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:
- [math]\displaystyle{ xy - (-1)^{|x||y|}yx }[/math]
- [math]\displaystyle{ x^2 }[/math] 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., [math]\displaystyle{ xy = (-1)^{|x||y|} yx }[/math] 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.
References
- David Eisenbud, Commutative Algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, vol 150, Springer-Verlag, New York, 1995. ISBN:0-387-94268-8
External links
- "rt.representation theory - Definition of the symmetric algebra in arbitrary characteristic for graded vector spaces". MathOverflow. https://mathoverflow.net/q/7080. Retrieved 2017-04-18.
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