The symmetric algebra is the quotient of by the ideal generated by . It is graded by symmetric powers:
The nth symmetric power is the quotient of that identifies tensors differing by a permutation of their factors. Equivalently, over characteristic zero it is the subspace of symmetric tensors.
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Symmetric algebra is a fundamental construction in algebra, particularly in the context of algebraic geometry and commutative algebra. Specifically, it is associated with the idea of forming polynomials from elements of a vector space or an algebra.