Graded-symmetric algebra

{{Short description|Type of algebra over a commutative ring}}

{{one source |date=May 2024}}

In algebra, given a commutative ring R, the graded-symmetric algebra of a graded R-module M is the quotient of the tensor algebra of M by the ideal I generated by elements of the form:

• $xy\; -\; (-1)^$

  x

  y

  yx
• $x^2$ when |x| is odd

for homogeneous elements x, y in M of degree |x|, |y|. By construction, a graded-symmetric algebra is graded-commutative; i.e., $xy\; =\; (-1)^$

x

y

yx and is universal for this.

In spite of the name, the notion is a common generalization of a symmetric algebra and an exterior algebra: indeed, if V is a (non-graded) R-module, then the graded-symmetric algebra of V with trivial grading is the usual symmetric algebra of V. Similarly, the graded-symmetric algebra of the graded module with V in degree one and zero elsewhere is the exterior algebra of V.