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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Hopf algebra} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra} [[!include higher algebra - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{relation_to_hopf_groups}{Relation to Hopf groups}\dotfill \pageref*{relation_to_hopf_groups} \linebreak \noindent\hyperlink{the_theorem_of_hopf_modules}{The theorem of Hopf modules}\dotfill \pageref*{the_theorem_of_hopf_modules} \linebreak \noindent\hyperlink{relation_to_lie_algebras}{Relation to Lie algebras}\dotfill \pageref*{relation_to_lie_algebras} \linebreak \noindent\hyperlink{TannakaDuality}{Tannaka duality}\dotfill \pageref*{TannakaDuality} \linebreak \noindent\hyperlink{as_3vector_spaces}{As 3-vector spaces}\dotfill \pageref*{as_3vector_spaces} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The notion of \emph{Hopf algebra} is an abstraction of the properties of \begin{itemize}% \item the [[group algebra]] of a [[group]]; \item the [[algebra of functions]] on a finite group, and more generally, the algebra of regular functions on an affine algebraic $k$-group; \item the [[universal enveloping algebra]] of a [[Lie algebra]], \end{itemize} where not only the [[associative algebra]] structure is remembered, but also the natural [[coalgebra]] structure, making it a [[bialgebra]], as well as the algebraic structure induced by the [[inverse]]-operation in the group, called the \emph{antipode}. More intrinsically, a Hopf algebra structure on an [[associative algebra]] is precisely the structure such as to make its [[category of modules]] into a [[rigid monoidal category]] equipped with a [[fiber functor]] -- this is the statement of \emph{[[Tannaka duality]]} for Hopf algebras. Hopf algebras and their generalization to [[Hopf algebroids]] arise notably as [[groupoid convolution algebras]]. Another important source of Hopf algebras is [[combinatorics]], see at \emph{[[combinatorial Hopf algebras]]}. There is a wide variety of variations of the notion of Hopf algebra, relaxing [[properties]] or adding [[structure]]. Examples are \emph{[[weak Hopf algebras]]}, \emph{[[quasi-Hopf algebras]]}, \emph{([[quasi-triangular Hopf algebra|quasi]]-)[[triangular Hopf algebras]]}, \emph{[[quantum groups]]}, \emph{[[hopfish algebras]]} etc. Most of these notions are systematized via [[Tannaka duality]] by the properties and structures on the coresponding [[categories of modules]], see at \emph{\hyperlink{TannakaDuality}{Tannaka duality}} below. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{}\hypertarget{}{} A $k$-[[bialgebra]] $(A,m,\eta,\Delta,\epsilon)$ with multiplication $m$, comultiplication $\Delta$, unit $\eta: k\to A$ and counit $\epsilon:A\to k$ is called a \textbf{Hopf algebra} if there exists a $k$-[[linear function]] \begin{displaymath} S : A \to A \end{displaymath} called the \textbf{antipode} or \textbf{coinverse} such that \begin{displaymath} m\circ(\mathrm{id}\otimes S)\circ \Delta = m\circ(S\otimes\mathrm{id})\circ\Delta = \eta\circ\epsilon \end{displaymath} (as a map $A\to A$). \end{defn} \begin{remark} \label{}\hypertarget{}{} If an antipode exists then it is unique, just the way that if inverses exist in a [[monoid]] then they are unique. One sometimes prefers to have a \textbf{skew-antipode} $\tilde{S}$ such that $h_{(2)}\tilde{S}(h_{(1)}) = \tilde{S}(h_{(2)}) h_{(1)} = (\eta\circ\epsilon) (h)$. If $S$ is an invertible antipode then $\tilde{S} - S^{-1}$ is a skew-antipode. $H$ is a bialgebra with a skew-antipode iff $H^{op}$ (the same vector space, opposite product, the same coproduct) is a Hopf algebra. The unit of a Hopf algebra is a [[grouplike element]], hence $S(1)1=1$, therefore $S(1)=1$. By linearity of $S$ this implies that $S\circ\eta\circ\epsilon = \eta\circ\epsilon$. \end{remark} \begin{prop} \label{}\hypertarget{}{} The antipode is an antihomomorphism both of algebras and coalgebras (i.e. a homomorphism $S:A\to A^{cop}_{op}$). \end{prop} \begin{proof} In Sweedler notation, for any $g,h\in A$, \begin{displaymath} S((h g)_{(1)}) (h g)_{(2)} = \epsilon(h g) \end{displaymath} \begin{displaymath} S((h g)_{(1)}) h_{(2)}g_{(2)} = \epsilon(h)\epsilon(g) \end{displaymath} \begin{displaymath} S((h g)_{(1)}) h_{(2)}g_{(2)} S g_{(3)} S h_{(3)} = \epsilon(h_{(1)})\epsilon(g_{(1)}) S g_{(2)} S h_{(2)} \end{displaymath} \begin{displaymath} S(h_{(1)}g_{(1)}) \epsilon(h_{(2)})\epsilon(g_{(2)}) = (S g) (S h) \end{displaymath} \begin{displaymath} S(h_{(1)}\epsilon(h_{(2)})g_{(1)}\epsilon(g_{(2)})) = (S g)(S h) \end{displaymath} Therefore $S(h g) = (S g) (S h)$. For the coalgebra part, notice first that $\epsilon(h)1\otimes 1 = \tau\circ\Delta(\epsilon(h)1)=\tau\circ\Delta(S h_{(1)}h_{(2)})$. Expand this as \begin{displaymath} (S h_{(1)}\otimes S h_{(2)})(h_{(4)}\otimes h_{(3)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(h_{(4)}\otimes h_{(3)}) \end{displaymath} \begin{displaymath} (S h_{(1)}\otimes S h_{(2)})(h_{(4)}\otimes h_{(3)}) (S h_{(5)}\otimes S h_{(6)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(h_{(3)}\otimes h_{(2)})(S h_{(4)}\otimes S h_{(5)}) \end{displaymath} \begin{displaymath} ((S h_{(1)}\otimes S h_{(2)})\epsilon(h_{(3)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(1\otimes\epsilon(h_{(2)})) \end{displaymath} \begin{displaymath} ((S h_{(1)}\otimes S h_{(2)})\epsilon(h_{(3)}) = (\tau\circ\Delta)(S h_{(1)})(1\otimes\epsilon(h_{(2)})1) = (\tau\circ\Delta)(S h_{(1)}\epsilon(h_{(2)})) \end{displaymath} \begin{displaymath} (S h_{(1)}\otimes S h_{(2)})=(\tau\circ\Delta)(S h) = (S h)_{(2)}\otimes (S h)_{(1)}. \end{displaymath} \end{proof} The axiom that must be satisfied by the antipode looks like a $k$-linear version of the identity satisfied by the inverse map in a group bimonoid: taking a group element $g$, duplicating by the diagonal map $\Delta$ to obtain $(g,g)$, taking the inverse of either component of this ordered pair, and then multiplying the two components, we obtain the identity element of our group. Just as an [[algebra]] is a [[monoid]] in [[Vect]] and a [[bialgebra]] is a [[bimonoid]] in $Vect$, a Hopf algebra is a [[Hopf monoid]] in $Vect$. \begin{remark} \label{}\hypertarget{}{} \textbf{Caution: convention in topology} In [[algebraic topology]], it is common to define Hopf algebras without mentioning the antipode, since in many topological cases of interest it exists automatically. For example, this is the case if it is [[graded object|graded]] and ``connected'' in the sense that its degree-0 part is just the [[ground field]] (a property possessed by the homology or cohomology of any connected space). In algebraic topology also the strict coassociativity is not always taken for granted. \end{remark} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{relation_to_hopf_groups}{}\subsubsection*{{Relation to Hopf groups}}\label{relation_to_hopf_groups} Note that the definition of Hopf algebra (or, really, of [[Hopf monoid]]) is [[duality|self-dual]]: a Hopf monoid in a [[symmetric monoidal category]] $V$ is the same as a Hopf monoid in the [[opposite category]] $V^{op}$ (i.e. a ``Hopf comonoid''). Thus we can view a Hopf algebra as ``like a group'' in two different ways, depending on whether the group multiplication corresponds to the multiplication or the comultiplication of the Hopf algebra. The formal connections between Hopf monoids and group objects are: \begin{enumerate}% \item A Hopf monoid in a [[cartesian monoidal category]] $V$ is the same as a group object in $V$. Such Hopf monoids are always \emph{cocommutative} (that is, their underlying comonoid is cocommutative). This is because every object of a cartesian monoidal category is a cocommutative comonoid object in a unique way, and every morphism is a comonoid homomorphism. \item A \emph{commutative} Hopf monoid in a [[symmetric monoidal category]] $V$ is the same as a [[group object]] in the [[opposite category]] $CMon(V)^{op}$, where $CMon(V)$ is the category of commutative monoids in $V$, hence a [[cogroup]] object in $CMon(V)$ (a point highlighted by [[Haynes Miller]], see (\hyperlink{Ravenel86}{Ravenel 86, appendix A1})). This works because the tensor product of commutative algebras is the categorical [[coproduct]], and hence the [[product]] in its [[opposite category]]. In particular, a [[commutative Hopf algebra]] is the same as a group object in the category $Alg^{op}$ of affine schemes. \end{enumerate} Corresponding to these two, an ordinary group $G$ gives us two different Hopf algebras (here $k$ is the [[ground ring]]): \begin{enumerate}% \item The [[group algebra]] $k[G]$ (the free vector space on the set $G$), with multiplication given by the group operation of $G$ and comultiplication given by the diagonal $g\mapsto g\otimes g$. This Hopf algebra is always cocommutative, and is commutative iff $G$ is abelian. It can be viewed as the result of applying the [[strong monoidal functor]] $k[-]:Set \to k Mod$ to the Hopf monoid $G$ in $Set$. \item The function algebra $k(G)$ (the set of functions $G\to k$), with comultiplication given by precomposition with the group operation \begin{displaymath} k(G) \to k(G\times G) \cong k(G)\otimes k(G), \end{displaymath} and multiplication given by pointwise multiplication in $k$. In this case we need some finiteness or algebraicity of $G$ in order to guarantee $k(G\times G) \cong k(G)\otimes k(G)$. This Hopf algebra is always commutative, and is cocommutative iff $G$ is abelian. \end{enumerate} Note that if $G$ is finite, then $k[G]\cong k(G)$ as $k$-modules, but the Hopf algebra structure is quite different. Mike, can you do something with these notes that I took at some point as a grad student? I don't know this stuff very well, which is why I don't incorporate them into the text, but at least I cleaned up the formatting a bit so that you can if you like it. ---Toby \begin{quote}% One can make a group into a Hopf algebra in at least $2$ very different ways. Both ways have a discrete version and a smooth version. Given a (finite, discrete) group $G$ and a ground ring (field?) $K$, then the group ring $K[G]$ is a cocommutative Hopf algebra, with $M(g_0,g_1) = g_0 g_1$, $I = 1$, $D(g) = g \otimes g$, $E(g) = 1$, and the nifty Hopf antipodal operator $S(g) = g^{-1}$. Notice that the coalgebra operations $D,E$ depend only on $Set|G|$. Given a (finite, discrete) group $G$ and a ground ring (field?) $K$, then the function ring $Fun(G,K)$ is a [[commutative Hopf algebra]], with $M(f_0,f_1)(g) = f_0(g)f_1(g)$, $I(g) = 1$, $D(f)(g,h) = f(g h)$, $E(f) = f(1)$, and the nifty Hopf antipodal operator $S(f)(g) = f(g^{-1})$. Notice that the algebra operations $M,I$ depend only on $Set|G|$. Given a (simply connected) Lie group $G$ and the complex (real?) field $K$, then the universal enveloping algebra $U(G)$ is a cocommutative Hopf algebra, with $M(\mathbf{g}_0,\mathbf{g}_1) = \mathbf{g}_0 \mathbf{g}_1$, $I = 1$, $D(\mathbf{g}) = \mathbf{g} \otimes 1 + 1 \otimes \mathbf{g}$, $E(\mathbf{g}) = 0$, and the nifty Hopf antipodal operator $S(\mathbf{g}) = -\mathbf{g}$. Notice that the coalgebra operation $D,E$ depend only on $K Vect|\mathfrak{g}|$. Given a (compact) Lie group $G$ and the complex (real?) field $K$, then the algebraic function ring $Anal(G)$ is a cocommutative Hopf algebra, with $M(f_0,f_1)(g) = f_0(g) f_1(g)$, $I(g) = 1$, $D(f)(g,h) = f(g h)$, $E(f) = f(1)$, and the nifty Hopf antipodal operator $S(f)(g) = f(g^{-1})$. Notice that the algebra operations $M,I$ depend only on $Anal Man|G|$. \end{quote} \hypertarget{the_theorem_of_hopf_modules}{}\subsubsection*{{The theorem of Hopf modules}}\label{the_theorem_of_hopf_modules} Hopf algebras can be characterized among bialgebras by the fundamental theorem on [[Hopf module|Hopf modules]]: the category of Hopf modules over a bialgebra is canonically equivalent to the category of vector spaces over the ground ring iff the bialgebra is a Hopf algebra. This categorical fact enables a definition of Hopf monoids in some setups that do not allow a sensible definition of antipode. \hypertarget{relation_to_lie_algebras}{}\subsubsection*{{Relation to Lie algebras}}\label{relation_to_lie_algebras} \begin{itemize}% \item [[Milnor-Moore theorem]] \end{itemize} \hypertarget{TannakaDuality}{}\subsubsection*{{Tannaka duality}}\label{TannakaDuality} The [[category of modules]] (finite dimensional) over the underlying [[associative algebra]] of a Hopf algebra canonically inherits the structure of an [[rigid monoidal category|rigid]] [[monoidal category]] such that the forgetful [[fiber functor]] to [[vector spaces]] over the ground field is a [[strict monoidal functor]]. The statement of [[Tannaka duality]] for Hopf algebras is that this property characterizes Hopf algebras. (See for instance (\hyperlink{Bakke}{Bakke})) For generalization of this characterization to [[quasi-Hopf algebras]] and [[hopfish algebras]] see (\hyperlink{Vercruysse}{Vercruysse}). [[!include structure on algebras and their module categories - table]] \hypertarget{as_3vector_spaces}{}\subsubsection*{{As 3-vector spaces}}\label{as_3vector_spaces} A Hopf algebra structure on an [[associative algebra]] $A$ canonically defines on $A$ the structure of an algebra object [[internalization|internal]] to the [[2-category]] of algebras, [[bimodule]]s and bimodule homomorphisms. By the discussion at [[n-vector space]] this allows to identify Hopf algebras with certain \emph{3-vector spaces} . (For instance (\href{http://ncatlab.org/nlab/show/Topological+Quantum+Field+Theories+from+Compact+Lie+Groups}{FHLT, p. 27})). More general 3-vector spaces are given by \emph{[[hopfish algebras]]} and generally by [[sesquiunital sesquialgebras]]. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item [[group algebra]] \item [[Steenrod algebra]] \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[quasi-Hopf algebra]] \item [[triangular Hopf algebra]] \item [[quasitriangular Hopf algebra]]/[[quantum group]] \item [[hopfish algebra]] \item [[Hopf algebroid]] \begin{itemize}% \item [[commutative Hopf algebroid]] \end{itemize} \item [[Hopf C-star algebra]] \item [[Hopf monoid]] \item [[Hopf monoidal category]] \item [[Hopf monad]] \item [[change of rings theorem]] \item [[Hopf ring spectrum]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Kenneth Brown]], \emph{Hopf algebras}, lectures, \href{http://www.maths.gla.ac.uk/~kab/Hopf%20lects%201-8.pdf}{pdf} \end{itemize} The diagrammatic definition of a Hopf algebra, is also in the \href{http://en.wikipedia.org/wiki/Hopf_algebra#Formal_definition}{Wikipedia entry}. \begin{itemize}% \item Eiichi Abe, \emph{Hopf algebras}, Cambridge UP 1980. \item [[Pierre Cartier]], \emph{A primer on Hopf algebras}, Frontiers in number theory, physics, and geometry II, 537--615 ; preprint IH\'E{}S 2006-40, 81p \href{http://preprints.ihes.fr/2006/M/M-06-40.pdf}{pdf} \href{http://www.math.osu.edu/~kerler.2/VIGRE/InvResPres-Sp07/Cartier-IHES.pdf}{pdf2loc} \item V. G. [[Drinfel'd]], \emph{Quantum groups}, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 2 798--820, AMS 1987, \href{http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.djvu}{djvu:1.3 M}, \href{http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.pdf}{pdf:2.5 M} \item G. Hochschild, \emph{Introduction to algebraic group schemes}, 1971 \item [[Shahn Majid]], \emph{Foundations of quantum group theory}, Cambridge University Press 1995, 2000. \item [[John Milnor]], [[John Moore]], \emph{On the structure of Hopf algebras}, Annals of Math. \textbf{81} (1965), 211-264 \href{http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf}{pdf} \item Susan Montgomery, \emph{Hopf algebras and their action on rings}, AMS 1994, 240p. \item B. Parshall, J.Wang, \emph{Quantum linear groups}, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp. \item [[Moss Sweedler]], \emph{Hopf algebras}, Benjamin 1969. \item William C. Waterhouse, \emph{Introduction to affine group schemes}, Graduate Texts in Mathematics \textbf{66}, Springer 1979. xi+164 pp. \end{itemize} [[Tannaka duality]] for Hopf algebras and their generalization is alluded to in \begin{itemize}% \item Joost Vercruysse, \emph{Hopf algebras---Variant notions and reconstruction theorems} (\href{http://arxiv.org/abs/1202.3613}{arXiv:1202.3613}) \end{itemize} and discussed in detail in \begin{itemize}% \item T\o{}rris Kol\o{}en Bakke, \emph{Hopf algebras and monoidal categories} (2007) (\href{https://munin.uit.no/bitstream/handle/10037/1084/finalthesis.pdf}{pdf}) \end{itemize} Discussion with an eye towards [[stable homotopy theory]] and the [[Steenrod algebra]] is in \begin{itemize}% \item [[Doug Ravenel]], appendix A1 of \emph{[[Complex cobordism and stable homotopy groups of spheres]]}, 1986 (\href{http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenelA1.pdf}{pdf}) \end{itemize} [[!redirects Hopf algebras]] [[!redirects antipode]] [[!redirects skew-antipode]] \end{document}