Lie algebras and differential operators
Robin K. S. Hankin
To cite this work or the weyl package in publications
please use Hankin (2022). In a very nice
youtube
video, Richard Borcherds discusses the fact that first-order
differential operators do not commute, but their commutator is itself
first-order; he says that they “almost” commute. Here I demonstrate
Borcherds’s observations in the context of the weyl
package. Symbolically, if
\[ D=\sum f_i\left(x_1,\dots,x_n\right)\frac{\partial}{\partial x_i}\qquad E=\sum g_i\left(x_1,\dots,x_n\right)\frac{\partial}{\partial x_i} \]
where \(f_i=f_i\left(x_1,\dots,x_n\right)\) and \(g_i=g_i\left(x_1,\dots,x_n\right)\) are functions, then
\[ DE=\sum_{i,j}f_i\frac{\partial}{\partial x_i}\,g_i\frac{\partial}{\partial x_j} =\sum_{i,j}f_ig_j\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j} + f_i\frac{\partial g_j}{\partial x_i}\,\frac{\partial}{\partial x_j} \]
\[ ED=\sum_{i,j}g_i\frac{\partial}{\partial x_i}\,f_i\frac{\partial}{\partial x_j} =\sum_{i,j}g_if_j\frac{\partial}{\partial x_j}\frac{\partial}{\partial x_i} + g_i\frac{\partial f_i}{\partial x_j}\,\frac{\partial}{\partial x_j} \]
so \(E\) and \(E\) “nearly” commute, in the sense that \(ED-DE\) is first order:
\[DE-ED= \sum_{i,j}f_i\frac{\partial g_j}{\partial x_i}\,\frac{\partial}{\partial x_j}-g_i\frac{\partial f_i}{\partial x_j}\,\frac{\partial}{\partial x_j} \]
Above we have used the fact that partial derivatives commute, which
leads to the cancellation of the second-order terms. We can verify this
using the weyl package:
D <- weyl(spray(cbind(matrix(sample(8),4,2),kronecker(diag(2),c(1,1))),1:4))
E <- weyl(spray(cbind(matrix(sample(8),4,2),kronecker(diag(2),c(1,1))),1:4))
F <- weyl(spray(cbind(matrix(sample(8),4,2),kronecker(diag(2),c(1,1))),1:4))
D## A member of the Weyl algebra:
## x y dx dy val
## 7 8 0 1 = 4
## 4 3 0 1 = 3
## 1 5 1 0 = 2
## 6 2 1 0 = 1(\(E\) and \(F\) are similar). Symbolically we would have
\[D= \left( x^6y^2 + 2xy^5\right)\frac{\partial}{\partial x}+ \left(4x^7y^8 + 3x^4y^3\right)\frac{\partial}{\partial y}. \]
The package allows us to compose \(E\) and \(D\), although the result is quite complicated:
summary(E*D)## A spray object. Summary of coefficients:
##
## a disord object with hash ab0b4d525d3ef5030fcdc229e07f50175ee9dc3f
##
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.0 4.0 8.5 20.5 22.0 128.0
##
##
## Representative selection of index and coefficients:
##
## x y dx dy val
## 11 3 1 0 = 6
## 9 12 1 1 = 16
## 9 4 0 1 = 12
## 8 6 2 0 = 2
## 10 4 1 1 = 3
## 12 3 2 0 = 1However, the Lie bracket, \(ED-DE\),
(.[E,D] in package idiom) is indeed first order:
.[E,D]## A member of the Weyl algebra:
## x y dx dy val
## 8 12 0 1 = -8
## 4 9 1 0 = 30
## 15 14 0 1 = 16
## 12 9 0 1 = -20
## 5 7 0 1 = 24
## 7 6 1 0 = 8
## 13 8 1 0 = -4
## 14 8 1 0 = 8
## 8 7 0 1 = -9
## 13 9 0 1 = -32
## 10 3 1 0 = -3
## 9 11 1 0 = 8
## 7 7 0 1 = -18
## 2 9 1 0 = -4
## 3 10 0 1 = -18
## 9 6 1 0 = 6
## 6 6 1 0 = -34
## 9 4 0 1 = 12
## 10 12 0 1 = 36Above, looking at the dx and dy columns, we
see that each row is either 1 0 or 0 1,
corresponding to either \(\partial/\partial
x\) or \(\partial/\partial y\)
respectively. Arguably this is easier to see with the other print
method:
options(polyform = TRUE)
.[E,D]## A member of the Weyl algebra:
## -8*x^8*y^12*dy +30*x^4*y^9*dx +16*x^15*y^14*dy -20*x^12*y^9*dy
## +24*x^5*y^7*dy +8*x^7*y^6*dx -4*x^13*y^8*dx +8*x^14*y^8*dx
## -9*x^8*y^7*dy -32*x^13*y^9*dy -3*x^10*y^3*dx +8*x^9*y^11*dx
## -18*x^7*y^7*dy -4*x^2*y^9*dx -18*x^3*y^10*dy +6*x^9*y^6*dx
## -34*x^6*y^6*dx +12*x^9*y^4*dy +36*x^10*y^12*dyoptions(polyform = FALSE) # revert to defaultWe may verify Jacobi’s identity:
.[D,.[E,F]] + .[F,.[D,E]] + .[E,.[F,D]]## A member of the Weyl algebra:
## empty sparse array with 4 columnsBorcherds goes on to consider the special case where the \(f_i\) and \(g_i\) are constant. In this case the operators commute (by repeated application of Schwarz’s theorem) and so their Lie bracket is identically zero. We can create constant operators easily:
(D <- as.weyl(spray(cbind(matrix(0,3,3),matrix(c(0,1,0,1,0,0,0,0,1),3,3,byrow=T)),1:3)))## A member of the Weyl algebra:
## x y z dx dy dz val
## 0 0 0 0 0 1 = 3
## 0 0 0 1 0 0 = 2
## 0 0 0 0 1 0 = 1(E <- as.weyl(spray(cbind(matrix(0,3,3),matrix(c(0,1,0,1,0,0,0,0,1),3,3,byrow=T)),5:7)))## A member of the Weyl algebra:
## x y z dx dy dz val
## 0 0 0 0 0 1 = 7
## 0 0 0 1 0 0 = 6
## 0 0 0 0 1 0 = 5(above, see how the first three columns of the index matrix are zero, corresponding to constant coefficients of the differential operator; symbolically \(D=2\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+3\frac{\partial}{\partial z}\) and \(E=6\frac{\partial}{\partial x}+5\frac{\partial}{\partial y}+7\frac{\partial}{\partial z}\). And indeed, their Lie bracket vanishes:
.[D,E]## [1] 0