Lie 代數
Lie algebra
括弧積$ \lbrack\cdot,\cdot\rbrack:{\frak g}\times{\frak g}\to{\frak g}
$ \lbrack ax+by,z\rbrack=a\lbrack x,z\rbrack+b\lbrack y,z\rbrack.
$ \lbrack x,ay+bz\rbrack=a\lbrack x,y\rbrack+b\lbrack x,z\rbrack.
交代性$ \lbrack x,x\rbrack=0
反交換性 (歪對稱性)$ \lbrack x,y\rbrack=-\lbrack y,x\rbrack
Jacobi 恆等式$ \lbrack x,\lbrack y,z\rbrack\rbrack+\lbrack z,\lbrack x,y\rbrack\rbrack+\lbrack y,\lbrack z,x\rbrack\rbrack=0
$ \lbrack x,y\rbrack:=xy-yx.
$ D_x:y\mapsto\lbrack x,y\rbrackと定めて、Leibniz 則$ D_x(yz)=D_x(y)z+yD_x(z)つまり$ \lbrack x,yz\rbrack=\lbrack x,y\rbrack z+y\lbrack x,z\rbrackを滿たす $ \lbrack A,B\rbrack:=AB-BA.
$ \{A,B\}:=\frac{\partial A}{\partial q}\frac{\partial B}{\partial p}-\frac{\partial A}{\partial p}\frac{\partial B}{\partial q}
Lie 代數$ \frak gに於いて、Leibniz 則$ D(\lbrack x,y\rbrack)=\lbrack D(x),y\rbrack+\lbrack x,D(y)\rbrackを滿たす寫像$ D:{\frak g}\to{\frak g}を「微分」と呼ぶ