Basic Palindrome check comparing chars
Your basic approach looks good.  However I noticed a few things:

You're not checking for strings with 0 characters.

The ternary statement adjusting for odd or even length is unnecessary.  Simply dividing by 2 and adding 1 will work for both odd or even lengths.

Inside the loop, you're calling toLower() twice on each iteration.  This is quite inefficient.  Storing the word converted to lower case in a variable and using that instead of the passed string would be much more efficient.

When testing for inequality it is usually better to use the inequality operator(!=) instead of not(!) and equals(==).  It's more concise and easier to read.