You do realize that hibernation power-saving mode means that not even the operating system is running, right? Hibernation means to copy the contents of system memory into a disk file and power off the computer (this is a special power-off state which events in Task Scheduler can still cause a power-on if they are configured to wake the computer when it is asleep). That means when in hibernation mode, you don't have YahooPOPs running, or your e-mail client, or Windows itself. In hibernate mode and with the power off, not even the hard disks are spinning (which is a good thing when you are transporting the computer).
When you power the computer back up, the contents of the save file get rewritten back into memory and it looks like you are back where you were before. You might want to put the computer into Standby mode instead of Hibernate mode. Make sure the disks spin down when Standby mode gets entered. However, some network cards will still disconnect when put into Standby mode so your network connection is lost. Your e-mail client and YahooPops will still be running but the network interface is dead when in Standby mode so there are errors when YahooPops tries to connect to Yahoo for mail.
Description of the Different Advanced Power Management States
http://support.microsoft.com/?id=308535
ACPI Specification
http://www.acpi.info/spec.htm (see page 20)