Editing Symmetric algebra (section)

{{Use American English|date = February 2019}}
{{short description|"Smallest" commutative algebra that contains a vector space}}
{{distinguish|Symmetric Frobenius algebra}}
In [[mathematics]], the '''symmetric algebra''' {{math|''S''(''V'')}} (also denoted {{math|Sym(''V''))}} on a [[vector space]] {{math|''V''}} over a [[field (mathematics)|field]] {{math|''K''}} is a [[commutative algebra (structure)|commutative algebra]] over {{mvar|K}} that contains {{mvar|V}}, and is, in some sense, minimal for this property. Here, "minimal" means that {{math|''S''(''V'')}} satisfies the following [[universal property]]: for every [[linear map]] {{mvar|f}} from {{mvar|V}} to a commutative algebra {{mvar|A}}, there is a unique [[algebra homomorphism]] {{math|''g'' : ''S''(''V'') → ''A''}} such that {{math|1=''f'' = ''g'' ∘ ''i''}}, where {{mvar|i}} is the [[inclusion map]] of {{mvar|V}} in {{math|''S''(''V'')}}.

If {{mvar|B}} is a basis of {{mvar|V}}, the symmetric algebra {{math|''S''(''V'')}} can be identified, through a [[canonical isomorphism]], to the [[polynomial ring]] {{math|''K''[''B'']}}, where the elements of {{mvar|B}} are considered as indeterminates. Therefore, the symmetric algebra over {{mvar|V}} can be viewed as a "coordinate free" polynomial ring over {{mvar|V}}.

The symmetric algebra {{math|''S''(''V'')}} can be built as the [[quotient ring|quotient]] of the [[tensor algebra]] {{math|''T''(''V'')}} by the [[two-sided ideal]] generated by the elements of the form {{math|''x'' ⊗ ''y'' − ''y'' ⊗ ''x''}}.

All these definitions and properties extend naturally to the case where {{mvar|V}} is a [[module (mathematics)|module]] (not necessarily a free one) over a [[commutative ring]].