▁of ▁ir reducible ▁unit ary ▁representations ▁of ▁, ▁and ▁it ▁is ▁even ▁not ▁clear ▁whether ▁this ▁set ▁is ▁a ▁good ▁choice ▁for ▁the ▁role ▁of ▁the ▁dual ▁object ▁for ▁. ▁So ▁the ▁problem ▁of ▁construct ing ▁du ality ▁in ▁this ▁situation ▁requires ▁complete ▁re th inking . ▁ ▁The ories ▁built ▁to ▁date ▁are ▁divided ▁into ▁two ▁main ▁groups : ▁the ▁theories ▁where ▁the ▁dual ▁object ▁has ▁the ▁same ▁nature ▁as ▁the ▁source ▁one ▁( like ▁in ▁the ▁Pon try agin ▁du ality ▁itself ), ▁and ▁the ▁theories ▁where ▁the ▁source ▁object ▁and ▁its ▁dual ▁differ ▁from ▁each ▁other ▁so ▁rad ically ▁that ▁it ▁is ▁impossible ▁to ▁count ▁them ▁as ▁objects ▁of ▁one ▁class . ▁ ▁The ▁second ▁type ▁theories ▁were ▁histor ically ▁the ▁first : ▁soon ▁after ▁Pon try agin ' s ▁work ▁T ada o ▁T ann aka ▁( 1 9 3 8 ) ▁and ▁Mark ▁Kre in ▁( 1 9 4 9 ) ▁constructed ▁a ▁du ality ▁theory ▁for ▁arbitrary ▁compact ▁groups ▁known ▁now ▁as ▁the ▁T ann aka – K re in ▁du ality . ▁In ▁this ▁theory ▁the ▁dual ▁object ▁for ▁a ▁group ▁ ▁is ▁not ▁a ▁group ▁but ▁a ▁category ▁of ▁its ▁representations ▁. ▁ ▁The ▁theories ▁of ▁first ▁type ▁appeared ▁later ▁and ▁the ▁key ▁example ▁for ▁them ▁was ▁the ▁du ality ▁theory ▁for ▁finite ▁groups . ▁In ▁this ▁theory ▁the ▁category ▁of ▁finite ▁groups ▁is ▁embedded ▁by ▁the ▁operation ▁ ▁of ▁taking ▁group ▁algebra ▁ ▁( over ▁) ▁into ▁the ▁category ▁of ▁finite ▁dimensional ▁H opf