(FWIW: Character entity ä is ä, not Ä.)

This has nothing to do with character encoding. You're outputting an HTML entity to a JavaScript string, and then asking the browser to display that JavaScript string without doing anything to interpret HTML (via alert). It's exactly as though you actually typed:

<h1>&#228;</h1>

...(which will show ä on the page), and

<script>

var a = &#228;;

alert(a);

</script>

...which won't. The HTML entity isn't being used anywhere that understands HTML entities. alert doesn't interpret HTML.

But if you did this:

<script>

var a = &#228;;

var div = document.createElement('div');

div.innerHTML = a;

document.body.appendChild(div);

</script>

...you'd see the character on the page, because we're giving the entity to something (innerHTML) that will interpret HTML. And so if you make that first line:

var a = @library.GetDictionaryItem(A);

...and then use a in an HTML context (as above), you'll get the ä in the document.

If you always get a decimal numeric character entity (like ä) from Umbraco, since those define unicode code points and JavaScript (mostly) uses unicode code points in its strings*, you can parse the entity easily enough:

function characterFromDecimalNumericEntity(str) {

    var decNumEntRex = /^&#(d+);$/;

    var match = decNumEntRex.exec(str);

    var codepoint = match ? parseInt(match[1], 10) : null;

    var character = codepoint ? String.fromCharCode(codepoint) : null;

    return character;

}

alert(characterFromDecimalNumericEntity(ä)); // ä

Live Example

* Why mostly: JavaScript strings are made up of 16-bit characters that correspond to UTF-16 code units, not Unicode code points (you can't store a Unicode code point in 16 bits, you need 21). All characters from the Basic Multilingual Plane fit within one UTF-16 code unit, but characters from the Supplementary Multilingual Plane, Supplementary Ideographic Plane, and so on require two UTF-16 code units for a character. One of those characters will occupy two characters in a JavaScript string. The function above would fail for them. More in the JavaScript spec and the Unicode FAQ.