How can i list the author of the category?

I think this will do what you want:

<?php
$sql = "
SELECT DISTINCT t.name as cat_name, um1.meta_value AS first_name, um2.meta_value AS last_name
FROM $wpdb->terms t
JOIN $wpdb->term_taxonomy tt ON (tt.term_id = t.term_id AND taxonomy = 'category')
JOIN $wpdb->term_relationships tr ON (tr.term_taxonomy_id = tt.term_taxonomy_id)
JOIN $wpdb->posts p ON p.ID = tr.object_id
JOIN $wpdb->usermeta um1 ON (um1.user_id = p.post_author AND um1.meta_key = 'first_name')
JOIN $wpdb->usermeta um2 ON (um2.user_id = p.post_author AND um2.meta_key = 'last_name')
WHERE p.post_type = 'post'
AND p.post_status IN ('publish','private')
AND p.post_date <= NOW()
ORDER BY cat_name, first_name, last_name
";
$rows = $wpdb->get_results($sql);
// Now output category name person1, person2, ...
$curr_cat = '';
foreach ($rows as $row) {
   if ($row->cat_name != $curr_cat) {
      if ($curr_cat !== '') echo "</p>\n</div>\n"; // Close previous div
      echo '<div class="cat_div">';        // Start new div
      $curr_cat = $row->cat_name;
      echo "CAT NAME: $row->cat_name";
      echo "<p>AUTHOR: ";
      $sep = '';
   }
   echo "$sep$row->first_name $row->last_name";
   $sep = ', ';
}
echo "</p>\n</div>\n"; // Close previous div
?>

Put this code in a template styled for your theme.

Worked! Many thanks

But when i use it in my sidebar php widget, its not worked.
I see this error:

Fatal error: Call to a member function get_results() on a non-object in yourwebsite.com\plugins\exec-php\includes\runtime.php(42) : eval()’d code on line 15

How can i fix it?

Regards

Try changing this:

<?php
$sql = "

to this:

<?php
global $wpdb;
$sql = "

Worked!. Thanks

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@vtzyzzy i have a question again.

This code listed authors category its worked but its not worked same php code widget.i see same error.

CODE:

<?php global $post; $wpdb;
$author_id = $post->post_author;
$categories = $wpdb->get_results("
	SELECT DISTINCT(terms.term_id) as ID, terms.name, terms.slug
	FROM $wpdb->posts as posts
	LEFT JOIN $wpdb->term_relationships as relationships ON posts.ID = relationships.object_ID
	LEFT JOIN $wpdb->term_taxonomy as tax ON relationships.term_taxonomy_id = tax.term_taxonomy_id
	LEFT JOIN $wpdb->terms as terms ON tax.term_id = terms.term_id
	WHERE 1=1 AND (
		posts.post_status = 'publish' AND
		posts.post_author = '$author_id' AND
		tax.taxonomy = 'category' )
	ORDER BY terms.name ASC
");

echo '<ul>';
foreach($categories as $category) :
echo'<li>';
echo '<a href="';
echo get_category_link( $category->ID );
echo'" title="';
echo $category->name;
echo'">';
echo $category->name;
echo '</a>';
echo'</li>';
endforeach;
echo'<br />';
echo'</ul>';
?>

The first semi-colon in this line:

<?php global $post; $wpdb;

should be a comma:

<?php global $post, $wpdb;

Yes its worked. Thank you for all your help.