Yes, you can access the user object in the template and check if the user is like this:
{% if user.is_staff %} <li> <a href="/admin/">Admin</a> </li> {% endif %}
This will be an example where your menu contains links to links. Admin link will be displayed only for users with is_staff status. You can do the same with is_authenticated.
Django is designed to separate the logic from the presentation, so if you want to have even finer control over the menu, I would suggest making the logic inside the view and then setting a variable that you can check in the template to determine which menu to show.
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