Hopf algebra

A Hopf algebra over a field $k$ is a bialgebra $H$ with an antipode $S$ satisfying $m(S\otimes 1)\Delta =u\epsilon =m(1\otimes S)\Delta$, where $m,u,\Delta ,\epsilon$ denote multiplication, unit, coproduct, and counit. These identities generalize taking an inverse.

For a group algebra $k[G]$, define $\Delta (g)=g\otimes g$, $\epsilon (g)=1$, and $S(g)=g^{-1}$, then extend linearly. The coproduct makes a group element act simultaneously on both factors of a tensor product of representations. A Hopf algebra therefore packages not only multiplication but also how symmetries act on products and how they are reversed.

A quantum group is a type of algebraic symmetry commonly encoded by a Hopf algebra deforming a classical symmetry algebra. The term covers several related constructions; it does not specify a single additional Hopf algebra axiom. A standard example over $\mathbb{C}$, for $q\ne 0,\pm 1$, is $U_q({\mathfrak{sl}}_2)$ with generators $E,F,K,K^{-1}$ and relationsIts coproduct can be taken as $\Delta K=K\otimes K$, $\Delta E=E\otimes K+1\otimes E$, and $\Delta F=F\otimes 1+K^{-1}\otimes F$.^[ref]

These formulas make the symmetry act on tensor products of representations. In particular, the coproduct is generally not unchanged by swapping its two tensor factors. Formally writing $K=q^h$ and letting $q$ approach one recovers the familiar Lie bracket relations for ${\mathfrak{sl}}_2$. Making that limit rigorous requires choosing a suitable algebra over a parameter ring.

An element $x$ of a Hopf algebra $H$ is primitive whenNo grading is required. Applying the counit to either tensor factor shows $\epsilon (x)=0$. If $x,y$ are primitive, the algebra-homomorphism property of comultiplication givesbecause elements in the two separate tensor factors commute. Thus the primitive elements form a Lie algebra under the commutator.

In a universal enveloping algebra over $\mathbb{Q}$, the primitive elements are exactly the original Lie algebra. The Poincaré-Birkhoff-Witt theorem identifies the associated graded algebra with a symmetric algebra; in characteristic zero, a nonzero homogeneous polynomial of degree greater than one cannot be additive under the diagonal. The highest-degree part of a primitive element must therefore have degree one. Positive characteristic of a field changes this conclusion: if $x$ is primitive in characteristic $p$, then $x^p$ is primitive too, since the intermediate binomial coefficients vanish modulo $p$.

An algebra over a field acted on by a bialgebra is a module algebra when multiplication and unit are module maps: the displayed product identity holds and $h1=\epsilon (h)1$. This expresses the product rule determined by comultiplication.

Over a field with $q\ne 0$ and $q^2\ne 1$, this Hopf algebra has generators $E,F,K,K^{-1}$ and relations $KE=q^{2}EK$, $KF=q^{-2}FK$ and $EF-FE=(K-K^{-1})/(q-q^{-1})$. One comultiplication convention is $\Delta E=E\otimes K+1\otimes E$, $\Delta F=F\otimes 1+K^{-1}\otimes F$ and $\Delta K=K\otimes K$. Its counit takes $E,F$ to zero and $K$ to one. The antipode takes $E$ to $-E{K}^{-1}$, $F$ to $-KF$ and $K$ to $K^{-1}$.

If $Ev=0$ and $Kv=\lambda v$, induction using the defining commutator gives the displayed identity. When $q$ is not a root of unity, a finite string of length $n+1$ in a simple module has $\lambda =ϵq^n$, $ϵ=\pm 1$. Its distinct $K$ eigenvalues make the string vectors independent, and the nonzero raising coefficients prove simplicity.

The symmetric quantum integer is the displayed Laurent polynomial, with $[0{]}_q=0$. It obeys $[n+1{]}_q=q^{-1}[n{]}_q+q^n$ and $[r{]}_q[s+1{]}_q-[s{]}_q[r+1{]}_q=[r-s{]}_q$. At a primitive odd $\ell$th root of unity, $[n{]}_q=0$ exactly when $\ell$ divides $n$.

Given a Yang-Baxter R-matrix, take generators $t_{ij}$ subject to the displayed matrix relation, with $T_1=T\otimes I$, $T_2=I\otimes T$. The coproduct $\Delta t_{ij}={\sum }_k{t}_{ik}\otimes t_{kj}$ and counit $\epsilon (t_{ij})={\delta }_{ij}$ give a bialgebra. An antipode requires additional inverses or an appropriate determinant quotient; an arbitrary RTT bialgebra is not automatically a Hopf algebra.

Given an invertible braiding $C$ on $V\otimes V$, the universal bialgebra coacting on $V$ for which $C$ is a comodule map is generated by the matrix coefficients $t_{ij}$. Its comultiplication is $\Delta t_{ij}={\sum }_k{t}_{ik}\otimes t_{kj}$. Colinearity gives quadratic relations. Writing $R=\tau C$ turns the relation into the familiar RTT construction with the appropriate ordered tensor factors. A Yang-Baxter R-matrix supplies a scalar coquasitriangular structure.

For the two-by-two RTT construction with the standard Yang-Baxter R-matrix, the displayed determinant commutes with the matrix generators and satisfies $\Delta D_q=D_q\otimes D_q$ and $\epsilon (D_q)=1$. Imposing $D_q=1$ gives the coordinate Hopf algebra of quantum SL2; adjoining its inverse instead gives the quantum general-linear coordinate algebra.

The generators $a,b,c,d$ obey $ab=qba$, $ac=qca$, $bd=qdb$, $cd=qdc$, $bc=cb$, $ad-da=(q-q^{-1})bc$ and $ad-qbc=1$. Matrix multiplication gives the coproduct, and the antipode matrix is $\left(\begin{array}{cc}d& -q^{-1}b\\ -qc& a\end{array}\right)$. For $q\ne 0,\pm 1$, its quotient with $c=0$, $a=u$, $d=u^{-1}$ and $b=v$ is the quantum Laurent-plane algebra $uv=qvu$, proving noncommutativity. The same quotient distinguishes $\Delta b$ from its flipped coproduct.

A Hopf algebra is cocommutative when flipping the two tensor factors leaves its coproduct unchanged. A group algebra has coproduct $g\mapsto g\otimes g$, so it is cocommutative even for a nonabelian group, whose group algebra is not commutative.

A Hopf algebra is commutative when its multiplication is commutative. This is independent of whether its coproduct is cocommutative; function algebras of finite nonabelian groups give commutative but noncocommutative examples.

Functions on a finite group form a Hopf algebra under pointwise multiplication, with $(\Delta f)(g,h)=f(gh)$, $\epsilon (f)=f(1)$ and $(Sf)(g)=f(g^{-1})$. The coproduct is cocommutative exactly when the group is abelian, since delta functions distinguish distinct products $gh,hg$.

For a left module over a Hopf algebra, a vector $v$ is invariant when $hv=\epsilon (h)v$. In the internal hom for modules over a Hopf algebra, invariant maps are exactly the R-module homomorphisms for the acting algebra.

The full ${\mathrm{Hom}}_k(M,N)$ carries the action $(hf)(u)=\sum h_{(1)}f(S(h_{(2)})u)$, making it the internal hom right adjoint to $-\otimes M$. No finite-dimensionality or inverse antipode is needed.

The convolution inverse $S$ of the identity on a bialgebra: $m(S\otimes 1)\Delta =j\epsilon =m(1\otimes S)\Delta$. Existence makes the bialgebra a Hopf algebra; $S$ reverses multiplication.