"This Core Storage operation is not allowed on a sparse logical volume group"

Use Cmd+R. You will have to make your HD a logical HFS partion as encryption did not go well. Open Disk Utilty and Verify the HD. Try Repair if recommended by Verify. If that does not work, make sure you backed up your personal data. In Disk Utility make sure your HD is GUID partion table and extended (journaled). Erase Macintosh HD and reinstall Yosemite. When the Welcome screen appears after install, turn off your Mac. Wait about a couple of minutes, then power up. Set up your Mac like first time out of box. Just make sure your personal data is backed up.

This procedure is a diagnostic test. It makes no changes to your data.

Please triple-click anywhere in the line below on this page to select it:

{ diskutil list; echo; diskutil cs list; } | pbcopy

Copy the selected text to the Clipboard by pressing the key combination command-C.

Launch the built-in Terminal application in any of the following ways:

☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)

☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.

☞ Open LaunchPad and start typing the name.

Paste into the Terminal window by pressing the key combination command-V. I've tested these instructions only with the Safari web browser. If you use another browser, you may have to press the return key after pasting.

Wait for a new line ending in a dollar sign ($) to appear below what you entered.

The output of the command will be automatically copied to the Clipboard. If the command produced no output, the Clipboard will be empty. Paste into a reply to this message.

The Terminal window doesn't show the output. Please don't copy anything from there.

If any personal information appears in the output, anonymize before posting, but don’t remove the context.

In response to your reply to me, no, you would not choose encrypted when reformatting to install OS X. It will install on Mac OS extended (journaled) only. Linc Davis has your Diagnostic Report and I'm sure will point you to the next step. Your issue may be overcome without reformatting and reinstall. Thank you Linc. This diagnostic instruction may help me at some point as well.

If you don't already have a current backup of all data, back up before proceeding. There are ways to back up a computer that isn't fully functional. Ask if you need guidance.

Start up in Recovery mode. When the OS X Utilities screen appears, select Disk Utility.

In the Disk Utility window, select the icon of the startup volume from the list on the left. It will be nested below another disk icon, usually with the same name. Click the Unlock button in the toolbar. When prompted, enter the login password of a user authorized to unlock the volume, or the alternate decryption key that was generated when you activated FileVault.

Then, from the menu bar, select

File ▹ Turn Off Encryption

Enter the password again.

You can then restart as usual, if the system is working. Decryption will be completed in the background. It may take several hours, and during that time performance will be reduced.

If you can't turn off encryption in Disk Utility because the menu item is grayed out, you'll have to erase the volume and then restore the data from a backup. Select the Erase tab, and then select

Mac OS Extended (Journaled)

from the Format menu.

You can then quit to be returned to the main Recovery screen. Follow these instructions if you back up with Time Machine. If you use other backup software, follow its developer's instructions.

Don't erase the volume unless you have at least two complete, independent backups. One is not enough to be safe.

Just a follow up. Are you up and running now? Hope all is well. Good luck!