Symmetric algebra
In
of V in S(V).If B is a basis of V, the symmetric algebra S(V) can be identified, through a
The symmetric algebra S(V) can be built as the
All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring.
Construction
From tensor algebra
It is possible to use the
It is straightforward to verify that the resulting algebra satisfies the universal property stated in the introduction. Because of the universal property of the tensor algebra, a linear map f from V to a commutative algebra A extends to an algebra homomorphism , which factors through S(V) because A is commutative. The extension of f to an algebra homomorphism is unique because V generates S(V) as a K-algebra.
This results also directly from a general result of
From polynomial ring
The symmetric algebra S(V) can also be built from polynomial rings.
If V is a K-vector space or a free K-module, with a basis B, let K[B] be the polynomial ring that has the elements of B as indeterminates. The homogeneous polynomials of degree one form a vector space or a free module that can be identified with V. It is straightforward to verify that this makes K[B] a solution to the universal problem stated in the introduction. This implies that K[B] and S(V) 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 respective categories.
If V is a module that is not free, it can be written where L is a free module, and M is a
where is the ideal generated by M. (Here, equals signs mean equality up to a canonical isomorphism.) 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. (Depending on the case, the kernel is a normal subgroup, a submodule or an ideal, and the usual definition of quotients can be viewed as a proof of the existence of a solution of the universal problem.)
Grading
The symmetric algebra is a
where called the nth symmetric power of V, is the vector subspace or submodule generated by the products of n elements of V. (The second symmetric power is sometimes called the symmetric square of V).
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
In the case of a vector space or a free module, the gradation is the gradation of the polynomials by the
One can also define as the solution of the universal problem for
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 n is an element of Tn(V) that is invariant under the
Let be the restriction to Symn(V) of the canonical surjection If n! is invertible in the ground field (or ring), then is an isomorphism. This is always the case with a ground field of characteristic zero. The inverse isomorphism is the linear map defined (on products of n vectors) by the symmetrization
The map is not injective if the characteristic is less than n+1; for example is zero in characteristic two. Over a ring of characteristic zero, can be non surjective; for example, over the integers, if x and y are two linearly independent elements of V = S1(V) that are not in 2V, 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 V over a commutative ring K, the symmetric algebra S(V) can be defined by the following universal property:
- For every K-algebra homomorphismsuch that where i is the inclusion of V in S(V).
- For every K-
As for every universal property, as soon as a solution exists, this defines uniquely the symmetric algebra,
The symmetric algebra is a functor from the category of K-modules to the category of K-commutative algebra, since the universal property implies that every module homomorphism can be uniquely extended to an
The universal property can be reformulated by saying that the symmetric algebra is a
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 (choosing a point turns an affine space into a vector space).
Analogy with exterior algebra
The Sk are
where 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 (for example over the complex field) [citation needed]
As a Hopf algebra
The symmetric algebra can be given the structure of a Hopf algebra. See Tensor algebra for details.
As a universal enveloping algebra
The symmetric algebra S(V) is the
See also
- exterior algebra, the alternating algebra analog
- graded-symmetric algebra, a common generalization of a symmetric algebra and an exterior algebra
- symplectic form
- Clifford algebra, a quantum deformation of the exterior algebra by a quadratic form
References
- ISBN 3-540-64243-9