²(''G'',''L''^*).

Crossed product algebras

Let us take the case that ''G'' is the Galois group of a field extension ''L''/''K''. A factor system ''c'' in H²(''G'',''L''^*) gives rise to a ''crossed product algebra'' ''A'', which is a ''K''- algebra containing ''L'' as a subfield, generated by the elements λ in ''L'' and ''u''_''g'' with multiplication

:$\lambda u_g = u_g \lambda^g ,$
:$u_g u_h = u_ c(g,h) .$

Equivalent factor systems correspond to a change of basis in ''A'' over ''K''. We may write

:$A = (L,G,c) .$

The crossed product algebra ''A'' is a central simple algebra (CSA) of degree equal to 'L'' : ''K''Jacobson (1996) p.57 The converse holds: every central simple algebra over ''K'' that splits over ''L'' and such that deg ''A'' = 'L'' : ''K'' arises in this way.Jacobson (1996) p.57 The tensor product of algebras corresponds to multiplication of the corresponding elements in H². We thus obtain an identification of the Brauer group, where the elements are classes of CSAs over ''K'', with H².Saltman (1999) p.44Jacobson (1996) p.59

Cyclic algebra

Let us further restrict to the case that ''L''/''K'' is cyclic with Galois group ''G'' of order ''n'' generated by ''t''. Let ''A'' be a crossed product (''L'',''G'',''c'') with factor set ''c''. Let ''u'' = ''u''_''t'' be the generator in ''A'' corresponding to ''t''. We can define the other generators

:$u_ = u^i \,$

and then we have ''u''^''n'' = ''a'' in ''K''. This element ''a'' specifies a cocycle ''c'' by

:$c(t^i,t^j) = \begin 1 & \text i+j < n, \\ a & \text i+j \ge n. \end$

It thus makes sense to denote ''A'' simply by (''L'',''t'',''a''). However ''a'' is not uniquely specified by ''A'' since we can multiply ''u'' by any element λ of ''L''^* and then ''a'' is multiplied by the product of the conjugates of λ. Hence ''A'' corresponds to an element of the norm residue group ''K''^*/N_''L''/''K''''L''^*. We obtain the isomorphisms

:$\operatorname(L/K) \equiv K^*/N_ L^* \equiv H^2(G,L^*) .$

References

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* {{cite book , last=Saltman , first=David J. , title=Lectures on division algebras , series=Regional Conference Series in Mathematics , volume=94 , location=Providence, RI , publisher= American Mathematical Society , year=1999 , isbn=0-8218-0979-2 , zbl=0934.16013

Cohomology theories
Group theory