show menu only when there is a live

I’m not sure what you mean about the menu, but if you want the live video to show only when there’s a live video and show nothing the rest of the time, set the fallback behavior in the plugin options to “Show nothing at all”, or add fallback_behavior="no_message" to the shortcode (for example, [youtube_live fallback_behavior="no_message"])

I would like to know if I have to know if there is any live video at the moment, if I have a display block in a div. And if you do not have it, I make a display none

Then you’ll want to set the containing div to display: none; and use the event listener to listen for a live video and change it to display: block;.

More info about the event listener here: https://github.com/macbookandrew/wp-youtube-live#event-listener (also on the WordPress plugin readme, but easier to link to a specific section on GitHub).

great, could you give me an example of how to use this function?

window.addEventListener('wpYouTubeLiveStarted', function() {
    /* your code here */
    console.log('stream started');
    /* your code here */
});

Instead of console.log(), you’d show/hide your div.

You’d either add this in a JS file included in your page or inside an inline <script></script> tag (not recommended).