This note discusses a framework for the investigation of the prime spectrum of an associative alg... more This note discusses a framework for the investigation of the prime spectrum of an associative algebra A that is equipped with an action of a Hopf algebra H . In particular, we study a notion of H -rationality for ideals of A and comment on a possible Dixmier-Moeglin equivalence for H -prime ideals of A .
Recently there has been some interest in so-called" Additivity Principles"(21 w... more Recently there has been some interest in so-called" Additivity Principles"(21 which, for a ring extension S c R and a prime ideal P of R, relate the Goldie rank of R/P to the Goldie ranks of S/Q, for all primes Q of S which are minimal over P n S. In this note, we prove such a ...
Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose ... more Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose center F contains k. We shall say that D can be reduced to r parameters if D = D_0 tensor_{F_0} F, where D_0 is a division algebra, the center F_0 of D_0 contains k and trdeg(F_0/k) = r. We show that every division algebra of odd degree n >= 5 can be reduced to at most (n-1)(n-2)/2 parameters. Moreover, every crossed product division algebra of degree n >= 4 can be reduced to at most (log_2(n) - 1)n + 1 parameters. Our proofs of these results rely on lattice-theoretic techniques.
Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose ... more Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose center F contains k. We shall say that D can be reduced to r parameters if we can write D ≃ D0 ⊗F 0 F , where D0 is a division algebra, the center F0 of D0 contains k and trdeg k (F0) = r. We show that every division algebra of odd degree n ≥ 5 can be reduced to ≤ 1 2 (n − 1)(n − 2) parameters. Moreover, every crossed product division algebra of degree n ≥ 4 can be reduced to ≤ (⌊log 2 (n)⌋ − 1)n + 1 parameters. Our proofs of these results rely on lattice-theoretic techniques.
Http Dx Doi Org 10 1080 00927879208824353, Jun 27, 2007
ABSTRACT Let be an algebra that is graded by a group G. Then the Hochschild and cyclic homologies... more ABSTRACT Let be an algebra that is graded by a group G. Then the Hochschild and cyclic homologies of S have canonical decompositions with components labeled by the set T(G) of conjugacy classes of G : for Hochschild homology and similarly for cyclic homology. In this article, we describe the components of Hochschild homology in the case where S is strongly G graded.The description is given in terms of a spectral sequence where Hq (R,Sg ) is the Hochschild homology of the identity component R = Se of S with coefficients in the bimodule Sg and Hp (CG (g),.) is the group homology of the centralizer CG (g) of g in G. If R is a separable algebra then the spectral sequence degenerates and yields an isomorphism
This note presents some results on projective modules and the Grothendieck groups K_0 and G_0 for... more This note presents some results on projective modules and the Grothendieck groups K_0 and G_0 for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions. Comment: 15 pages
Proceedings of the American Mathematical Society, 2014
We study rational actions of an algebraic torus G by automorphisms on an associative algebra R. T... more We study rational actions of an algebraic torus G by automorphisms on an associative algebra R. The G-action on R induces a stratification of the prime spectrum Spec R which was introduced by Goodearl and Letzter. For a noetherian algebra R, Goodearl and Letzter showed that the strata of Spec R are isomorphic to the spectra of certain commutative Laurent polynomial algebras. The purpose of this note is to give a new proof of this result which works for arbitrary algebras R.
Proceedings of the American Mathematical Society, 1995
Let B = A#σH denote a crossed product of the associative algebra A with the Hopf algebra H. We in... more Let B = A#σH denote a crossed product of the associative algebra A with the Hopf algebra H. We investigate the weak dimension and the global dimension of B and show that wdim B ≤ wdim H + wdim A and l.gldim B ≤ r.gldim H + l.gldim A.
This note presents some results on projective modules and the Grothendieck groups K0 and G0 for F... more This note presents some results on projective modules and the Grothendieck groups K0 and G0 for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions.
We give a simple proof of the Kac-Zhu class equation for semisimple Hopf algebras over an algebra... more We give a simple proof of the Kac-Zhu class equation for semisimple Hopf algebras over an algebraically closed field of characteristic 0.
This note discusses a framework for the investigation of the prime spectrum of an associative alg... more This note discusses a framework for the investigation of the prime spectrum of an associative algebra A that is equipped with an action of a Hopf algebra H . In particular, we study a notion of H -rationality for ideals of A and comment on a possible Dixmier-Moeglin equivalence for H -prime ideals of A .
Recently there has been some interest in so-called" Additivity Principles"(21 w... more Recently there has been some interest in so-called" Additivity Principles"(21 which, for a ring extension S c R and a prime ideal P of R, relate the Goldie rank of R/P to the Goldie ranks of S/Q, for all primes Q of S which are minimal over P n S. In this note, we prove such a ...
Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose ... more Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose center F contains k. We shall say that D can be reduced to r parameters if D = D_0 tensor_{F_0} F, where D_0 is a division algebra, the center F_0 of D_0 contains k and trdeg(F_0/k) = r. We show that every division algebra of odd degree n >= 5 can be reduced to at most (n-1)(n-2)/2 parameters. Moreover, every crossed product division algebra of degree n >= 4 can be reduced to at most (log_2(n) - 1)n + 1 parameters. Our proofs of these results rely on lattice-theoretic techniques.
Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose ... more Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose center F contains k. We shall say that D can be reduced to r parameters if we can write D ≃ D0 ⊗F 0 F , where D0 is a division algebra, the center F0 of D0 contains k and trdeg k (F0) = r. We show that every division algebra of odd degree n ≥ 5 can be reduced to ≤ 1 2 (n − 1)(n − 2) parameters. Moreover, every crossed product division algebra of degree n ≥ 4 can be reduced to ≤ (⌊log 2 (n)⌋ − 1)n + 1 parameters. Our proofs of these results rely on lattice-theoretic techniques.
Http Dx Doi Org 10 1080 00927879208824353, Jun 27, 2007
ABSTRACT Let be an algebra that is graded by a group G. Then the Hochschild and cyclic homologies... more ABSTRACT Let be an algebra that is graded by a group G. Then the Hochschild and cyclic homologies of S have canonical decompositions with components labeled by the set T(G) of conjugacy classes of G : for Hochschild homology and similarly for cyclic homology. In this article, we describe the components of Hochschild homology in the case where S is strongly G graded.The description is given in terms of a spectral sequence where Hq (R,Sg ) is the Hochschild homology of the identity component R = Se of S with coefficients in the bimodule Sg and Hp (CG (g),.) is the group homology of the centralizer CG (g) of g in G. If R is a separable algebra then the spectral sequence degenerates and yields an isomorphism
This note presents some results on projective modules and the Grothendieck groups K_0 and G_0 for... more This note presents some results on projective modules and the Grothendieck groups K_0 and G_0 for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions. Comment: 15 pages
Proceedings of the American Mathematical Society, 2014
We study rational actions of an algebraic torus G by automorphisms on an associative algebra R. T... more We study rational actions of an algebraic torus G by automorphisms on an associative algebra R. The G-action on R induces a stratification of the prime spectrum Spec R which was introduced by Goodearl and Letzter. For a noetherian algebra R, Goodearl and Letzter showed that the strata of Spec R are isomorphic to the spectra of certain commutative Laurent polynomial algebras. The purpose of this note is to give a new proof of this result which works for arbitrary algebras R.
Proceedings of the American Mathematical Society, 1995
Let B = A#σH denote a crossed product of the associative algebra A with the Hopf algebra H. We in... more Let B = A#σH denote a crossed product of the associative algebra A with the Hopf algebra H. We investigate the weak dimension and the global dimension of B and show that wdim B ≤ wdim H + wdim A and l.gldim B ≤ r.gldim H + l.gldim A.
This note presents some results on projective modules and the Grothendieck groups K0 and G0 for F... more This note presents some results on projective modules and the Grothendieck groups K0 and G0 for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions.
We give a simple proof of the Kac-Zhu class equation for semisimple Hopf algebras over an algebra... more We give a simple proof of the Kac-Zhu class equation for semisimple Hopf algebras over an algebraically closed field of characteristic 0.
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