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  "body": "\n## Example\n\nLet $n \\ge 2$ and let $k$ be a field that contains a [[primitive $n$-th root of unity|0J817RTVG6]] $\\omega$.\nThe *Taft algebra* $T_n(\\omega)$ is the $k$-algebra generated by $g$ and $x$ with the relations\n$$g^n = 1, \\qquad x^n = 0, \\qquad x g = \\omega\\, g x.$$\nIt has the $k$-basis $g^i x^j$ with $0 \\le i, j \\le n - 1$, so its dimension is $n^2$.\nIt is a [[Hopf algebra|7XFYP0N84H]] with\n$$\\Delta(g) = g \\otimes g, \\quad \\Delta(x) = x \\otimes 1 + g \\otimes x, \\quad \\varepsilon(g) = 1, \\quad \\varepsilon(x) = 0, \\quad S(g) = g^{-1}, \\quad S(x) = -g^{-1} x.$$\nThe check that $\\Delta$ respects the relations: $\\Delta(g)^n = g^n \\otimes g^n = 1 \\otimes 1$.\nFor $\\Delta(x) \\Delta(g) = \\omega\\, \\Delta(g) \\Delta(x)$, both sides equal $\\omega(gx \\otimes g + g^2 \\otimes gx)$ after using $xg = \\omega gx$.\nFor $\\Delta(x)^n = 0$, put $A = g \\otimes x$ and $B = x \\otimes 1$; then $BA = \\omega AB$, so the $q$-binomial formula with $q = \\omega$ gives $(A + B)^n = A^n + B^n$, and $A^n = g^n \\otimes x^n = 0$, $B^n = x^n \\otimes 1 = 0$.\nThe antipode check on $x$: $S(x) \\cdot 1 + S(g) x = -g^{-1}x + g^{-1}x = 0 = \\varepsilon(x)$, and $x S(1) + g S(x) = x - x = 0$.\n\n## Remarks\n\n$T_n(\\omega)$ is not commutative, since $xg = \\omega gx \\ne gx$.\nIt is not cocommutative, since the flip of $\\Delta(x)$ is $1 \\otimes x + x \\otimes g \\ne \\Delta(x)$.\nThe antipode satisfies $S^2(x) = g^{-1} x g = \\omega x$, so $S^2 \\ne \\id$; this shows that the antipode of a finite-dimensional Hopf algebra need not be an involution.\nFor $n = 2$ and $\\omega = -1$ (with $\\operatorname{char} k \\ne 2$) it is Sweedler's $4$-dimensional Hopf algebra, the smallest Hopf algebra that is neither commutative nor cocommutative.\n\n## Explanations\n\n### L1 human\n\nThe Taft algebra is a small, explicit Hopf algebra that is \"quantum\" in both directions.\nIt is built from a cyclic group element $g$ of order $n$ and a nilpotent element $x$ that do not commute but $q$-commute.\nMoving $x$ past $g$ costs a root of unity.\nBecause everything is finite and explicit, it is the standard first test case for statements about finite-dimensional Hopf algebras.\n",
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