Symmetric algebra - Citizendia

In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is the free commutative unital associative K-algebra containing V. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the idea of a free object is one of the basic concepts of Abstract algebra. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive

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 tensors in V. In Mathematics, a symmetric tensor is a Tensor that is invariant under a Permutation of its vector arguments A Frobenius algebra whose bilinear form is symmetric is also called a symmetric algebra, but is not discussed here. In Mathematics, especially in the fields of Representation theory and Module theory, a Frobenius algebra is a finite dimensional unital In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric

1 Construction
- 1.1 Grading
- 1.2 Distinction with symmetric tensors
2 Interpretation as polynomials
- 2.1 Symmetric algebra of an affine space
3 Categorical properties
4 Analogy with exterior algebra
5 Module analog
6 As a universal enveloping algebra
7 See also

Construction

It turns out that S(V) is in effect the same as the polynomial ring, over K, in indeterminates that are basis vectors for V. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables Basis vector redirects here For basis vector in the context of crystals see Crystal structure. 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 Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra 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 the quotient algebra of T(V) by the ideal generated by all differences of products

$v\otimes w - w\otimes v.$

for v and w in V. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

Grading

Just as with a polynomial ring, there is a direct sum decomposition of S(V) as a graded algebra, into summands

S^k(V)

which consist of the linear span of the monomials in vectors of V of degree k, for k = 0, 1, 2, . The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure In Mathematics, the word monomial means two different things in the context of Polynomials The first meaning is a product of powers of Variables . . (with S⁰(V) = K and S¹(V)=V). The K-vector space S^k(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 symmetric multilinear operators defined on V^k. In Linear algebra, a multilinear map is a Mathematical function of several vector variables that is linear in each variable

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). In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism

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. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying 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 forms (symmetric 2-tensors) and quadratic forms (elements of the symmetric square), as described in ε-quadratic forms. A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables

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:

$v_1\cdots v_k \mapsto \sum_{\sigma \in S_n} v_{\sigma(1)}\otimes \cdots \otimes v_{\sigma(k)}.$

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 kth 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 map:

$v_1\cdots v_k \mapsto \frac{1}{k!} \sum_{\sigma \in S_n} v_{\sigma(1)}\otimes \cdots \otimes v_{\sigma(k)}.$

For instance, $vw \mapsto \frac{1}{2}(v\otimes w + w \otimes v)$ . In Category theory, a branch of Mathematics, a section is a right inverse of a morphism

This is related to the representation theory of the symmetric group: in characteristic 0, over an algebraically closed field, the group algebra is semisimple, so every representation splits into a direct sum of irreducible representations, and if $T=S\oplus V$ , one can identify S as either a subspace of T or as the quotient T/V. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach In mathematics the term semisimple is used in a number of related ways within different subjects

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 $S(V^*) \times V \to K$ .

For instance, given the plane with a basis $K 2$ , the (homogeneous) linear polynomials on $K 2$ are generated by the coordinate functionals x and y. These coordinates are covectors: given a vector, they evaluate to their coordinate, for instance:

x (2,3) = 2, and y (2,3) = 3. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional

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 $v 2 - v w + u v$ .

Symmetric 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). In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean 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. In Mathematics, a filtered algebra is a generalization of the notion of a Graded algebra.

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"). In Mathematics, the idea of a free object is one of the basic concepts of Abstract algebra.

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 the forgetful functor that sends a commutative algebra to its underlying vector space. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, in the area of Category theory, a forgetful functor is a type of Functor.

The unit of the adjunction is the map $V \to S(V)$ 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 $a b - b a$ , obtaining a commutative algebra, analogously to abelianization of a group. In Mathematics, a Subcategory A of a category B is said to be reflective in B when the Inclusion functor from In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup 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 functors comparable to the exterior powers; here, though, the dimension grows with k; it is given by

$\operatorname{dim}(S^k(V)) = \binom{n+k-1}{k}$

where n is the dimension of V. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, the dimension of a Vector space V is the cardinality (i

Module analog

The construction of the symmetric algebra generalizes to the symmetric algebra S(M) of a module M over a commutative ring. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property If M is a free 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. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction 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 an abelian Lie algebra in which the Lie bracket is identically 0. In Mathematics, for any Lie algebra L one can construct its universal enveloping algebra U ( L) In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

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