After much debate and political maneuvering, Cathleen Black, the veteran publishing executive, is the city’s new schools chancellor. She replaces Joel Klein.

Black was put forward for the job by Mayor Bloomberg in early November, but her candidacy had been in jeopardy after the State Education Commissioner David Steiner questioned her lack of related work experience.

Schools chancellors need a professional certificate in educational leadership and other qualifications. Black has had a successful career in business and publishing — she’s been called the “First Lady of glamorous glossies” — but she’s never been a teacher, and her resume boasts just a single day as a guest principal in the Bronx.

To get around this, Black needed Steiner to grant her a waiver, and yesterday he did just that, having been convinced that her lack of experience would be mitigated by the appointment of a chief academic officer to serve as her deputy.

On Monday night, Bronx Assembly members Vanessa Gibson and Marco Crespo released a joint statement slamming Steiner’s decision. “This is not the time to compromise on who will lead New York’s schools or to adopt a ‘Let’s Make A Deal’ approach regarding the person who will be leading public education in our City,” Gibson said.

Crespo has co-sponsored legislation which would require the state legislature to sign off on all waivers. “This type of backdoor game will only undermine the public’s confidence in our schools and I will continue to fight for legislation that would require the full state Legislature to vote on such a waiver in joint session before it can take effect,” he said in a statement.

Assemblyman Jeff Dinowitz echoed Crespo and Gibson: “I can’t think of another example where the head of a key city agency was appointed only on the condition that someone with actual qualifications was appointed to serve as a top deputy,” he said in a statement.

Other Black critics, including the Coalition for Public Education and civil rights lawyer Norman Siegel, are now threatening to take the matter to court.

Bronx politicians as a whole are split on Black. Councilman Joel Rivera, for one, is a staunch supporter. In a letter sent to Steiner, Rivera said Black was “committed to keeping the focus on ending inequality in education and closing the achievement gap.”

In a phone interview, Rivera said Black had called him and convinced him that she would work closely with local elected officials — something Klein’s opponents said he failed to do. Not that Rivera is critical of Klein. “[He’s] done a pretty remarkable job with the Department of Education,” he said.

Rivera said there are two schools of thought: one, that you need an educator to run a school system, and two, that when you have a 135,000-employee agency, with a multi-billion dollar budget, you need, first and foremost, a strong manager.

Rivera is firmly in the second camp. “It’s not as if she’ll be in the classroom,” he said. “Give her an opportunity to show what she can do . . . instead of ridiculing.”

Black’s chief academic officer will be Shael Polakow-Suransky, a former teacher and principal who founded the Bronx International High School, a small school located on the Morris High School Campus.