- Symmetric algebra
In
mathematics , the symmetric algebra "S"("V") (also denoted "Sym"("V")) on avector space "V" over a field "K" is the freecommutative unital associative "K"-algebra containing "V".It corresponds to polynomials with indeterminates in "V", without choosing coordinates.The dual, S(V^*) corresponds to polynomials "on" "V".
It should not be confused with
symmetric tensor s in "V". AFrobenius algebra whosebilinear form is symmetric is also called a symmetric algebra, but is not discussed here.Construction
It turns out that "S"("V") is in effect the same as the
polynomial ring , over "K", in indeterminates that arebasis vector s for "V". Therefore this construction only brings something extra when the "naturality" of looking at polynomials this way has some advantage.It is possible to use the
tensor algebra "T"("V") to describe the symmetric algebra "S"("V"). In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative; if elements of "V" commute, then tensors in them must, so that we construct the symmetric algebra from the tensor algebra by taking thequotient algebra of "T"("V") by the ideal generated by all differences of products:votimes w - wotimes v.
for "v" and "w" in "V".
Grading
Just as with a polynomial ring, there is a
direct sum decomposition of "S"("V") as agraded algebra , into summands:"Sk"("V")
which consist of the linear span of the
monomial s in vectors of "V" of degree "k", for "k" = 0, 1, 2, ... (with "S"0("V") = "K" and "S"1("V")="V"). The "K"-vector space "Sk"("V") is the "k"-th symmetric power of "V". The case "k" = 2, for example, is the symmetric square. It has a universal property with respect to symmetricmultilinear operators defined on "V""k".Distinction with symmetric tensors
The symmetric algebra and symmetric tensors are easily confused: the symmetric algebra is a "quotient" of the tensor algebra, while the symmetric tensors are a "subspace" of the tensor algebra.
The symmetric algebra must be a quotient to satisfy its
universal property (since every symmetric algebra is an algebra, the tensor algebra maps to the symmetric algebra).Conversely, symmetric tensors are defined as invariants: given the natural action of the
symmetric group on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially. Note that under the tensor product, symmetric tensors are not a subalgebra: given vectors "v" and "w", they are trivially symmetric 1-tensors, but v otimes w is not a symmetric 2-tensor.The grade 2 part of this distinction is the difference between
symmetric bilinear form s (symmetric 2-tensors) andquadratic form s (elements of the symmetric square), as described inε-quadratic form s.In characteristic 0 symmetric tensors and the symmetric algebra can be identified. In any characteristic, there is a
symmetrization map from the symmetric algebra to the symmetric tensors, given by::v_1cdots v_k mapstosum_{sigma in S_n} v_{sigma(1)}otimes cdots otimes v_{sigma(k)}.The composition with the inclusion of the symmetric tensors in the tensor algebra and the quotient to the symmetric algebra is multiplication by k! on the "k"th graded component.Thus in characteristic 0, the symmetrization map is an isomorphism of graded vector spaces, and one can identify symmetric tensors with elements of the symmetric algebra. One divides by frac{1}{k!} to make this a section of the quotient
v_1cdots v_k mapsto frac{1}{k!} sum_{sigma in S_n} v_{sigma(1)}otimes cdots otimes v_{sigma(k)}.For instance, vw mapsto frac{1}{2}(votimes w + w otimes v).This is related to the
representation theory of the symmetric group:in characteristic 0, over an algebraically closed field, thegroup algebra issemisimple , so every representation splits into a direct sum of irreducible representations, and if T=Soplus V, one can identify "S" as either a subspace of "T" or as the quotient "T/V".Interpretation as polynomials
Given a vector space "V", the polynomials on this space are S(V^*), the symmetric algebra of the "dual" space: a polynomial on a space "evaluates" vectors on the space, via the pairing S(V^*) imes V o K.
For instance, given the plane with a basis K^2, the (homogeneous) linear polynomials on K^2 are generated by the coordinate
functional s "x" and "y". These coordinates arecovector s: given a vector, they evaluate to their coordinate, for instance::x(2,3) = 2, ext{ and } y(2,3)=3.Given monomials of higher degree, these are elements of various symmetric powers, and a general polynomial is an element of the symmetric algebra. Without a choice of basis for the vector space, the same holds, but one has a polynomial algebra without choice of basis.Conversely, the symmetric algebra of the vector space itself can be interpreted, not as polynomials "on" the vector space (since one cannot evaluate an element of the symmetric algebra of a vector space against a vector in that space: there is no pairing between S(V) and V), but polynomials "in" the vectors, such as v^2-vw+uv.
ymmetric algebra of an affine space
One can analogously construct the symmetric algebra on an
affine space (or its dual, which corresponds to polynomials on that affine space).The key difference is that the symmetric algebra of an affine space is not a graded algebra, but afiltered algebra : one can determine the degree of a polynomial on an affine space, but not its homogeneous parts.For instance, given a linear polynomial on a vector space, one can determine its constant part by evaluating at 0. On an affine space, there is no distinguished point, so one cannot do this (choosing a point turns an affine space into a vector space).
Categorical properties
The symmetric algebra on a vector space is a
free object in the category of commutative unital associative algebras (in the sequel, "commutative algebras").Formally, the map that sends a vector space to its symmetric algebra is a
functor from vector spaces over "K" to commutative algebras over "K",and is a "free object", meaning that it is left adjoint to theforgetful functor that sends a commutative algebra to its underlying vector space.The unit of the adjunction is the map V o S(V) that embeds a vector space in its symmetric algebra.
Commutative algebras are a
reflective subcategory of algebras;given an algebra "A", one can quotient out by its commutator ideal generated by ab-ba, obtaining a commutative algebra, analogously toabelianization of a group. The construction of the symmetric algebra as a quotient of the tensor algebra is, as functors, a composition of the free algebra functor with this reflection.Analogy with exterior algebra
The "S""k" are
functor s comparable to theexterior power s; here, though, the dimension grows with "k"; it is given by:operatorname{dim}(S^k(V)) = inom{n+k-1}{k}where "n" is the dimension of "V".Module analog
The construction of the symmetric algebra generalizes to the symmetric algebra "S"("M") of a
module "M" over acommutative ring . If "M" is afree module over the ring "R", then its symmetric algebra is isomorphic to the polynomial algebra over "R" whose indeterminates are a basis of "M", just like the symmetric algebra of a vector space. However, that is not true if "M" is not free; then "S"("M") is more complicated.As a universal enveloping algebra
The symmetric algebra "S"("V") is the
universal enveloping algebra of anabelian Lie algebra in which the Lie bracket is identically 0.ee also
*
exterior algebra , the anti-symmetric analog
*Weyl algebra , a of the symmetric algebra by asymplectic form
Wikimedia Foundation. 2010.