Can any one explain the language L = {w: w = uu, u in La (1 * 01 *)}

No, your simplification is wrong.

Consider, as a simpler case, $ A = {a, b } $ and $ L = {w | w = uu, u in A } $. Then, $ L = {aa, bb } $.

Instead, if we apply your simplification and write $ The & # 39; = AA $ then, we have $ The & # 39; = {aa, ab, ba, bb } $ which is a bigger language, since we have forgotten that both $ u $& # 39; sin $ w = uu $ must be the even word.

Keep this in mind, your language of origin $ L $ seems to be $ {1 ^ n01 ^ m1 ^ n01 ^ m | n, m leq 0 } $