@prefix psr: . @prefix skos: . psr: a skos:ConceptScheme . psr:-H6268KGD-3 skos:exactMatch , ; skos:definition """In mathematics, the symmetric algebra

S (V)

(also denoted

Sym(V))

on a vector space

V

over a field

K

is a commutative algebra over

K

that contains

V

, and is, in some sense, minimal for this property. Here, "minimal" means that

S (V)

satisfies the following universal property: for every linear map

f

from

V

to a commutative algebra

A

, there is a unique algebra homomorphism

g : S (V) \to A

such that

f = g \circ i

, where

i

is the inclusion map of

V

S (V)

.
If

B

is a basis of

V

, the symmetric algebra

S (V)

can be identified, through a canonical isomorphism, to the polynomial ring

K [B]

, where the elements of

B

are considered as indeterminates. Therefore, the symmetric algebra over

V

can be viewed as a "coordinate free" polynomial ring over

V

.
The symmetric algebra

S (V)

can be built as the quotient of the tensor algebra

T (V)

by the two-sided ideal generated by the elements of the form

x \otimes y - y \otimes x

.
All these definitions and properties extend naturally to the case where

V

is a module (not necessarily a free one) over a commutative ring.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Symmetric_algebra)"""@en, """En mathématiques, l'algèbre symétrique est une algèbre sur un corps associative, commutative et unifère utilisée pour définir des polynômes sur un espace vectoriel. L'algèbre symétrique est un outil important dans la théorie des algèbres de Lie et en topologie algébrique dans la théorie des classes caractéristiques.
(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/Alg%C3%A8bre_sym%C3%A9trique)"""@fr ; skos:prefLabel "symmetric algebra"@en, "algèbre symétrique"@fr ; skos:broader psr:-FTGGBTC5-X ; a skos:Concept ; skos:narrower psr:-HS1X95S1-9 ; skos:inScheme psr: . psr:-FTGGBTC5-X skos:prefLabel "algèbre commutative"@fr, "commutative algebra"@en ; a skos:Concept ; skos:narrower psr:-H6268KGD-3 . psr:-HS1X95S1-9 skos:prefLabel "symmetric polynomial"@en, "polynôme symétrique"@fr ; a skos:Concept ; skos:broader psr:-H6268KGD-3 .