ALEXANDRIA, Va. — One month after a judge declared Google’s search engine an illegal monopoly, the tech giant faces another antitrust lawsuit that threatens to break up the company, this time over its advertising technology.

The Justice Department and a coalition of states contend that Google built and maintains a monopoly over the technology that matches online publishers to advertisers. Dominance over the software on both the buy side and the sell side of the transaction enables Google to keep as much as 36 cents on the dollar when it brokers sales between publishers and advertisers, the government contends in court papers.