In the theoretical situation you described, my very theoretical and non-legal opinion is that you should immediately notify the board's attorney. Don't take it upon yourself to decide how to proceed because you may be entering a minefield of conflicting and ambiguous requirements and prohibitions.

If you think the co-op's attorney is a party to the theoretically questionable activity, I'm not sure where you would start. Try looking on the Attorney General's website for a place where you can report suspected questionable attorney activity.

What if the Board is informed by a shareholder via email that another shareholder is engaged in illegal activity in the apartment, or a shareholder is making illegal no permits) renovations. Can authorities be notified?

I agree with Steven's advice about contacting the co-op's attorney. He will ask what proof do you have of criminal activity.

So, before you contact the attorney, you need to be able to specifically provide examples of what illegal activity is taking place in the apartment. You'll also be asked how do you know the shareholder has no permits.

Steven is correct when he advises you to tread lightly. An email from a shareholder claiming there is illegal activity is not proof of wrongdoing.

Knowing it and proving it are 2 different things.

Good luck.

I'm sensing here that there are two separate issues being questioned - illegal and unauthorized.

If the board becomes or is made aware of illegal activity such as growing pot, cooking methamphetamine, prostitution, financial "boiler room", etc, the board's attorney should be notified immediately so she/he can notify the the authorities. If, in the opinion of the board the illegal activity represents an immediate danger to shareholders or the building, in my opinion the police should be contacted immediately.

If the activity is simply unauthorized, like the example given of unapproved or undocumented alterations, the board has the power to assess the situation and unilaterally shut it down. The board can enter the apartment without permission if they believe an unauthorized activity is taking place. If there is no immediate danger (such as an unlicensed plumber making a gas hookup), the co-op's attorney should be contacted first if possible. If someone smells gas or other fumes, safety is the primary concern.