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Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras. (English. French summary) Zbl 1292.16024

Let \(k\) be an infinite field, \(H\) a Hopf \(k\)-algebra with coproduct \(\Delta(x)=x_1\otimes x_2\) and \(A\) an \(H\)-comodule algebra. For each positive integer \(i\), let \(X_i^H\) be a copy of \(H\). The identity map from \(H\) to \(X_i^H\) sends \(x\in H\) to \(X_i^x\). Then the tensor algebra \(T\) on \(X_H=\bigoplus_iX_i^H\) is isomorphic to the algebra of non-commutative polynomials in the indeterminates \(X_i^{x_r}\) where \(\{x_r\}_r\) is a linear basis of \(H\), and \(T\) has an \(H\)-comodule algebra structure with a coaction \(\delta\colon T\to T\otimes H\) by \(\delta(X_i^x)=X_i^{x_1}\otimes x_2\).
An element \(P\in T\) is a polynomial \(H\)-identity for \(A\) if \(\mu(P)=0\) for all \(H\)-comodule algebra maps \(\mu\colon T\to A\). Denote the set of all polynomial \(H\)-identities for \(A\) by \(Id_H(A)\). Let \(\alpha\) be a two-cocycle on \(H\), and \(\mu_H\) a copy of the underlying vector space \(H\). Denote the identity map \(\mu\colon H\to\mu_H\) by \(x\to\mu_x\) for \(x\in H\). Then the algebra \(^\alpha H\) is the algebra with multiplication \(\mu_x\mu_y=\alpha(x_1,y_1)\mu_{x_2y_2}\) for all \(x,y\in H\), and \(^\alpha H\) is an \(H\)-comodule algebra with coaction \(\delta(\mu_x)=\mu_{x_1}\otimes x_2\), and a Galois object over \(H\). Moreover, let \(t_i^H\) be a copy of \(H\) by identifying \(x\in H\) linearly with \(t_i^x\in t_i^H\) and \(S\) the symmetric algebra on \(\bigoplus_it_i^H\). Then the map \(\mu_\alpha\colon T\to S\otimes^\alpha H\) by \(X_i^x\to t_i^{x_1}\otimes\mu_{x_2}\) is an \(H\)-comodule algebra map where the coaction \(\delta(t_i^x\mu_y)=t^x\mu_{y_1}\otimes y_2\).
Then it is shown that \(P\in T\) is a polynomial \(H\)-identity for \(^\alpha H\) if and only if \(\mu_\alpha(P)=0\). For \(H=H_{n^2}\), (the Taft algebra), and \(E(n)\) generated by \(x,y_1,\ldots,y_n\) subject to \(x^2=1\), \(y_i^2=0\), \(y_ix+xy_i=0\), \(y_iy_j+y_jy_i=0\) for all \(i,j\), the Galois objects over \(H\) are described, and for Galois objects \(A,B\), \(Id_H(A)=Id_H(B)\) if and only if \(A\cong B\) as comodule algebras.

MSC:

16T05 Hopf algebras and their applications
16T15 Coalgebras and comodules; corings
16R50 Other kinds of identities (generalized polynomial, rational, involution)
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References:

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