Kenneth Brown - Academia.edu (original) (raw)

Papers by Kenneth Brown

arXiv (Cornell University), Oct 22, 2018

Let C be a decomposable plane curve over an algebraically closed field k of characteristic 0. Tha... more Let C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, C is defined in k 2 by an equation of the form g(x) = f (y), where g and f are polynomials of degree at least 2. We use this data to construct 3 pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, and the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring O(C) of C as a quantum homogeneous space, and prove this when each of g and f has degree at most 5 or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree 3 A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(−1, −1, 0).

Transactions of the American Mathematical Society, 1996

We study the prime ideal spaces of the quantized function algebras R q [ G ] R_{q}[G] , for G G a... more We study the prime ideal spaces of the quantized function algebras R q [ G ] R_{q}[G] , for G G a semisimple Lie group and q q an indeterminate. Our method is to examine the structure of algebras satisfying a set of seven hypotheses, and then to demonstrate, using work of Joseph, Hodges and Levasseur, that the algebras R q [ G ] R_{q}[G] satisfy this list of assumptions. Rings satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkar’s strong second layer condition. For such rings much representation-theoretic information is carried by the graph of links of the prime spectrum, and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions. Homogeneity is a key feature – it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms. Bounds on the ranks of these groups are obtained in the case of R q [ G ] R_{q}[G] . In the final section the re...

Lectures on Algebraic Quantum Groups, 2002

We define bialgebras, Hopf Algebras, and related algebraic structures, largely following the orig... more We define bialgebras, Hopf Algebras, and related algebraic structures, largely following the original paper [?] of Milnor and Moore but incorporating various simplifications and amplifications. The reader is urged to recall our conventions on grading and commutativity from ??. The theme is the definition of algebraic structures by use of dual commutative diagrams. Thus the familiar concepts of algebra and module dualize to concepts of coalgebra and comodule, and the structures of algebra and coalgebra combine to give the notion of a bialgebra. Incorporating antipodes (sometimes called conjugations), we obtain the notion of a Hopf algebra. In the cocommutative case, bialgebras and Hopf algebras can be viewed as monoids and groups in the symmetric monoidal category of cocommutative coalgebras.

Thi s work is subjec t to copyright. Al l rights are reserved , whether the whol e or part of the... more Thi s work is subjec t to copyright. Al l rights are reserved , whether the whol e or part of the material is concerned, specificall y the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting , reproduction on microfilms or in other ways , and storage in data banks. For any kind of use permissio n of the copyright owner must be obtained.

Proceedings of the London Mathematical Society, 2002

Physical Review A, 2005

The role of mixed state entanglement in liquid-state nuclear magnetic resonance (NMR) quantum com... more The role of mixed state entanglement in liquid-state nuclear magnetic resonance (NMR) quantum computation is not yet well-understood. In particular, despite the success of quantum information processing with NMR, recent work has shown that quantum states used in most of those experiments were not entangled. This is because these states, derived by unitary transforms from the thermal equilibrium state, were too close to the maximally mixed state. We are thus motivated to determine whether a given NMR state is entanglable-that is, does there exist a unitary transform that entangles the state? The boundary between entanglable and nonentanglable thermal states is a function of the spin system size N and its temperature T. We provide new bounds on the location of this boundary using analytical and numerical methods; our tightest bound scales as N ∼ T , giving a lower bound requiring at least N ∼ 22, 000 proton spins to realize an entanglable thermal state at typical laboratory NMR magnetic fields. These bounds are tighter than known bounds on the entanglability of effective pure states.

Journal of Algebra, 2008

We show how the existence of a PBW-basis and a large enough central subalgebra can be used to ded... more We show how the existence of a PBW-basis and a large enough central subalgebra can be used to deduce that an algebra is Frobenius. This is done by considering the examples of rational Cherednik algebras, Hecke algebras, quantised universal enveloping algebras, quantum Borels and quantised function algebras. In particular, we give a positive answer to [34, Problem 6] stating that the restricted rational Cherednik algebra at the value t = 0 is symmetric.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2003

Trends in Representation Theory of Algebras and Related Topics

Advances in Mathematics, 2006

The standard Poisson structure on the rectangular matrix variety M m,n (C) is investigated, via t... more The standard Poisson structure on the rectangular matrix variety M m,n (C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T ⊂ GL m+n (C). These orbits, finite in number, are shown to be smooth irreducible locally closed subvarieties of M m,n (C), isomorphic to intersections of dual Schubert cells in the full flag variety of GL m+n (C). Three different presentations of the T-orbits of symplectic leaves in M m,n (C) are obtained-(a) as pullbacks of Bruhat cells in GL m+n (C) under a particular map; (b) in terms of rank conditions on rectangular submatrices; and (c) as matrix products of sets similar to double Bruhat cells in GL m (C) and GL n (C). In presentation (a), the orbits of leaves are parametrized by a subset of the Weyl group S m+n , such that inclusions of Zariski closures correspond to the Bruhat order. Presentation (b) allows explicit calculations of orbits. From presentation (c) it follows that, up to Zariski closure, each orbit of leaves is a matrix product of one orbit with a fixed column-echelon form and one with a fixed rowechelon form. Finally, decompositions of generalized double Bruhat cells in M m,n (C) (with respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves are obtained.

Transactions of the American Mathematical Society, 2006

Necessary and sufficient conditions are given for the completed group algebras of a compact p p -... more Necessary and sufficient conditions are given for the completed group algebras of a compact p p -adic analytic group with coefficient ring the p p -adic integers or the field of p p elements to be prime, semiprime and a domain. Necessary and sufficient conditions are found for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coates et al.

Pacific Journal of Mathematics, 1979

We are concerned in this paper with the following question: When is the maximal right quotient ri... more We are concerned in this paper with the following question: When is the maximal right quotient ring of the group algebra kG a right self-injective ring? In general, the maximal right quotient ring Q(R) of a ring R is a right iέ-submodule of the right injective hull E{R) of R, and we may rephrase our question as: When does Q(kG) =E{kG)1 Of course, a sufficient condition for this to occur is that kG be right nonsingular, so that, for example, E(kG)-Q(kG) when k is a field of characteristic zero. However, Q(kG) is often injective even when kG is a singular ring; for example, when G is finite, it is well-known that kG is itself an injective ring.

Mathematische Zeitschrift, 2001

Let H be a Hopf algebra over the field k which is a finite module over a central affine sub-Hopf ... more Let H be a Hopf algebra over the field k which is a finite module over a central affine sub-Hopf algebra R. Examples include enveloping algebras U (g) of finite dimensional k-Lie algebras g in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity. The ramification behaviour of the maximal ideals of Z(H) with respect to the subalgebra R is studied, and the conclusions are then applied to the cases of classical and quantised enveloping algebras. In the case of U (g) for g semisimple a conjecture of Humphreys [27] on the block structure of U (g) is confirmed. In the case of Uǫ(g) for g semisimple and ǫ an odd root of unity we obtain a quantum analogue of a result of Mirković and Rumynin, [34], and we fully describe the factor algebras lying over the regular sheet, [9]. The blocks of Uǫ(g) are determined, and a necessary condition (which may also be sufficient) for a baby Verma Uǫ(g)-module to be simple is obtained.

Journal of Algebra, 1997

We prove that a Noetherian Hopf algebra of finite global dimension possesses further attractive h... more We prove that a Noetherian Hopf algebra of finite global dimension possesses further attractive homological properties, at least when it satisfies a polynomial identity. This applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity, as well as to classical enveloping algebras in positive characteristic. In all three cases we show that these algebras are Auslander-regular and Macaulay. We derive representation theoretic consequences concerning the coincidence of the non-Azumaya and singular loci for each of the above three classes of algebras.

American Heart Journal, 1998

The prognostic value of exercise thallium-201 imaging has been well established in referral patie... more The prognostic value of exercise thallium-201 imaging has been well established in referral patient populations at tertiary care centers, but these results may be influenced by referral bias. This study was performed to evaluate the prognostic value of thallium imaging in a community-based population of 446 residents of Olmsted County, Minn. Eleven variables were prospectively selected and tested for their associations with outcome end points. Four variables (age, history of myocardial infarction, number of abnormal thallium segments on the postexercise images, and increased thallium lung uptake) contained the most independent prognostic information. For the end point overall mortality rate, the multivariate chi-square values were 17.2 (p &amp;amp;amp;lt; 0.0001) for age and 20.9 (p &amp;amp;amp;lt; 0.0001) for the number of abnormal thallium segments on the postexercise images. Five-year survival rate for patients older than the median age of 59 years with an abnormal scan was 84% versus 97% for patients &amp;amp;amp;lt; or = 59 years of age with a normal scan. Exercise thallium imaging was useful for prognostic purposes in this relatively low-risk community population, confirming the findings of referral population studies.

International journal for parasitology

Apicomplexan parasites secrete transmembrane (TM) adhesive proteins as part of the process leadin... more Apicomplexan parasites secrete transmembrane (TM) adhesive proteins as part of the process leading to host cell attachment and invasion. These microneme proteins are cleaved in their TM domains by an unidentified protease termed ...