Symmetric algebra
In mathematics, the symmetric algebra over a commutative ring.
Construction
From tensor algebra
It is possible to use the tensor algebra to describe the symmetric algebra. In fact, can be defined as the quotient algebra of by the two sided ideal generated by the commutatorsIt is straightforward, but rather boring, to verify that the resulting algebra satisfies the universal property stated in the introduction.
This results also directly from a general result of category theory, which asserts that the composition of two left adjoint functors is also a left adjoint functor. Here, the forgetful functor from commutative algebras to vector spaces or modules is the composition of the forgetful functors from commutative algebras to associative algebras, and from associative algebras to vectors or modules. As the tensor algebra and the quotient by commutators are left adjoint to these forgetful functors, their composition is left adjoint to the forgetful functor from commutative algebra to vectors or modules, and this proves the desired universal property.
From polynomial ring
The symmetric algebra can also be built from polynomial rings.If is a -vector space or a free -module, with a basis, let be the polynomial ring that has the elements of as indeterminates. The homogeneous polynomials of degree one form a vector space or a free module that can be identified with. It is straightforward to verify that this makes a solution to the universal problem stated in the introduction. This implies that and are canonically isomorphic, and can therefore be identified. This results also immediately from general considerations of category theory, since free modules and polynomial rings are free objects of their respetive categories.
If is a module that is not free, it can be written where is a free module, and is a submodule of. In this case, one has
where is the ideal generated by. Again this can be proved by showing that one has a solution of the universal property, and this can be done either by a straightforward but boring computation, or by using category theory, and more specifically, the fact that a quotient is the solution of the universal problem for morphisms that map to zero a given subset.
Grading
The symmetric algebra is a graded algebra. That is, it is a direct sumwhere called the th symmetric power of, is the vector subspace or submodule generated by the products of elements of..
This can be proved by various means. One follows from the tensor-algebra construction: since the tensor algebra is graded, and the symmetric algebra is its quotient by a homogeneous ideal, the ideal generated by all where and are in, that is, homogeneous of degree one.
In the case of a vector space or a free module, the gradation is the gradation of the polynomials by the total degree. A non-free module can be written as, where is a free module of base ; its symmetric algebra is the quotient of the symmetric algebra of by the homogeneous ideal generated by the elemens of, which are homogeneous of degree one.
One can also define as the solution of the universal problem for -linear symmetric functions from into a vector space or a module, and then verify that the direct sum of all satisfies the universal problem for the symmetric algebra.
Relationship with symmetric tensors
As the symmetric algebra of a vector space is a quotient of the tensor algebra, an element of the symmetric algebra is not a tensor, and, in particular, is not a symmetric tensor. However, symmetric tensors are strongly related to the symmetric algebra.A symmetric tensor of degree is an element of that is invariant under the action of the symmetric group More precisely, given the transformation defines a linear endomorphism of. A symmetric tensor is a tensor that is invariant under all these endomorphisms. The symmetric tensors of degree form a vector subspace . The symmetric tensors are the elements of the direct sum which is a graded vector space. It is not an algebra, as the tensor product of two symmetric tensors is not symmetric in general.
Let be the restriction to of the canonical surjection If is invertible in the ground field, then is an isomorphism. This is always the case with a ground field of characteristic zero. The inverse isomorphism is the linear map defined by the symmetrization
The map is not injective if divides the characteristic; for example is zero in characteristic two. Over a ring of characteristic zero, can be non surjective; for example, over the integers, if and are two linearly independent elements of that are not in, then since
In summary, over a field of characteristic zero, the symmetric tensors and the symmetric algebra form two isomorphic graded vector spaces. They can thus be identified as far as only the vector space structure is concerned, but they cannot be identified as soon as products are involved. Moreover, this isomorphism does not extend to the cases of fields of positive characteristic and rings that do not contain the rational numbers.
Categorical properties
Given a module over a commutative ring, the symmetric algebra can be defined by the following universal property:For every linear map from to a commutative algebra, there is a unique algebra homomorphism such that where is the inclusion of in.
As for every universal property, as soon as a solution exists, this defines uniquely the symmetric algebra, up to a canonical isomorphism. It follows that all properties of the symmetric algebra can be deduced from the universal property. This section is devoted to the main properties that belong to category theory.
The symmetric algebra is a functor from the category of -modules to the category of -commutative algebra, since the universal property implies that every module homomorphism can be uniquely extended to an algebra homomorphism
The universal property can be reformulated by saying that the symmetric algebra is a left adjoint to the forgetful functor that sends a commutative algebra to its underlying module.
Symmetric algebra of an affine space
One can analogously construct the symmetric algebra on an affine space. The key difference is that the symmetric algebra of an affine space is not a graded algebra, but a filtered 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.
Analogy with exterior algebra
The Sk are functors comparable to the exterior powers; here, though, the dimension grows with k; it is given bywhere n is the dimension of V. This binomial coefficient is the number of n-variable monomials of degree k.
In fact, the symmetric algebra and the exterior algebra appear as the isotypical components of the trivial and sign representation of the action of acting on the tensor product