Zoran Skoda gl2 linear realization

The following shows that the Gurevich-Saponov differential calculus for $gl(2)$, linear in derivatives is just an example of our framework – the right hand side are the components of matrix $\phi = (\phi^i_j(\partial^{1,1},\ldots,\partial^{2,2}))_{i,j = 1,,2}$ satisfying our differential equation for realizations hence it comes from some isomorphism of coalgebras between $U(gl_2)$ and $S(gl_2)$. Here the matrix elements, which are in general formal power series, is linear in partial derivatives.

Let us put $h = 1$ (it is just a rescale). If we postulate commutative coordinates

$X_1 = A, X_2 = B, X_3 = C, X_4 = D$ and set

$$a = A(1+\partial_a) + B\partial_b$$

$$b = A \partial_c + B (\partial_d +1)$$

$$c = C (\partial_a +1) + D \partial_b$$

$$d = C \partial_c + D (\partial_d+1)$$

and $[\partial_a,A] = 1, [\partial_b, B] = 1, [\partial_c,C] = 1, [\partial_d,D] = 1,$ then we obtain

1) $[a,b] = b, [a,c] = -c, [a,d] = 0, [b,c] = a-d, [b,d] = b, [c,d] = -c$

2) $a = \sum X_i \phi^i_a$

$b = \sum X_i \phi^i_b$ etc. and $\phi$ satisfies our differential equation

and

3) your relations follow

$$\array{ \partial_a a - a \partial_a = \partial_a +1\\ \partial_a b - b \partial_a = \partial_c\\ \partial_a c - c \partial_a = 0\\ \partial_a d - d \partial_a = 0\\ \partial_b a - a \partial_b = \partial_b\\ \partial_b b - b \partial_b = \partial_d +1\\ \partial_b c - c \partial_b = 0\\ \partial_b d - d \partial_b = 0\\ \partial_c a - a \partial_c = 0\\ \partial_c b - b \partial_c = 0\\ \partial_c c - c \partial_c = \partial_a + 1\\ \partial_c d - d \partial_c = \partial_c\\ \partial_d a - a \partial_d = 0\\ \partial_d b - b \partial_a = 0\\ \partial_d c - c\partial_d = \partial_b \\ \partial_d d - d \partial_d = \partial_d + 1}$$

what is just the form $[\partial_j, \hat{x}_i] = \phi^j_i$

category: valenciennes