How to share the author’s archive?

That’s get_the_author_posts_link():

<?php echo get_the_author_posts_link(); ?>

If you want the author link for someone who isn’t the author of this page in WordPress terms – I don’t know what the bbpress state here is – then you can construct this from IDs instead (based on a snippet from twentytwentyone as you can see):

$author_id = bbp_get_user_id();
$author_data = get_userdata( $author_id );
if ( $author_data ) {
    printf(
        '<a class="author-link" href="%1$s" rel="author">%2$s</a>',
        esc_url( get_author_posts_url( $author_id ) ),
        sprintf(
            /* translators: %s: Author name. */
            esc_html__( 'View all of %s\'s posts.', 'twentytwentyone' ),
            $author_data->display_name
        )
    );
}