Author: Robin K. S. Hankin

# 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  =    1

However, 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  =   36

Above, 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*dy

    options(polyform = FALSE) # revert to default

We 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 columns

Borcherds 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

Hankin, R. K. S. 2022. “Quantum Algebra in R: The Weyl Package.” arXiv. https://doi.org/10.48550/ARXIV.2212.09230.