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1
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Actions of Hopf algebras on pro-semisimple noetherian algebras and their invariants

100%
Tyc A.
Colloquium Mathematicum
|
2001
|
tom 88
|
nr 1
39-55
EN
Let H be a Hopf algebra over a field k such that every finite-dimensional (left) H-module is semisimple. We give a counterpart of the first fundamental theorem of the classical invariant theory for locally finite, finitely generated (commutative) H-module algebras, and for local, complete H-module algebras. Also, we prove that if H acts on the k-algebra A = k[[X₁,...,Xₙ]] in such a way that the unique maximal ideal in A is invariant, then the algebra of invariants $A^{H}$ is a noetherian Cohen-Macaulay ring.
2
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An elementary proof of the Weitzenböck Theorem

100%
Tyc A.
Colloquium Mathematicum
|
1998
|
tom 78
|
nr 1
123-132
3
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A decomposition theorem for complete comodule algebras over complete Hopf algebras

100%
Tyc A.
Colloquium Mathematicum
|
1997
|
tom 74
|
nr 2
243-252
4
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A note on semisimple derivations of commutative algebras

100%
Tyc A.
Colloquium Mathematicum
|
2005
|
tom 102
|
nr 2
263-270
EN
A concept of a slice of a semisimple derivation is introduced. Moreover, it is shown that a semisimple derivation d of a finitely generated commutative algebra A over an algebraically closed field of characteristic 0 is nothing other than an algebraic action of a torus on Max(A), and, using this, that in some cases the derivation d is linearizable or admits a maximal invariant ideal.
5
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Connected sequences of stable derived functors and their applications

68%
Simson D., Tyc A.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 1. Introduction........................................................................................................................................................................................................ 5 2. Category of complexes.................................................................................................................................................................................... 7 3. Left stable derived functors of covariant functors....................................................................................................................................... 11 4. Functors with, extensions............................................................................................................................................................................... 24 5. On the exactness of connected sequences................................................................................................................................................ 37 6. Right and left stable derived functors of contravariant functors. Right stable derived functors of covariant functors................... 39 7. Symmetric power functor $SP^n$ and exterior power functor $Λ^n$..................................................................................................... 43 8. On J. H. C. Whitehead's functor Γ.................................................................................................................................................................. 48 9. Computation of the modules $L^s_qSP^2(R)$, $L^s_qΛ^2(R)$ and $L^s_qΓ(R)$........................................................................... 53 10. Computation of the functors $L^s_qSP^2$, $L^s_qΛ^2$ and $L^s_qΓ$............................................................................................ 59 11. Eilenberg-MacLane's stable homology and cohomology functors...................................................................................................... 64 References............................................................................................................................................................................................................ 67
6
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The geometric reductivity of the quantum group $SL_{q}(2)$

64%
Kępa M., Tyc A.
Colloquium Mathematicum
|
2011
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tom 124
|
nr 2
169-190
EN
We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants $A^{G}$ is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple, and that in positive characteristic every finite-dimensional quantum group is geometrically reductive. Both the concept of geometrically reductive quantum group and the above mentioned theorems are formulated in the language of Hopf algebras and generalize the results of Borsai and Ferrer Santos. The main theorem of the paper says that in positive characteristic the quantum group $SL_{q}(2)$ is geometrically reductive for any parameter q.
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