Sweedler's Hopf algebra

This is an interesting exercise in Etingof's book Introduction to Representation Theory.

(Problem 9.3.2) Let \(A\) be the algebra over complex numbers generated by elements \(g,x\) with defining relations \(gx = −xg,\, x^{2} = 0,\, g^{2} = 1\). Find the simple modules, the indecomposable projective modules, and the Cartan matrix of \(A\).

Proof

Recall that for any finite dimensional algebra over \(\CC\) with irreducible modules \(M_{i}\), we have the decomposition (Theorem 3.5.4)

\[\begin{equation} A/\radical{A}\simeq \End M_{i} \, . \end{equation}\]

We choose the basis of \(A\) as \(\set{1,g,x,gx}\), then the radical is \(\CC\linearspan{x,gx}\), and the decomposition is

\[\begin{equation} A/\radical{A}\simeq \CC p_{+} \os \CC p_{-} \, , \end{equation}\]

where \(p_{\pm}=\frac{1}{2}(1\pm g)\) are projectors of \(A/\radical{A}\). Hence there are two irreducible modules \(M_{\pm}\eqqq \CC p_{\pm}\) with the actions \(x=0, \, g=\pm 1\).

We can choose the lifting of \(p_{\pm}\) in \(A\) as \(p_{\pm}\), hence by Theorem 9.2.1 the projective covers of \(M_{\pm}\) are \(P_{\pm}\eqqq A p_{\pm}=\CC\linearspan{p_{\pm},xp_{\pm}}\), and the actions are

\[\begin{equation} x=\begin{bmatrix} 0&0\\1&0 \end{bmatrix} \, , \quad g=\begin{bmatrix} \pm1&0\\0&\mp 1 \end{bmatrix} \, . \end{equation}\]

Then we have the short exact sequences

\[\begin{equation} 0\to M_{\mp} \to P_{\pm} \to M_{\pm} \to 0 \, , \end{equation}\]

and the Cartan matrix of \(A\) is

\[\begin{equation} (\dim\Hom(P_{i},P_{j}))= \begin{bmatrix} 1&1\\1&1 \end{bmatrix} \, . \end{equation}\]

Remarks

Clifford algebra

\(A\) is the Clifford algebra with degenerate metric \(\diag(0,1)\). The projective covers are nontrivial spinors on the \(2d\) Newton-Cartan geometry.

Hopf algebra

\(A\) is the Sweedler's Hopf algebra, the smallest noncommutative noncocommutative Hopf algebra. Taft generalizes it to a family of finite dimensional Hopf algebras, called Taft algebra.

The comultiplication is

\[\begin{equation} \Delta g=g\ox g \, , \quad \Delta x=x\ox 1+g\ox x \, , \end{equation}\]

the counit is

\[\begin{equation} \epsilon(g)=1 \, , \quad \epsilon(x)=0 \, , \end{equation}\]

and the antipode is

\[\begin{equation} S(g)=g \, , \quad S(x)=-gx \, . \end{equation}\]

Quiver algebra

\(A\) is the quotient of the quiver with adjacent matrix

\[\begin{equation} \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix} \end{equation}\]

by the ideal generated by all length two paths, see https://arxiv.org/pdf/math/0009214. The path algebra is specified by

\[\begin{equation} e_{\pm}v_{\pm}=e_{\pm} \, , \quad v_{\mp}e_{\pm}=e_{\pm} \, , \quad v_{\pm}^{2}=v_{\pm} \, , \end{equation}\]

and after the quotient we have

\[\begin{equation} e_{i}e_{j}=0 \, . \end{equation}\]

The isomorphism is given by

\[\begin{equation} v_{\pm}=\frac{1\pm g}{2} \, , \quad e_{\pm}=x\frac{1\pm g}{2} \, . \end{equation}\]