Lie Group Analysis of a Forced KdV Equation

Table 3

Table of adjoint representations.

Ad(𝑒^𝜖𝑋𝑖)𝑋𝑗	𝑋1	𝑋2	𝑋3
𝑋1	𝑋1	𝑋2	𝑋3−𝛼𝜖𝑋1
𝑋2	𝑋1	𝑋2	𝑋3+𝜖(3𝑛𝛼^2𝑋1+2𝛼𝑋2)
𝑋3	𝑒^𝜖𝛼𝑋1	𝑒^−2𝛼𝜖𝑋2+3𝑛𝛼𝑒^−𝜖𝛼−1𝑋1	𝑋3

Here Ad(𝑒^𝜖𝑋𝑖)𝑋_𝑗=𝑋_𝑗−𝜖[𝑋_𝑖,𝑋_𝑗]+(1/2!)𝜖²[𝑋_𝑖,[𝑋_𝑖,𝑋_𝑗]]−⋯ is the adjoint representation where 𝜖 is a real number.