Main Content

Research in the RG Algebraic Lie Theory

Research topics

The main research topics in our group are the following.

Hopf algebras
Nichols algebras
Quantum groups
Representation theory
Coxeter groups and -groupoids
Noncommutative algebra
Kac-Moody algebras

Topics of the current PhD students

Johannes Herrendorf: Nichols algebras of low dimensional Yetter-Drinfeld modules

Katharina Schäfer: Left coideal subalgebras of Nichols algebras

Fengchang Li: Nichols algebras with affine root system

Publications

Preprint
Heckenberger, I.; Vendramin, L. Bosonization of curved Lie bialgebras. arXiv:2209.02115

Preprint
Andruskiewitsch, N.; Angiono, I.; Heckenberger, I. Corrigendum: Liftings of the Jordan plane. arxiv:2203.03350 

Preprint
Alia, A.; Heckenberger, I. On the representation theory of non-semisimple (graded) deformed Fomin-Kirillov algebras. arXiv:2011.14777

Preprint
Heckenberger, I.; Wolf, K. Two-cocycles and cleft extensions in left braided categories. arXiv:1906.05109

Preprint
Andruskiewitsch, N.; Angiono, I.; Heckenberger, I. Examples of finite-dimensional pointed Hopf algebras in positive characteristic. arXiv:1905.03074

Master Thesis
Wolf, K. PBW deformations of algebras. Marburg, September 2018

Preprint
Heckenberger, I.; Zheng, Y. A characterization of Nichols algebras of diagonal type which are free algebras. arXiv:1806.05903

Preprint
Andruskiewitsch, N.; Angiono, I.; Heckenberger, I. On Nichols algebras of infinite rank with finite Gelfand-Kirillov dimension. arXiv:1805.12000

MR4034793 Article Preprint
Heckenberger, I.; Vendramin, L. PBW deformations of a Fomin-Kirillov algebra and other examples. Algebr. Represent. Theory 22 (2019), no. 6, 1513–1532.

MR3968806 Article Preprint
Heckenberger, I.; Shareshian, J.; Welker, V. On the lattice of subracks of the rack of a finite group. Trans. Amer. Math. Soc. 372 (2019), no. 2, 1407–1427.

MR3953068 Article Preprint
Heckenberger, I.; Sawada, J. A Pascal-like bound for the number of necklaces with fixed density. Theoret. Comput. Sci. 778 (2019), 73–77.

MR3943743 Article Preprint
Andruskiewitsch, N.; Angiono, I.; Heckenberger, I. On finite GK-dimensional Nichols algebras of diagonal type. Tensor categories and Hopf algebras, 1–23, Contemp. Math., 728, Amer. Math. Soc., Providence, RI, 2019.

MR3834727 Article Preprint
Andruskiewitsch, Nicolás; Angiono, Iván; Heckenberger, István Liftings of Jordan and super Jordan planes. Proc. Edinb. Math. Soc. (2) 61 (2018), no. 3, 661–672.

MR3737835 Article Preprint
Heckenberger, I.; Zheng, Y. Root multiplicities for Nichols algebras of diagonal type of rank two. J. Algebra 496 (2018), 91–115.

MR3656477 Article Preprint
Heckenberger, I.; Vendramin, L. The classification of Nichols algebras over groups with finite root system of rank two. J. Eur. Math. Soc. (JEMS) 19 (2017), no. 7, 1977–2017.

MR3605018 Article Preprint
Heckenberger, I.; Vendramin, L. A classification of Nichols algebras of semisimple Yetter-Drinfeld modules over non-abelian groups. J. Eur. Math. Soc. (JEMS) 19 (2017), no. 2, 299–356.

MR3515188 Article Preprint
Heckenberger, I.; Welker, V. A deformation of the Orlik-Solomon algebra. Math. Scand. 118 (2016), no. 2, 183–202.

MR3356939 Article Preprint
Heckenberger, I.; Lochmann, A.; Vendramin, L. Nichols algebras with many cubic relations. Trans. Amer. Math. Soc. 367 (2015), no. 9, 6315–6356.

MR3341467 Article Preprint
Cuntz, M.; Heckenberger, I. Finite Weyl groupoids. J. Reine Angew. Math. 702 (2015), 77–108.

MR3313687 Article Preprint
Wang, J.; Heckenberger, I. Rank 2 Nichols algebras of diagonal type over fields of positive characteristic. SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 011, 24 pp.

MR3272075 Article Preprint
Heckenberger, I.; Vendramin, L. Nichols algebras over groups with finite root system of rank two III. J. Algebra 422 (2015), 223–256.

MR3284337
Heckenberger, I.; Joseph, A. Erratum to the article "On the left and right Brylinski-Kostant filtrations'' [MR2501195]. Algebr. Represent. Theory 17 (2014), no. 6, 1925–1926.

MR3276225 Article Preprint
Heckenberger, I.; Vendramin, L. Nichols algebras over groups with finite root system of rank two II. J. Group Theory 17 (2014), no. 6, 1009–1034.

MR3096611 Article Preprint
Heckenberger, I.; Schneider, H.-J. Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid. Israel J. Math. 197 (2013), no. 1, 139–187.

MR3077876 Article Preprint
Heckenberger, I.; Schneider, H.-J. Yetter-Drinfeld modules over bosonizations of dually paired Hopf algebras. Adv. Math. 244 (2013), 354–394.

MR2967250 Article Preprint
Heckenberger, I.; Kolb, S. Homogeneous right coideal subalgebras of quantized enveloping algebras. Bull. Lond. Math. Soc. 44 (2012), no. 4, 837–848.

MR2891215 Article Preprint
Heckenberger, I.; Lochmann, A.; Vendramin, L. Braided racks, Hurwitz actions and Nichols algebras with many cubic relations. Transform. Groups 17 (2012), no. 1, 157–194.

MR2869179 Article Preprint
Cuntz, M.; Heckenberger, I. Finite Weyl groupoids of rank three. Trans. Amer. Math. Soc. 364 (2012), no. 3, 1369–1393.

MR2805202 Article Preprint
Heckenberger, István; Welker, Volkmar Geometric combinatorics of Weyl groupoids. J. Algebraic Combin. 34 (2011), no. 1, 115–139.

MR2803792 Article Preprint
Graña, M.; Heckenberger, I.; Vendramin, L. Nichols algebras of group type with many quadratic relations. Adv. Math. 227 (2011), no. 5, 1956–1989.

MR2764869
Heckenberger, István; Kolb, Stefan Right coideal subalgebras of the Borel part of a quantized enveloping algebra. Int. Math. Res. Not. IMRN 2011, no. 2, 419–451.

MR2755086
Cuntz, M.; Heckenberger, I. Reflection groupoids of rank two and cluster algebras of type A. J. Combin. Theory Ser. A 118 (2011), no. 4, 1350–1363.

MR2840165
Heckenberger, I.; Yamane, H. Drinfel'd doubles and Shapovalov determinants. Rev. Un. Mat. Argentina 51 (2010), no. 2, 107–146.

MR2766176
Andruskiewitsch, Nicolás; Heckenberger, István; Schneider, Hans-Jürgen The Nichols algebra of a semisimple Yetter-Drinfeld module. Amer. J. Math. 132 (2010), no. 6, 1493–1547.

MR2734956
Heckenberger, I.; Schneider, H.-J. Root systems and Weyl groupoids for Nichols algebras. Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 623–654.

MR2732989
Heckenberger, I.; Schneider, H.-J. Nichols algebras over groups with finite root system of rank two I. J. Algebra 324 (2010), no. 11, 3090–3114.

MR2596372
Heckenberger, I. Lusztig isomorphisms for Drinfelʹd doubles of bosonizations of Nichols algebras of diagonal type. J. Algebra 323 (2010), no. 8, 2130–2182.

MR2525553
Cuntz, Michael; Heckenberger, István Weyl groupoids of rank two and continued fractions. Algebra Number Theory 3 (2009), no. 3, 317–340.

MR2501195
Heckenberger, Istvan; Joseph, Anthony On the left and right Brylinski-Kostant filtrations. Algebr. Represent. Theory 12 (2009), no. 2-5, 417–442.

MR2498801
Cuntz, M.; Heckenberger, I. Weyl groupoids with at most three objects. J. Pure Appl. Algebra 213 (2009), no. 6, 1112–1128.

MR2462836
Heckenberger, I. Classification of arithmetic root systems. Adv. Math. 220 (2009), no. 1, 59–124.

MR2467721
Heckenberger, István; Spill, Fabian; Torrielli, Alessandro; Yamane, Hiroyuki Drinfeld second realization of the quantum affine superalgebras of D(1)(2,1;x) via the Weyl groupoid. Combinatorial representation theory and related topics, 171–216, RIMS Kôkyûroku Bessatsu, B8, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008.

MR2390080
Heckenberger, István; Yamane, Hiroyuki A generalization of Coxeter groups, root systems, and Matsumoto's theorem. Math. Z. 259 (2008), no. 2, 255–276.

MR2379892
Heckenberger, I. Rank 2 Nichols algebras with finite arithmetic root system. Algebr. Represent. Theory 11 (2008), no. 2, 115–132.

MR2500361
Heckenberger, István Classification of arithmetic root systems of rank 3. Proceedings of the XVIth Latin American Algebra Colloquium (Spanish), 227–252, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2007.

MR2365438
Heckenberger, I.; Kolb, S. On the Bernstein-Gelfand-Gelfand resolution for Kac-Moody algebras and quantized enveloping algebras. Transform. Groups 12 (2007), no. 4, 647–655.

MR2360244
Heckenberger, István; Kolb, Stefan Differential forms via the Bernstein-Gelfand-Gelfand resolution for quantized irreducible flag manifolds. J. Geom. Phys. 57 (2007), no. 11, 2316–2344.

MR2331765
Graña, M.; Heckenberger, I. On a factorization of graded Hopf algebras using Lyndon words. J. Algebra 314 (2007), no. 1, 324–343.

MR2295200
Heckenberger, I. Examples of finite-dimensional rank 2 Nichols algebras of diagonal type. Compos. Math. 143 (2007), no. 1, 165–190.

MR2266849
Heckenberger, István; Kolb, Stefan De Rham complex for quantized irreducible flag manifolds. J. Algebra 305 (2006), no. 2, 704–741.

MR2207786
Heckenberger, I. The Weyl groupoid of a Nichols algebra of diagonal type. Invent. Math. 164 (2006), no. 1, 175–188.

MR2294771
Heckenberger, I. Weyl equivalence for rank 2 Nichols algebras of diagonal type. Ann. Univ. Ferrara Sez. VII (N.S.) 51 (2005), 281–289.

MR2078702
Heckenberger, I.; Kolb, S. The locally finite part of the dual coalgebra of quantized irreducible flag manifolds. Proc. London Math. Soc. (3) 89 (2004), no. 2, 457–484.

MR1992890
Heckenberger, István; Kolb, Stefan Differential calculus on quantum homogeneous spaces. Lett. Math. Phys. 63 (2003), no. 3, 255–264.

MR1978646
Heckenberger, István; Kolb, Stefan Podleś' quantum sphere: dual coalgebra and classification of covariant first-order differential calculus. J. Algebra 263 (2003), no. 2, 193–214.

MR1972632
Heckenberger, István Spin geometry on quantum groups via covariant differential calculi. Adv. Math. 175 (2003), no. 2, 197–242.

MR1917623
Heckenberger, István; Schüler, Axel Laplace operator and Hodge decomposition for quantum groups and quantum spaces. Quantum groups and integrable systems (Prague, 2001). Czechoslovak J. Phys. 51 (2001), no. 12, 1342–1347.

MR1851089
Heckenberger, István; Schüler, Axel De Rham cohomology and Hodge decomposition for quantum groups. Proc. London Math. Soc. (3) 83 (2001), no. 3, 743–768.

MR1813897
Heckenberger, István Classification of left-covariant differential calculi on the quantum group SLq(2). J. Algebra 237 (2001), no. 1, 203–237.

MR1851791
Heckenberger, I.; Schüler, A. On FRT-Clifford algebras. Adv. Appl. Clifford Algebras 10 (2000), no. 2, 267–296 (2001). (Reviewer: Adriana M. Criscuolo) 17B37 (15A66 

MR1795294
Heckenberger, István Hodge and Laplace-Beltrami operators for bicovariant differential calculi on quantum groups. Compositio Math. 123 (2000), no. 3, 329–354.

MR1751618
Heckenberger, István; Schüler, Axel Symmetrizer and antisymmetrizer of the Birman-Wenzl-Murakami algebras. Lett. Math. Phys. 50 (1999), no. 1, 45–51.

MR1681983
Heckenberger, István; Schüler, Axel Exterior algebras related to the quantum group O(Oq(3)). Quantum groups and integrable systems (Prague, 1998). Czechoslovak J. Phys. 48 (1998), no. 11, 1355–1362.

MR1647563
Heckenberger, I.; Schmüdgen, K. Classification of bicovariant differential calculi on the quantum groups SLq(n+1) and Spq(2n). J. Reine Angew. Math. 502 (1998), 141–162.

MR1615877
Heckenberger, I.; Schmüdgen, K. Bicovariant differential calculi on SLq(N) and Spq(N). Quantum groups and integrable systems, Part I (Prague, 1997). Czechoslovak J. Phys. 47 (1997), no. 11, 1145–1151.

MR1615873
Heckenberger, I.; Schüler, A. Higher order differential calculus on SLq(N). Quantum groups and integrable systems, Part I (Prague, 1997). Czechoslovak J. Phys. 47 (1997), no. 11, 1153–1161.

MR1463038
Heckenberger, István; Schmüdgen, Konrad Levi-Civita connections on the quantum groups SLq(N), Oq(N) and Spq(N). Comm. Math. Phys. 185 (1997), no. 1, 177–196.