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The tensor algebra

The tensor algebra - definition

Suppose that R is a commutative ring with unity. Let M be an R module.

If Tj(M)=MRRMj

then we can define a (non-commutative) product map

Tj(M)×Tk(M)Tj+k(M)

by defining

(a1aj)(b1bk)=(a1ajb1bk)

This allows us to view the infinite direct sum

T(M)=RMT2(M)T3(M)

as a (non-commutative) R-algebra. This ring is called the tensor algebra of M.

Universal property

If A is any R-algebra and M is any R module, then given a map f:MA there is a unique map T(M)A which, when restricted to M gives f.

Graded rings

The tensor algebra T(M) is an example of a graded ring. A general graded ring is any ring S that is a direct sum of additive subgroups Si where SiSjSi+j. Each Sj is called the subset of homogeneous elements of degree j.

A graded ideal I is an ideal of S that is the direct sum of its homogeneous components ISj.

A map f:ST between graded rings is a homomorphism of graded rings if f(Sj)Tj for every j.

Example: The polynomial ring F[x1,,xn] is graded where the homogeneous components are spanned by the monomials of a given degree. The ideal generated by x1,,xn is graded because any element of this ideal – that is, any polynomiial with zero constant term – can be written as a sum of monomials of a fixed degree in a unique way.

The ring R=Z[x] is graded by degree, and the ideal generated by J=(1+x) is not graded because 1 and x are not in J.

An ideal generated by homogeneous elements in a graded ring is graded.

Quotients of graded rings

If S is a graded ring and I is a graded ideal, then S/I is graded with homogeneous components Sj/Ij.

The symmetric algebra

The symmetric algebra - definition

A bilinear map f:M×ML is symmetric if f(m1,m2)=f(m2,m1) for all pairs (m1,m2). The symmetric tensor product S2(M) has the universal property that any symmetric bilinar map f:M×ML “factors through” S2(M) in a unique way. S2(M) is constructed from MM by imposing the relation mnnm=0 for all pairs (m,n).

The symmetric algebra S(M) is obtained from the tensor algebra by modding out by the ideal C(M) generated by all mnnm in T(M). The map

T(M)S(M)

is a map of graded rings since C(M) is a graded ideal, so S(M) is a graded ring. S0(M)=R and S1(M)=M.

More on the symmetric algebra

In S(M), the degree k terms are spanned by tensors m1mk modulo the relation which allows you to freely permute the terms.

Any k-symmetric multilinear map M××ML factors through Sk(M) in a unique way.

If A is any commutative R algebra, and f:MA is a module map, then there is a unique R-algebra map S(M)A which restricts to f on M.

If R is a field and V is a vector space of dimension k then S(V) is the (graded) polynomial ring in k variables over R. If R is a commutative ring and V is a free module of rank k then S(V) is the (graded) polynomial ring over R in k variables.

The exterior algebra

Alternating maps and the wedge product

A bilinear map f:M×ML is alternating if f(m,m)=0 for all mM. The exterior product (or wedge product) MM has the property that there is a map M×MMM such that if f:M×ML is an alternating map, then there is a unique map MML making the usual triangle commute.

The map MMMM is written (m1,m2)m1m2 and MM is spanned by all m1m2. We have m1m2=m2m1 for elementary tensors where m1,m2M.

MM is the quotient of MM by the elementary tensors mm.

The exterior algebra

The quotient of the tensor algebra by the ideal generated by all tensors of the form mm is called the exterior algebra, written M. It is graded so that that the degree k terms consist of linear combinations of the “wedge” of k elements of M.

If A is an R-algebra such that a2=0 for all aA, and f:MA is a module homomorphism, then there is a unique map MA which restricts to f.

If V is a finite dimensional vector space over a field F, then V is finite dimensional. This holds for any free, finite rank module over a ring R.

If R=Z[x,y], if M=R then M is spanned by 11=0 so M=0. If I=(x,y), then there is an alternating map from I to Z defined by

f(ax+by,cx+dy)=adbc(mod(x,y))

so xy is not zero.

The cross product in R3.

The coefficients of

(ae0+be1+ce2)(xe0+ye1+ze2)

in terms of the two-forms eiej give the cross product in R3.

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