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Register a new userRationality of the Hilbert series of Hopf-invariants of free algebras
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It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function, whenever the ground field has zero characteristic.
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We show that the invariants of a free associative algebra of finite rank under a linear action of a finite-dimensional Hopf algebra generated by group-like and skew-primitive elements form a finitely generated algebra exactly when the action is scalar. This generalizes an analogous result for group actions by automorphisms obtained by Dicks and Formanek, and Kharchenko.
Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}s variables that correspond to the g_is, i=1,...,r. Let F<x_{1,g_1},...,x_{r,g_r}> be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F<x_{1,g_1},...,x_{r,g_r}> which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function.
We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples.
In this paper we study the theory of cleft extensions for a weak bialgebra H. Among other results, we determine when two unitary crossed products of an algebra A by H are equivalent and we prove that if H is a weak Hopf algebra, then the categories of H-cleft extensions of an algebra A, and of unitary crossed products of A by H, are equivalent.
In this paper, we study the representations of the Hopf-Ore extensions $kG(chi^{-1}, a, 0)$ of group algebra $kG$, where $k$ is an algebraically closed field. We classify all finite dimensional simple $kG(chi^{-1}, a, 0)$-modules under the assumption $|chi|=infty$ and $|chi|=|chi(a)|<infty$ respectively, and all finite dimensional indecomposable $kG(chi^{-1}, a, 0)$-modules under the assumption that $kG$ is finite dimensional and semisimple, and $|chi|=|chi(a)|$. Moreover, we investigate the decomposition rules for the tensor product modules over $kG(chi^{-1}, a, 0)$ when char$(k)$=0. Finally, we consider the representations of some Hopf-Ore extension of the dihedral group algebra $kD_n$, where $n=2m$, $m>1$ odd, and char$(k)$=0. The Grothendieck ring and the Green ring of the Hopf-Ore extension are described respectively in terms of generators and relations.
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