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- Symmetric algebra

Symmetric algebra should not be confused with Symmetric Frobenius algebra.

In mathematics, the **symmetric algebra** (also denoted on a vector space over a field is a commutative algebra over that contains, and is, in some sense, minimal for this property. Here, "minimal" means that satisfies 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 map of in .

If is a basis of, the symmetric algebra can be identified, through a canonical isomorphism, to the polynomial ring, where the elements of are considered as indeterminates. Therefore, the symmetric algebra over can be viewed as a "coordinate free" polynomial ring over .

The symmetric algebra can be built as the quotient of the tensor algebra by the two-sided ideal generated by the elements of the form .

All these definitions and properties extend naturally to the case where is a module (not necessarily a free one) over a commutative ring.

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 commutators

*v* ⊗ *w*-*w* ⊗ *v.*

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 from to a commutative algebra extends to an algebra homomorphism

*T(V)* → *A*

*S(V)* → *A*

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 (forgetting the multiplication) is the composition of the forgetful functors from commutative algebras to associative algebras (forgetting commutativity), and from associative algebras to vectors or modules (forgetting the multiplication). 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.

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 respective categories.

If is a module that is not free, it can be written

*V*=*L/M,*

*S(V)*=*S(L/M)*=*S(L)/\langle**M\rangle,*

*\langle**M\rangle*

The symmetric algebra is a graded algebra. That is, it is a direct sum

infty | |

S(V)=oplus | |

n=0 |

*S*^{n(V),}

*S*^{n(V),}

*S*^{2(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 homogeneous ideal: the ideal generated by all

*x* ⊗ *y*-*y* ⊗ *x,*

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 (graded) symmetric algebra of (a polynomial ring) by the homogeneous ideal generated by the elements of, which are homogeneous of degree one.

One can also define

*S*^{n(V)}

*S*^{n(V)}

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.

l*S*_{n.}

*\sigma\in*l*S*_{n,}

*v*_{1 ⊗ } … ⊗ *v*_{n}*\mapsto**v*_{\sigma(1)} ⊗ … ⊗ *v*_{\sigma(n)}

infty | |

styleoplus | |

n=0 |

*\operatorname{Sym}*^{n(V),}

Let

*\pi*_{n}

*T*^{n(V)\to}*S*^{n(V).}

*\pi*_{n}

*v*_{1 … }*v*_{n}*\mapsto*

1{n!} | |

\sum |

\sigma\inS_{n} |

*v*_{\sigma(1)} ⊗ … ⊗ *v*_{\sigma(n)}*.*

The map

*\pi*_{n}

*\pi*_{n(x ⊗ }*y*+*y* ⊗ *x)*=2*xy*

*\pi*_{n}

*xy\not\in*

2(V)), | |

\pi | |

n(\operatorname{Sym} |

12 | |

(x ⊗ |

*y*+*y* ⊗ *x)**\not\in**\operatorname{Sym}*^{2(V).}

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.

Given a module over a commutative ring, the symmetric algebra can be defined by the following universal property:

*g:S(V)\to**A*

*f*=*g\circ**i,*

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.

*f:V\to**W*

*S(f):S(V)\to**S(W).*

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.

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).

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}*

*S*_{n}

*V*^{ ⊗ }

The symmetric algebra can be given the structure of a Hopf algebra. See Tensor algebra for details.

The symmetric algebra *S*(*V*) is the universal enveloping algebra of an abelian Lie algebra, i.e. one in which the Lie bracket is identically 0.

- exterior algebra, the alternating algebra analog
- graded-symmetric algebra, a common generalization of a symmetric algebra and an exterior algebra
- Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic form
- Clifford algebra, a quantum deformation of the exterior algebra by a quadratic form