Yes, you can do one of two things:

1) If you are on your private computer and do not mind being constantly logged in or can remember to log out after each session, you can select "Remember Me" when you log in so that you will stay logged in even if you leave the LttD site. Otherwise you will be logged out as soon as you leave the site, as you experience.

2) If you are on a public computer or have children and do not wish to remain logged in, or cannot remember to log yourself out every time if that is your desire, right click the link in question and choose to open the link in a new browser window, or if you are using a browser with tabs, a new tab.
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"The devil I'll bring you," answered Hagen. "I have enough to carry with my shield and breastplate; my helm is bright, the sword is in my hand, therefore I bring you naught."