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Representation theory of Hopf algebras
In abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra ''H'' over a field ''K'' is a ''K''-vector space ''V'' with an action ''H'' × ''V'' → ''V'' usually denoted by juxtaposition ( that is, the image of (''h'',''v'') is written ''hv'' ). The vector space ''V'' is called an ''H''-module.
==Properties==
The module structure of a representation of a Hopf algebra ''H'' is simply its structure as a module for the underlying associative algebra. The main use of considering the additional structure of a Hopf algebra is when considering all ''H''-modules as a category. The additional structure is also used to define invariant elements of an ''H''-module ''V''. An element ''v'' in ''V'' is invariant under ''H'' if for all ''h'' in ''H'', ''hv'' = ε(''h'')''v'', where ε is the counit of ''H''. The subset of all invariant elements of ''V'' forms a submodule of ''V''.
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