A div by default cannot be given focus. However, you can change that by adding a tabindex attribute to the div:
<div tabindex="0" id="example"></div>
You can then give the div focus, and also blur it with the hover event:
$("#example").hover(function() {
this.focus();
}, function() {
this.blur();
}).keydown(function(e) {
alert(e.keyCode);
});
When the div has focus, it will accept keyboard events. You can see an example of this working here.