**Professor of Mathematics**

### Dr. Kirkman’s

**Office:**Manchester 386**Phone:**(336) 758-5351**Email:**kirkman 'at' wfu.edu

PhD in Mathematics and M.S. in Statistics Michigan State University, 1975

Noncommutative Algebra, Representation Theory, and Homological Algebra

Computational invariant theory, computational algebraic geometry, and quantum calculus.

Publications

- Inertial coeffcient rings and the idempotent lifting property, Proc. Amer. Math. Soc. 61 (1976), 217-222.
- The Pierce representation of an inertial coeffcient ring, Rocky Mountain J. Math. 8 (1978), 533-538.
- Orders over hereditary rings, J. Algebra 55 (1978), 1-27 (with James Kuz- manovich).
- Hereditary module finite algebras, J. London Math. Soc. (2) 19 (1979), 268-276 (with John Fuelberth and James Kuzmanovich).
- Hereditary finitely generated algebras satisfying a polynomial identity, Proc. Amer. Math. Soc. 83 (1981), 461-466 (with James Kuzmanovich).
- Hereditary rings integral over their centers, J. Algebra 102 (1986), 119-128 (with James Kuzmanovich).
- Right hereditary affne PI rings are left hereditary, Glasgow Math. J. 30 (1988), 115-120 (with James Kuzmanovich).
- Matrix subrings having finite global dimension, J. Algebra 109 (1987), 74-92 (with James Kuzmanovich).
- On the global dimension of a ring modulo its nilpotent radical, Proc. Amer. Math. Soc. 102 (1988), 25-28 (with James Kuzmanovich).
- On the global dimension of fibre products, Pacific J. Math. 134 (1988), 121-132 (with James Kuzmanovich)
- Global dimensions of a class of tiled orders, J. Algebra 127 (1989), 57-72 (with James Kuzmanovich).
- Finitistic dimension of finite dimensional monomial algebras, J. Algebra 136 (1991), 37-50 (with Edward Green and James Kuzmanovich).
- Algebras with large homological dimensions, Proc. Amer. Math. Soc. 109 (1990), 903-6 (with James Kuzmanovich).
- Finitistic dimension in Noetherian rings, J. Algebra 147 (1992), 350-364 (with James Kuzmanovich and Lance Small).
- Q-analogs of harmonic oscillators and related rings, Israel J. Math. 81 (1993), 111-127 (with Lance Small).
- Constructing projective resolutions, Comm. Alg. 21 (1993), 1869-1887 (with Charles D. Feustel, Edward L. Green, and James Kuzmanovich).
- On the finitistic dimension of fixed subrings, Comm. Algebra 22 (10) (1994), 3755-5774 (with James Kuzmanovich).
- A q-analog of the Virasora algebra, Comm. Algebra 22 (1994), 3755-3774 (with Claudio Procesi and Lance Small).
- Minimal prime ideals in enveloping algebras of Lie superalgebras, Proc. Amer. Math. Soc. 124 (1996), no. 6, 16931702 (with James Kuzmanovich)
- Global and Krull dimensions of quantum Weyl algebras, J. Algebra. 216 (1999) 405-416 (with Hisaaki Fujita and James Kuzmanovich).
- Noetherian down-up algebras, Proc. Amer. Math. Soc. 127 (1999), 3161-3167 (with Ian Musson and Donald Passman).
- The primitivity of Noetherian down-up algebras, Comm. Algebra 28 (2000), no. 6, 2983{2997 (with James Kuzmanovich).
- Non-Noetherian down-up algebras, Comm. Algebra 28 (2000), no. 11, 5255{ 5268 (with James Kuzmanovich).
- Examples of FCR algebras, Comm. Algebra 30 (2002)(7), 3311{3326 (with Lance Small).
- Hopf down-up algebras, J. Algebra 262 (2003), 42-53 (with Ian Musson).
- Fixed subrings of Noetherian graded regular rings, Journal of Algebra 288 (2005) no. 2, 463{484 (with James Kuzmanovich).
- Down-up algebras from trees, Comm. Algebra 34 (2006), no. 10, 3785{3799 (with James Kuzmanovich).
- Rigidity of graded regular algebras, Trans. Amer. Math. Soc. 360 (2008), 6331-6369 (with James Kuzmanovich and James Zhang).
- Gorenstein subrings of invariants under Hopf algebra actions, J. Algebra. 322(2009) no. 10, 3640{3669 (with James Kuzmanovich and James Zhang).
- Shephard-Todd-Chevalley theorem for skew polynomial rings, Algebr. Repre- sent. Theory 13 (2010) no. 2, 127{158 (with James Kuzmanovich and James Zhang).
- Finiteness conditions on the Yoneda algebra of a monomial algebra, J. Pure Appl. Algebra 218 (2014), no. 1, 52{64 (with Andrew Conner, James Kuz- manovich, and W Frank Moore).
- Invariants of (-1)-skew polynomial rings under permutation representations. Recent advances in representation theory, quantum groups, algebraic geome- try, and related topics, 155{192, Contemp. Math., 623, Amer. Math. Soc., Providence, RI, 2014 (with James Kuzmanovich and James Zhang).
- Invariant theory of finite group actions on down-up algebras, Transformation Groups 20 (2015) no. 1, 113-165 (with James Kuzmanovich and James Zhang).
- Periodic free resolutions from twisted matrix factorizations, J. Algebra 455 (2016), 137{163 (with Thomas Cassidy, Andrew Conner, and W. Frank Moore).
- Quantum binary polyhedral groups and their actions on quantum planes, J. Reine Angew. Math. 719 (2016), 211{252 (with Kenneth Chan, Chelsea Walton, James Zhang).
- Noncommutative complete intersections, J. Algebra 429 (2015), 253{286 (with James Kuzmanovich and James Zhang).
- The invariant thoery of Artin-Schelter regular algebras: A survey, Recent Devel- opments in Representation Theory (Maurice Auslander Distinguished Lectures and International Conference, May 1-6, 2014, Woods Hole Oceanographic In- stitute, Woods Hole, MA), Contemp. Math. 673, p. 25-50, Amer. Math. Soc., Providence, RI., 2016.
- On the discriminant of twisted tensor products, J. Algebra 477 (2017), 29-55 (with Jason Gaddis and W. Frank Moore).
- Nakayama automorphism and rigidity of dual re ection group coactions, J. Al- gebra 487 (2017), 60-92. (with James Kuzmanovich and James Zhang).
- Rigidity of down-up algebras with respect to finite group coactions, J. Pure and Applied Algebra 221 (2017) No. 12 3089-3103 (with Jianmin Chen and James Zhang).
- McKay correspondence for semisimple Hopf actions on regular graded algebras I, J. Algebra 508 (2018), 512{538 (with Kenneth Chan, Chelsea Walton, James Zhang).
- McKay correspondence for semisimple Hopf actions on regular graded algebras II, J. Noncommut. Geom. 13 (2019), no. 1, 87{114. (with Kenneth Chan, Chelsea Walton, James Zhang).
- Auslander's Theorem for permutation actions on noncommutative algebras, Proc. Amer. Math. Soc. 147 (2019), no. 5, 1881{1896. (with Jason Gaddis, W. Frank Moore, and Robert Won).
- Color Lie algebras and PBW deformations of skew group algebras, J. Algebra 518 (2019), 211{236. (with S. Fryer, T. Kanstrup, A.V. Shepler, and W. With- erspoon).
- Auslander's Theorem for group coactions on Noetherian down-up algebras Trans- form. Groups 25 (2020) No. 4, 1037{1062, (with J. Chen and James Zhang). 46. Noncommutative Knorrer periodicity and noncommutative Kleinian singulari- ties. J. Algebra 540 (2019), 234{273. (with Andrew Conner, W. Frank Moore, and Chelsea Walton).
- Three infinite families of re ection Hopf algebras, J. Pure Appl. Algebra, 224 (2020), no. 8, 106315, 34 (with Luigi Ferraro, W. Frank Moore, and Robert Won).
- Semisimple re ection Hopf algebras of dimension sixteen (with Luigi Ferraro, W. Frank Moore, and Robert Won), submitted, arXiv 1907.06763.
- The Jacobian, re ection arrangement and discriminant for re ection Hopf al- gebras (with James Zhang), to appear Int. Math. Res. Not., arXiv 1902.00421.
- On the Noether bound for noncommutative rings (with Luigi Ferraro, W. Frank Moore and Kewen Peng), to appear in Proc. Amer. Math. Soc., arXiv 1907.06761.
- McKay Matrices for Finite-dimensional Hopf Algebras (with Georgia Benkart, Rekha Biswall, Van Nguyen, and Jieru Zhu), submitted, arXiv 2007.05510.
- Degree bounds for Hopf actions on Artin-Schelter regular algebras (with Robert Won and James Zhang), submitted, arXiv 2008.05047.

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