Women in Noncommutative Algebra and Representation Theory (WINART) 2016
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Quantum Group Actions, Twisting Elements, and Deformations of Algebras
QUANTUM GROUP ACTIONS, TWISTING ELEMENTS, AND DEFORMATIONS OF ALGEBRAS GEORGIA BENKART AND SARAH WITHERSPOON Abstract. We construct twisting elements for module algebras of restricted two-parameter quantum groups from factors of their R-matrices. We general- ize the theory of Giaquinto and Zhang to universal deformation formulas for categories of module algebras and give examples arising from R-matrices of two-parameter quantum groups. 1. Introduction The algebraic deformation theory of Giaquinto and Zhang [GZ] laid out a frame- work for certain types of deformations of associative algebras, namely those arising from actions of bialgebras on algebras. This work generalized known examples of deformations coming from the action of a Lie algebra (equivalently of its universal enveloping algebra). Giaquinto and Zhang illustrated their ideas by constructing a new deformation formula from the action of a noncommutative bialgebra, namely, the universal enveloping algebra of a certain nonabelian Lie algebra. This Lie algebra is spanned by the n £ n matrix units E1;p; Ep;n for p = 1; : : : ; n, along with the n £ n diagonal matrices of trace 0, and so it is an abelian extension of a Heisenberg Lie algebra when n ¸ 3, and it is a two-dimensional nonabelian Lie al- gebra when n = 2. In recent work, Grunspan [G] applied the deformation formula for the n = 2 case to solve the open problem of giving an explicit deformation of the Witt algebra. In their study of deformations and orbifolds, C¸ald¸araru, Giaquinto, and the second author [CGW] constructed a deformation from a noncocommutative (and noncommutative) bialgebra { the Drinfeld double of a Taft algebra. -
Yetter-Drinfeld Modules Under Cocycle Twists
YETTER-DRINFELD MODULES UNDER COCYCLE TWISTS GEORGIA BENKART, MARIANA PEREIRA, AND SARAH WITHERSPOON To Susan Montgomery in honor of her distinguished career Abstract. We give an explicit formula for the correspondence between simple Yetter-Drinfeld modules for certain finite-dimensional pointed Hopf algebras H and those for cocycle twists Hσ of H. This implies an equivalence between modules for their Drinfeld doubles. To illustrate our results, we consider the restricted two-parameter quantum groups ur;s(sln) under conditions on the pa- rameters guaranteeing that ur;s(sln) is a Drinfeld double of its Borel subalgebra. We determine explicit correspondences between ur;s(sln)-modules for different values of r and s and provide examples where no such correspondence can exist. Our examples were obtained via the computer algebra system Singu- lar::Plural. 1. Introduction Radford [24] gave a construction of simple Yetter-Drinfeld modules for a pointed Hopf algebra H, whose group G(H) of grouplike elements is abelian, under fairly general hypotheses on H. These simple modules are in one-to-one correspondence with the Cartesian product of G(H) with its dual group, and are realized as vector subspaces of the Hopf algebra H itself. Radford and Schneider [25] generalized this method to include a much wider class of Hopf algebras given by cocycle twists of tensor product Hopf algebras. Again the simple modules are in one-to-one corre- spondence with a set of characters, but this time each simple module is realized as the quotient of a Verma module by its unique maximal submodule (equivalently, by the radical of the Shapovalov form) reminiscent of standard Lie-theoretic meth- ods. -
Indecomposable Coalgebras, Simple Comodules, And
proceedings of the american mathematical society Volume 123, Number 8, August 1995 INDECOMPOSABLE COALGEBRAS,SIMPLE COMODULES, AND POINTED HOPF ALGEBRAS SUSAN MONTGOMERY (Communicated by Lance W. Small) Abstract. We prove that every coalgebra C is a direct sum of coalgebras in such a way that the summands correspond to the connected components of the Ext quiver of the simple comodules of C . This result is used to prove that every pointed Hopf algebra is a crossed product of a group over the indecomposable component of the identity element. Introduction A basic structure theorem for cocommutative coalgebras asserts that any such coalgebra C is a direct sum of its irreducible components; as a consequence, it can be shown that any pointed cocommutative Hopf algebra is a skew group ring of the group G of group-like elements of H over the irreducible component of the identity element. These results were proved independently by Cartier and Gabriel and by Kostant in the early 1960's; see [Di, SI]. This paper is concerned with versions of these results for arbitrary coalgebras and for arbitrary (pointed) Hopf algebras. In fact much is already known about the coalgebra problem. In 1975 Ka- plansky [K] showed that any coalgebra C is (uniquely) a direct sum of inde- composable ones; moreover when C is cocommutative, the indecomposable components are irreducible. In 1978 Shudo and Miyamoto [ShM] defined an equivalence relation on the set of simple subcoalgebras of C and showed that the equivalence classes correspond to the indecomposable summands. A weaker version of this equivalence relation was studied recently in [XF]. -
Noncommutative Algebraic Geometry and Representation Theory January 24 to January 25, 2013 MSRI, Berkeley, CA, USA
Connections for Women: Noncommutative Algebraic Geometry and Representation Theory January 24 to January 25, 2013 MSRI, Berkeley, CA, USA Organizers: Georgia Benkart (University of Wisconsin) Ellen Kirkman* (Wake Forest University) Susan Sierra (Princeton University & University of Edinburgh) 1 of 22 Connections for Women: Noncommutative Algebraic Geometry and Representation Theory, January 24 - 25, 2013 at MSRI, Berkeley, CA USA REPORT ON THE MSRI WORKSHOP \CONNECTIONS FOR WOMEN: NONCOMMUTATIVE ALGEBRAIC GEOMETRY AND REPRESENTATION THEORY" JANUARY 24-25, 2013 Organizers: • Georgia Benkart (University of Wisconsin-Madison) • Ellen Kirkman (Wake Forest University) • Susan Sierra (University of Edinburgh) 1 Scientific description The Connections for Women Workshop had three overarching goals: (1) to provide an accessible introduction to the main themes of the MSRI semester- long program \Noncommutative Algebraic Geometry and Representation Theory" (NAGRT); (2) to bring together researchers in this program as well as in the year-long program in commutative algebra and the fall pro- gram in cluster algebras; and (3) to connect junior researchers, especially women and minorities, to senior researchers. The workshop preceded the five-day \Introductory Workshop" for the NAGRT program that was held at MSRI January 28-February 1, 2013. Noncommutative algebra impacts virtually every area of algebra and combinatorics as well as geometry, mathematical physics, and statistical mechanics, and has played a crucial role in solving open problems in these areas. The Connections Workshop featured eight hour-long lectures, which focused on fundamental ideas and open problems in noncommutative alge- bra, geometry, and representation theory. There were four poster sessions, each featuring the work of three or four early-career researchers. -
Meetings & Conferences of The
MEETINGS & CONFERENCES OF THE AMS MAY TABLE OF CONTENTS The Meetings and Conferences section of The most up-to-date meeting and confer- necessary to submit an electronic form, the Notices gives information on all AMS ence information can be found online at: although those who use L ATEX may submit meetings and conferences approved by www.ams.org/meetings/. abstracts with such coding, and all math press time for this issue. Please refer to Important Information About AMS displays and similarily coded material the page numbers cited on this page for Meetings: Potential organizers, (such as accent marks in text) must more detailed information on each event. speakers, and hosts should refer to be typeset in LATEX. Visit www.ams.org/ Invited Speakers and Special Sessions are page 88 in the January 2018 issue of the cgi-bin/abstracts/abstract.pl/. Ques- listed as soon as they are approved by the Notices for general information regard- tions about abstracts may be sent to abs- cognizant program committee; the codes ing participation in AMS meetings and [email protected]. Close attention should be listed are needed for electronic abstract conferences. paid to specified deadlines in this issue. submission. For some meetings the list Abstracts: Speakers should submit ab- Unfortunately, late abstracts cannot be may be incomplete. Information in this stracts on the easy-to-use interactive accommodated. A issue may be dated. Web form. No knowledge of LTEX is MEETINGS IN THIS ISSUE –––––– 2018 –––––––– –––––––– 2020 –––––––– January 15–18 Denver, Colorado p. 629 April 21–22 Boston, Massachusetts p. 620 June 11–14 Shanghai, People's Republic –––––––– 2021 –––––––– of China p. -
Poincaré-Birkhoff-Witt Theorems
POINCARE-BIRKHOFF-WITT´ THEOREMS ANNE V. SHEPLER AND SARAH WITHERSPOON Abstract. We sample some Poincar´e-Birkhoff-Witttheorems appearing in mathematics. Along the way, we compare modern techniques used to estab- lish such results, for example, the Composition-Diamond Lemma, Gr¨obner ba- sis theory, and the homological approaches of Braverman and Gaitsgory and of Polishchuk and Positselski. We discuss several contexts for PBW theorems and their applications, such as Drinfeld-Jimbo quantum groups, graded Hecke algebras, and symplectic reflection and related algebras. 1. Introduction In 1900, Poincar´e[76] published a fundamental result on Lie algebras that would prove a powerful tool in representation theory: A Lie algebra embeds into an asso- ciative algebra that behaves in many ways like a polynomial ring. Capelli [20] had proven a special case of this theorem, for the general linear Lie algebra, ten years earlier. In 1937, Birkhoff [10] and Witt [97] independently formulated and proved versions of the theorem that we use today, although neither author cited this earlier work. The result was called the Birkhoff-Witt Theorem for years and then later the Poincar´e-WittTheorem (see Cartan and Eilenberg [21]) before Bourbaki [14] prompted use of its current name, the Poincar´e-Birkhoff-WittTheorem. The original theorem on Lie algebras was greatly expanded over time by a num- ber of authors to describe various algebras, especially those defined by quadratic- type relations (including Koszul rings over semisimple algebras). Poincar´e-Birkhoff- Witt theorems are often used as a springboard for investigating the representation theory of algebras. These theorems are used to • reveal an algebra as a deformation of another, well-behaved algebra, • posit a convenient basis (of \monomials") for an algebra, and • endow an algebra with a canonical homogeneous (or graded) version. -
Proceedings JL
VOLUME 115 NUMBER 2 JUNE 1992 WHOLE NUMBER 396 Proceedings JL. of the I C A N M athematici SOCIETY EDITED BY William Adams Jeffry N. Kahn Dale Alspach Barbara Lee Keyfitz, J. Marshall Ash Coordinating Editor for ODE, Maurice Auslander PDE, Global Analysis, and Albert Baernstein Dynamical Systems Eric Bedford Irwin Kra, Managing Editor Andreas R. Blass Peter Li, Coordinating Editor Andrew M. Bruckner for Lie Groups, Topology, Christopher Croke and Geometry Clifford J. Earle, Jr., Wei Y. Loh Coordinating Editor for Joseph S. B. Mitchell Analysis and Operator Theory M. Susan Montgomery, Eric Friedlander Coordinating Editor for Theodore W. Gamelin Algebra, Number Theory, James Glimm, Coordinating and Combinatorics Editor for Applied Charles Pugh Mathematics, Probability, David Sharp and Statistics Hal L. Smith Roe Goodman Ronald M. Solomon Thomas Goodwillie Ronald Stern Lawrence Gray Franklin D. Tall Dennis A. Hejhal John Trangenstein Palle E. T. Jorgensen James West PROVIDENCE.RHODE ISLAND USA ISSN0002-9939 Proceedings of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Subscription information. Proceedings of the American Mathematical Society is pub- lished monthly. Subscription prices for Volumes 114-116 (1992) are $508 list; $406 institutional member; $305 individual member. A late charge of 10% of the subscrip- tion price will be imposed upon orders received from nonmember after January 1 of the subscription year. Subscribers outside the United States and India must pay a postage surcharge of $25; subscribers in India must pay a postage surcharge of $38. Expedited delivery to destinations in North America—$38; elsewhere—$90. -
CURRICULUM VITAE for SUSAN MONTGOMERY (October 16, 2019)
CURRICULUM VITAE for SUSAN MONTGOMERY (October 16, 2019) Education 1965 B. A. (summa cum laude) University of Michigan 1966 M. S. University of Chicago 1969 Ph.D. University of Chicago Academic Positions 1969 - 70 DePaul University (Assistant Professor) 1970 - University of Southern California (Assistant Professor 1970-75, Associate Professor 1975-82, Professor 1982-present, Chair 1996-99) 2017 - 20 Gabilan Distinguished Professor of Science and Engineering. Visiting Positions 1973 Visiting Assistant Professor, Hebrew University, Jerusalem (February-May) 1977 Visiting Associate Professor, University of Chicago (February-August); Visiting Research Associate, University of Leeds (September 1 - October 15); Fellow, Institute for Advanced Studies, Hebrew University (Oct. 15 - Jan. 15, 1978) 1981 Visiting Research Associate, University of Leeds (April-June) 1985 Visiting Scholar, UC San Diego (February-March); Member, MSRI, Berkeley (April- June); Honorary Fellow, University of Wisconsin, Madison (September-December) 1988 Visiting Professor, University of New South Wales, Sydney (July) 1993 Member, MSRI, Berkeley (June) 1996 Gastprofessor, Mathematisches Institut, University of Munich, Germany (January- June) 1999-2000 Co-organizer and member, Non-commutative Algebra Year, MSRI, Berkeley 2004 Member, Mittag-Leffler Institute, Sweden (March-April) Awards and Recognition 1964 Phi Beta Kappa (junior year) 1965 Woodrow Wilson Fellow 1965-69 NSF Graduate Fellow 1984 AMS Invited Address, Louisville, January 1984-85 John S. Guggenheim Memorial Foundation -
Meetings & Conferences of The
MEETINGS & CONFERENCES OF THE AMS MARCH TABLE OF CONTENTS The Meetings and Conferences section of The most up-to-date meeting and confer- necessary to submit an electronic form, the Notices gives information on all AMS ence information can be found online at: although those who use L ATEX may submit meetings and conferences approved by www.ams.org/meetings/. abstracts with such coding, and all math press time for this issue. Please refer to Important Information About AMS displays and similarily coded material the page numbers cited on this page for Meetings: Potential organizers, (such as accent marks in text) must more detailed information on each event. speakers, and hosts should refer to be typeset in LATEX. Visit www.ams.org/ Invited Speakers and Special Sessions are page 88 in the January 2018 issue of the cgi-bin/abstracts/abstract.pl/. Ques- listed as soon as they are approved by the Notices for general information regard- tions about abstracts may be sent to abs- cognizant program committee; the codes ing participation in AMS meetings and [email protected]. Close attention should be listed are needed for electronic abstract conferences. paid to specified deadlines in this issue. submission. For some meetings the list Abstracts: Speakers should submit ab- Unfortunately, late abstracts cannot be may be incomplete. Information in this stracts on the easy-to-use interactive accommodated. A issue may be dated. Web form. No knowledge of LTEX is MEETINGS IN THIS ISSUE –––––– 2018 –––––––– –––––––– 2019 –––––––– January 16–19 Baltimore, Maryland p. 380 March 17–18 Columbus, Ohio p. 373 March 15–17 Auburn, Alabama p. -
A-Inner Automorphisms of Filtered Algebras. Ii
PROCEEDINGS OK THE AMERICAN MATHEMATICAL SOCIETY Volume K7. Number 4. April I'M A-INNER AUTOMORPHISMS OF FILTERED ALGEBRAS. II SUSAN MONTGOMERY1 Abstract We commue earlier work and compute the A'-inner automorphisms of the ring of differential polynomials in one variable over an arbitrary domain. This is then applied to iterated Ore extensions. We also show that the ring of generic matrices has no nonidentity automorphisms which fix the center. In this paper we continue the study of A'-inner automorphisms of filtered algebras begun in [7]. which will be referred to as (I) in what follows. In (1) we described the A-inner automorphisms of the enveloping algebra of a Lie algebra, and of the ring of differential polynomials A = R[x; d] in one variable over a commutative domain R. Here we first show that the ring of generic matrices has only the trivial A-inner automorphism, equivalently, any automorphism fixing the center is the identity automorphism. We then extend our result from (I) on differential polynomials, allowing an arbitrary prime ring R as the coefficient ring. When R is a domain, we explicitly determine the group of A'-inner automorphisms of A = R[x; d] in terms of R and d. We then apply these results to certain iterated Ore extensions; in particular, if the original coefficient ring is commutative, then the group of all A'-inner automorphisms is abelian. In what follows, A will always denote a prime ring. Recall that o G Aul(A) is X-inner if it becomes inner when extended to the (right) Martindale quotient ring Q0( A) of A; when A is an Ore domain, o is A'-inner if and only if it becomes inner on the quotient division ring D of A. -
Reference, Volume 45, Number 5
reference.qxp 4/9/98 11:40 AM Page 632 Reference The Reference section of the Notices Ronald Coifman Board on Mathematical Sciences is intended to provide the reader with Yale University National Research Council frequently sought information in an Robert Fefferman 2101 Constitution Avenue, NW easily accessible manner. New infor- Washington, DC 20418 University of Chicago mation is printed as it becomes avail- telephone: 202-334-2421 able and is referenced after the first C. William Gear e-mail: [email protected] printing. As soon as information is up- NEC Research Institute Web: http://www2.nas.edu/ dated or otherwise changed, it will be M. Susan Montgomery bms/. noted in this section. University of Southern California Dianne O’Leary Mathematical Sciences Upcoming Deadlines Education Board, National University of Maryland August 15, 1998: Deadlines for ap- Research Council plications for NRC Associateship Pro- Alan Perelson Sherry Baca grams. Information available at the Santa Fe Institute Prescott Unified School District Web site http://rap.nas.edu/. William Pulleyblank Prescott, Arizona December 1, 1998: Deadline for IBM T. J. Watson Research Center applications for fellowship opportu- Deborah Loewenberg Ball nities in Asia offered by the NSF. Donald Richards University of Michigan World Wide Web: http://www. University of Virginia Hyman Bass (chair) twics.com/~nsftokyo/home.html Daniel Stroock Columbia University or by e-mail: [email protected]. Massachusetts Institute of Benjamin Blackhawk Technology St. Paul Academy and Summit Board on Mathematical DeWitt Sumners School Sciences, National Research St. Paul, Minnesota Council Florida State University Gail F. Burrill Robert MacPherson (chair) William Ziemer National Center for Research in Institute for Advanced Study Indiana University Mathematical Sciences Education Lynne Billard Daryl Pregibon (ex-officio) University of Wisconsin at University of Georgia AT&T Laboratories Madison Mary Ellen Bock Patricia Campbell Purdue University BMS Staff University of Maryland Fan Chung Graham John R. -
Proceedings JL
VOLUME 115 NUMBER 1 MAY 1992 WHOLE NUMBER 395 Proceedings JL. OFTHE AMERICAN MATHEMATICAL SOCIETY EDITED BY William Adams Jeffry N. Kahn Dale Alspach Barbara Lee Keyfitz, J. Marshall Ash Coordinating Editor for ODE, Maurice Auslander PDE, Global Analysis, and Albert Baernstein Dynamical Systems Eric Bedford Irwin Kra, Managing Editor Andreas R. Blass Peter Li, Coordinating Editor Andrew M. Bruckner for Lie Groups, Topology, Christopher Croke and Geometry Clifford J. Earle, Jr., Wei Y. Loh Coordinating Editor for Joseph S. B. Mitchell Analysis and Operator Theory M. Susan Montgomery, Eric Friedlander Coordinating Editor for Theodore W. Gamelin Algebra, Number Theory, James Glimm, Coordinating and Combinatorics Editor for Applied Charles Pugh Mathematics, Probability, David Sharp and Statistics Hal L. Smith Roe Goodman Ronald M. Solomon Thomas Goodwillie Ronald Stern Lawrence Gray Franklin D. Tall Dennis A. Hejhal John Trangenstein Palle E. T. Jorgensen James West PROVIDENCE,RHODE ISLAND USA ISSN0002-9939 Proceedings of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Subscription information. Proceedings of the American Mathematical Society is pub- lished monthly. Subscription prices for Volumes 114-116 (1992) are $508 list; $406 institutional member; $305 individual member. A late charge of 10% of the subscrip- tion price will be imposed upon orders received from nonmember after January 1 of the subscription year. Subscribers outside the United States and India must pay a postage surcharge of $25; subscribers in India must pay a postage surcharge of $38. Expedited delivery to destinations in North America—$38; elsewhere—$90. Back number information. For back issues see the AMS Catalogue of Publications.