Graded-symmetric algebra

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:

• ${\displaystyle xy-(-1)^{|x||y|}yx}$
• ${\displaystyle x^{2}}$ 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., ${\displaystyle xy=(-1)^{|x||y|}yx}$ 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.