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Noncommutative invariants of bialgebras. (English. Russian original) Zbl 0842.16025

Algebra Logic 33, No. 6, 366-380 (1994); translation from Algebra Logika 33, No. 6, 654-680 (1994).
Summary: Suppose that \(H\) is a bialgebra over a field \(C\) and \(R = C\langle V\rangle\) is the tensor algebra of the \(C\)-space \(V\) endowed with the structure of an \(H\)-module algebra, so that \(V\) is a submodule of the \(H\)-module \(R\), \(R^H\) is the algebra of \(H\)-invariants, and \(W\), the support of the algebra \(R^H\), is the smallest subspace of the \(C\)-space \(V\) such that \(R^H \subseteq C\langle W\rangle\). The main result of the paper is the theorem stating that if the algebra of \(H\)-invariants \(R^H\) is finitely generated, then the support of \(R^H\) is a finite-dimensional submodule of the \(H\)-module \(V\), whose elements are \(H\)-semi-invariants of the same weight.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
15A72 Vector and tensor algebra, theory of invariants
16W20 Automorphisms and endomorphisms
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References:

[1] W. Dicks and E. Formanek, ?Poincare series and a problem of S. Montgomery,?Lin. Mult. Algebra,12, No. 1, 21-30 (1983). · Zbl 0493.15020 · doi:10.1080/03081088208817467
[2] V. K. Kharchenko, ?Noncommutative invariants of finite groups and Noetherian varieties,?J. Pure Appl. Algebra,31, Nos. 1-3, 83-90 (1984). · Zbl 0529.16027 · doi:10.1016/0022-4049(84)90079-3
[3] A. N. Koryukin, ?Noncommutative invariants of reductive groups,?Algebra Logika,23, No. 4, 419-429 (1984). · Zbl 0587.20023
[4] T. Tambour,Connections between Commutative and Noncommutative Rings of Invariants, Preprint, Nos. 6 and 8, University of Lund and Lund Institute of Technology (1990).
[5] M. C. Wolf, ?Symmetric functions on noncommuting elements,?Duke Math. J.,2, No. 4, 626-637 (1936). · JFM 62.1103.01 · doi:10.1215/S0012-7094-36-00253-3
[6] A. T. Kolotov, ?Free subalgebras of free associative algebras,?Sib. Mat. Zh.,19, No. 2, 328-335 (1978). · Zbl 0412.49013 · doi:10.1007/BF00970518
[7] V. K. Kharchenko, ?Algebras of invariants of free algebras,?Algebra Logika,17, No. 4, 478-487 (1978).
[8] V. K. Kharchenko,Automorphisms and Derivations of Associative Rings, Kluwer (1991). · Zbl 0746.16002
[9] P. M. Cohn,Free Rings and Their Relations, Acad. Press, New York (1971). · Zbl 0232.16003
[10] E. Abe,Hopf Algebras, Cambridge University Press (1977).
[11] M. Sweedler,Hopf Algebras, Benjamin, New York (1969).
[12] J. A. Dieudonne, J. B. Carrell, and D. Mumford,Geometric Invariant Theory, Springer, Berlin (1965).
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