Firstly, a dollar bill is just a promissory note, an IOU. It is a promise to pay the holder the value of one dollar. It is easy to use but the dollar bill remains the property of the people that printed it. In the United States it is the Federal Reserve.
The Federal Reserve is not Government.
The Federal Reserve is comprised of 12 Banks, represented by a board of 7 people.
These Banks are owned by shareholders.
Major shareholders in all these Banks are a group of 4 companies; I will call them the “big four”.
The big four are: State Street Corporation, Vanguard Group, BlackRock and FMR [Fidelity].
These companies are major shareholders in the top ten financial groups in the US: Bank of America, JP Morgan, Citigroup, Wells Fargo, Goldman Sachs, Morgan Stanley and others.
So the big four not only have representatives on the boards of all the major US financial Corporations but also on the board of the Fed.