In mathematics, a Poisson superalgebra is a –graded associative unital algebra that is equipped with a second bilinear map,
- .
![{\displaystyle [\cdot ,\cdot ]:A\times A\to A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ca455c7bce22a9824f8ff5afb5858b6381a173)
Let denote the parity of a homogeneous element , then the bracket satisfies:
- Graded Antisymmetry: .
- Graded Jacobi Idenitity: .
- Graded Leibniz Rule: .
![{\displaystyle [x,y]=-(-1)^{|x||y|}[y,x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc58253639a099b9a89bf1c834811c0fba43ac9)
![{\displaystyle [x,[y,z]]=[[x,y],z]+(-1)^{|x||y|}[y,[x,z]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e19473c95943aae6b2beb5d8dc791cc5cc83f4)
![{\displaystyle [x,yz]=[x,y]z+(-1)^{|x||y|}y[x,z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/279f145d57dcbecdeee47ce3c5bf9d65bb062a0a)
This is one of two possible ways of “super”izing the Poisson algebra. This gives the classical dynamics of fermion fields and classical spin-1/2 particles. The other way is to define an antibracket algebra or Gerstenhaber algebra, used in the BRST and Batalin-Vilkovisky formalism. The difference between these two is in the grading of the Lie bracket. In the Poisson superalgebra, the grading of the bracket is zero:
![{\displaystyle |[a,b]|=|a|+|b|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c833bec9efeab59d548883dca31c99a5de08177)
whereas in the Gerstenhaber algebra, the bracket decreases the grading by one:
![{\displaystyle |[a,b]|=|a|+|b|-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc58b1352053eefa40df0d55ff1f55e9c6264ae8)
Examples
- If is any associative -graded algebra, then, defining a new product , called the super-commutator, by for any pure graded x, y, turns into a Poisson superalgebra.
- The algebra of smooth functions of a symplectic manifold is a Poisson Superalgebra if we set .
![{\displaystyle [\cdot ,\cdot ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28dd4c22d60192519c1c12cf645b040f368db9e9)
![{\displaystyle [x,y]:=xy-(-1)^{|x||y|}yx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/559fa633d6bb26a4a098b35d6ea63f8ed53920c1)
See also
References
- Y. Kosmann-Schwarzbach (2001) [1994], “Poisson algebra”, Encyclopedia of Mathematics, EMS Press
- Henneaux, Marc; Teitelboim, Claudio (1992). Quantization of Gauge System. Princeton University Press. ISBN 9780691037691.