Generating string field theory solutions with matter operators from $KBc$ algebra

The $KBc$ algebra is a subalgebra that has been used to construct classical solutions in Witten's open string field theory, such as the tachyon vacuum solution. The main purpose of this paper is to give various operator sets that satisfy the $KBc$ algebra. In addition, since those sets can contain matter operators arbitrarily, we can reproduce the KOS and the Erler-Maccaferri solutions. Starting with a single D-brane solution on the tachyon vacuum, we replace the original $KBc$ in it with an appropriate set to generate each of the above solutions. Thus, it is expected that the $KBc$ algebra, combined with the single D-brane solution, leads to a more unified description of classical solutions.