Kac-Moody groups and cluster algebras. (English) Zbl 1232.17035
This article can be regarded as the culmination of a deep, involved project of the authors, making significant progress in the understanding of Kac-Moody Lie algebras, cluster structures on Weyl group-indexed subgroups the of the unipotent group and the dual semicanonical basis. It is a wide generalisation of their earlier work on the Dynkin cases A, D and E [C. Geiss, B. Leclerc and J. Schröer, Invent. Math. 165, No. 3, 589–632 (2006; Zbl 1167.16009)].
Fix a finite quiver \(Q\) with no oriented cycles. Let \(\Lambda=\Lambda_Q\) be the corresponding preprojective algebra (which may be infinite dimensional). Let \(W_Q\) be the Weyl group of \(Q\). For each element \(w\in W\), a Frobenius subcategory \(\mathcal{C}_w\) of the category \(nil(\Lambda)\) of all finite dimensional nilpotent representations of \(\Lambda\) was associated to \(w\) in [A. Buan, O. Iyama, I. Reiten and J. Scott, Compos. Math. 145, 1035–1079 (2009; Zbl 1181.18006)], where it was shown that the corresponding stable category, \(\underline{\mathcal{C}_w}\) is \(2\)-Calabi-Yau. These results were also discovered independently in [C. Geiss, B. Leclerc and J. Schröer, Cluster algebra structures and semicanonical bases for unipotent groups, Preprint arXiv:math/0703039v4 [math.RT], (2007)], in the special case where \(w\) is adaptable. Each reduced expression i for \(w\) gives rise to corresponding maximal rigid \(\Lambda\)-module \(V_{\mathbf{i}}\).
A cluster algebra \(\mathcal{A}(\mathcal{C}_w)\) is associated to \(\mathcal{C}_w\), with initial seed given by \(V_{\mathbf{i}}\) (for any choice of reduced expression i). It is shown that \(\mathcal{A}(\mathcal{C}_w)\) has a natural realisation as a certain subalgebra of the graded dual of \(U(\mathbf{n})\), where n is the positive part of the symmetric Kac-Moody Lie algebra g of the same type as \(\Lambda\). It is shown that all the cluster monomials lie in the dual of Lusztig’s semicanonical basis of \(U(\mathbf{n})\) and that the intersection of the dual semicanonical basis with \(\mathcal{A}(\mathcal{C}_w)\) is a basis for \(\mathcal{A}(\mathcal{C}_w)\). It is further shown that \(\mathcal{A}(\mathcal{C}_w)\) is isomorphic to the coordinate ring of of the finite dimensional unipotent subgroup \(N(w)\) (associated to \(w\)) of the symmetric Kac-Moody group attached to g.
It is also shown that inverting the generators of the coefficient ring gives rise to the algebra of regular functions on the unipotent cell associated to \(w\), solving a conjecture of A. Buan, O. Iyama, I. Reiten and J. Scott [loc. cit.].
Also interesting is that the endomorphism algebra of \(V_{\mathbf{i}}\) is shown to be quasihereditary, allowing a description of mutation of maximal rigid modules in \(\mathcal{C}_w\) in terms of \(\Delta\)-dimension vectors of corresponding modules over the endomorphism algebra.
Another spin-off is a new categorification of every acyclic cluster algebra with a skew-symmetric exchange matrix and a certain choice of coefficients.
This thorough, well-written paper covers a lot of ground and establishes very general strong, results, and is likely to be a standard reference in the future.
Fix a finite quiver \(Q\) with no oriented cycles. Let \(\Lambda=\Lambda_Q\) be the corresponding preprojective algebra (which may be infinite dimensional). Let \(W_Q\) be the Weyl group of \(Q\). For each element \(w\in W\), a Frobenius subcategory \(\mathcal{C}_w\) of the category \(nil(\Lambda)\) of all finite dimensional nilpotent representations of \(\Lambda\) was associated to \(w\) in [A. Buan, O. Iyama, I. Reiten and J. Scott, Compos. Math. 145, 1035–1079 (2009; Zbl 1181.18006)], where it was shown that the corresponding stable category, \(\underline{\mathcal{C}_w}\) is \(2\)-Calabi-Yau. These results were also discovered independently in [C. Geiss, B. Leclerc and J. Schröer, Cluster algebra structures and semicanonical bases for unipotent groups, Preprint arXiv:math/0703039v4 [math.RT], (2007)], in the special case where \(w\) is adaptable. Each reduced expression i for \(w\) gives rise to corresponding maximal rigid \(\Lambda\)-module \(V_{\mathbf{i}}\).
A cluster algebra \(\mathcal{A}(\mathcal{C}_w)\) is associated to \(\mathcal{C}_w\), with initial seed given by \(V_{\mathbf{i}}\) (for any choice of reduced expression i). It is shown that \(\mathcal{A}(\mathcal{C}_w)\) has a natural realisation as a certain subalgebra of the graded dual of \(U(\mathbf{n})\), where n is the positive part of the symmetric Kac-Moody Lie algebra g of the same type as \(\Lambda\). It is shown that all the cluster monomials lie in the dual of Lusztig’s semicanonical basis of \(U(\mathbf{n})\) and that the intersection of the dual semicanonical basis with \(\mathcal{A}(\mathcal{C}_w)\) is a basis for \(\mathcal{A}(\mathcal{C}_w)\). It is further shown that \(\mathcal{A}(\mathcal{C}_w)\) is isomorphic to the coordinate ring of of the finite dimensional unipotent subgroup \(N(w)\) (associated to \(w\)) of the symmetric Kac-Moody group attached to g.
It is also shown that inverting the generators of the coefficient ring gives rise to the algebra of regular functions on the unipotent cell associated to \(w\), solving a conjecture of A. Buan, O. Iyama, I. Reiten and J. Scott [loc. cit.].
Also interesting is that the endomorphism algebra of \(V_{\mathbf{i}}\) is shown to be quasihereditary, allowing a description of mutation of maximal rigid modules in \(\mathcal{C}_w\) in terms of \(\Delta\)-dimension vectors of corresponding modules over the endomorphism algebra.
Another spin-off is a new categorification of every acyclic cluster algebra with a skew-symmetric exchange matrix and a certain choice of coefficients.
This thorough, well-written paper covers a lot of ground and establishes very general strong, results, and is likely to be a standard reference in the future.
Reviewer: Robert Marsh (Leeds)
MSC:
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 13F60 | Cluster algebras |
| 14M99 | Special varieties |
| 16G20 | Representations of quivers and partially ordered sets |
| 17B35 | Universal enveloping (super)algebras |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 20G05 | Representation theory for linear algebraic groups |
| 22E67 | Loop groups and related constructions, group-theoretic treatment |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
Keywords:
Cluster algebra; Kac-Moody group; Kac-Moody algebra; Semicanonical basis; Frobenius category; Preprojective algebra; Representation theory; Unipotent group; Algebraic group; Calabi-Yau category; Quasihereditary algebraReferences:
| [1] | Abe, E., Hopf Algebras, Cambridge Tracts in Math., vol. 74 (1980), Cambridge University Press · Zbl 0476.16008 |
| [2] | Assem, I.; Simson, D.; Skowroński, A., Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory, London Math. Soc. Stud. Texts, vol. 65 (2006), Cambridge University Press: Cambridge University Press Cambridge, x+458 pp · Zbl 1092.16001 |
| [3] | Auslander, M., Representation theory of artin algebras. I, Comm. Algebra, 1, 177-268 (1974) · Zbl 0285.16028 |
| [4] | Auslander, M.; Solberg, Ø., Relative homology and representation theory. I. Relative homology and homologically finite subcategories, Comm. Algebra, 21, 9, 2995-3031 (1993) · Zbl 0792.16017 |
| [5] | Auslander, M.; Solberg, Ø., Relative homology and representation theory. II. Relative cotilting theory, Comm. Algebra, 21, 9, 3033-3079 (1993) · Zbl 0792.16018 |
| [6] | Auslander, M.; Platzeck, M.; Reiten, I., Coxeter functors without diagrams, Trans. Amer. Math. Soc., 250, 1-46 (1979) · Zbl 0421.16016 |
| [7] | Auslander, M.; Reiten, I.; Smalø, S., Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., vol. 36 (1997), Cambridge University Press: Cambridge University Press Cambridge, xiv+425 pp |
| [8] | Berenstein, A.; Zelevinsky, A., Total positivity in Schubert varieties, Comment. Math. Helv., 72, 128-166 (1997) · Zbl 0891.20030 |
| [9] | Berenstein, A.; Fomin, S.; Zelevinsky, A., Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. J., 126, 1, 1-52 (2005) · Zbl 1135.16013 |
| [10] | Bongartz, K., Minimal singularities for representations of Dynkin quivers, Comment. Math. Helv., 69, 4, 575-611 (1994) · Zbl 0832.16008 |
| [11] | Buan, A.; Marsh, R., Cluster-tilting theory, (Trends in Representation Theory of Algebras and Related Topics. Trends in Representation Theory of Algebras and Related Topics, Contemp. Math., vol. 406 (2006)), 1-30 · Zbl 1111.16014 |
| [12] | Buan, A.; Marsh, R.; Reineke, M.; Reiten, I.; Todorov, G., Tilting theory and cluster combinatorics, Adv. Math., 204, 2, 572-618 (2006) · Zbl 1127.16011 |
| [13] | Buan, A.; Iyama, O.; Reiten, I.; Scott, J., Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math., 145, 1035-1079 (2009) · Zbl 1181.18006 |
| [14] | Caldero, P.; Keller, B., From triangulated categories to cluster algebras II, Ann. Sci. Éc. Norm. Super. (4), 39, 6, 983-1009 (2006) · Zbl 1115.18301 |
| [15] | Caldero, P.; Keller, B., From triangulated categories to cluster algebras, Invent. Math., 172, 169-211 (2008) · Zbl 1141.18012 |
| [16] | Cline, E.; Parshall, B.; Scott, L., Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math., 391, 85-99 (1988) · Zbl 0657.18005 |
| [17] | Crawley-Boevey, W., On the exceptional fibres of Kleinian singularities, Amer. J. Math., 122, 1027-1037 (2000) · Zbl 1001.14001 |
| [18] | Crawley-Boevey, W.; Schröer, J., Irreducible components of varieties of modules, J. Reine Angew. Math., 553, 201-220 (2002) · Zbl 1062.16019 |
| [19] | Derksen, H.; Weyman, J.; Zelevinsky, A., Quivers with potentials and their representations I: Mutations, Selecta Math. (N.S.), 14, 1, 59-119 (2008) · Zbl 1204.16008 |
| [20] | Derksen, H.; Weyman, J.; Zelevinsky, A., Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc., 23, 3, 749-790 (2010) · Zbl 1208.16017 |
| [21] | Di Francesco, P.; Kedem, R., Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property, Lett. Math. Phys., 89, 3, 183-216 (2009) · Zbl 1195.81077 |
| [22] | Fomin, S.; Zelevinsky, A., Double Bruhat cells and total positivity, J. Amer. Math. Soc., 12, 2, 335-380 (1999) · Zbl 0913.22011 |
| [23] | Fomin, S.; Zelevinsky, A., Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15, 2, 497-529 (2002) · Zbl 1021.16017 |
| [24] | Fomin, S.; Zelevinsky, A., Cluster algebras. II. Finite type classification, Invent. Math., 154, 1, 63-121 (2003) · Zbl 1054.17024 |
| [25] | Fomin, S.; Zelevinsky, A., Cluster algebras: notes for the CDM-03 conference, (Current Developments in Mathematics (2003), Int. Press: Int. Press Somerville, MA), 1-34 · Zbl 1119.05108 |
| [26] | Fomin, S.; Zelevinsky, A., Cluster algebras. IV. Coefficients, Compos. Math., 143, 112-164 (2007) · Zbl 1127.16023 |
| [27] | Fu, C.; Keller, B., On cluster algebras with coefficients and 2-Calabi-Yau categories, Trans. Amer. Math. Soc., 362, 859-895 (2010) · Zbl 1201.18007 |
| [28] | Gabriel, P., Finite representation type is open. Representations of algebras, (Proc. Internat. Conf.. Proc. Internat. Conf., Carleton Univ., Ottawa, Ont., 1974. Proc. Internat. Conf.. Proc. Internat. Conf., Carleton Univ., Ottawa, Ont., 1974, Lecture Notes in Math., vol. 488 (1975), Springer-Verlag: Springer-Verlag Berlin), 132-155 · Zbl 0313.16034 |
| [29] | Gabriel, P.; Roiter, A. V., Representations of Finite-Dimensional Algebras (1997), Springer-Verlag: Springer-Verlag Berlin, iv+177 pp., translated from the Russian, with a chapter by B. Keller. Reprint of the 1992 English translation |
| [30] | Geiß, C., Geometric methods in representation theory of finite-dimensional algebras, (Representation Theory of Algebras and Related Topics. Representation Theory of Algebras and Related Topics, Mexico City, 1994. Representation Theory of Algebras and Related Topics. Representation Theory of Algebras and Related Topics, Mexico City, 1994, CMS Conf. Proc., vol. 19 (1996), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 53-63 · Zbl 0854.16009 |
| [31] | Geiß, C.; Leclerc, B.; Schröer, J., Semicanonical bases and preprojective algebras, Ann. Sci. Éc. Norm. Super. (4), 38, 2, 193-253 (2005) · Zbl 1131.17006 |
| [32] | Geiß, C.; Leclerc, B.; Schröer, J., Verma modules and preprojective algebras, Nagoya Math. J., 182, 241-258 (2006) · Zbl 1137.17021 |
| [33] | Geiß, C.; Leclerc, B.; Schröer, J., Rigid modules over preprojective algebras, Invent. Math., 165, 3, 589-632 (2006) · Zbl 1167.16009 |
| [34] | Geiß, C.; Leclerc, B.; Schröer, J., Auslander algebras and initial seeds for cluster algebras, J. Lond. Math. Soc. (2), 75, 718-740 (2007) · Zbl 1135.17007 |
| [35] | Geiß, C.; Leclerc, B.; Schröer, J., Semicanonical bases and preprojective algebras II: A multiplication formula, Compos. Math., 143, 1313-1334 (2007) · Zbl 1132.17004 |
| [36] | Geiß, C.; Leclerc, B.; Schröer, J., Cluster algebra structures and semicanonical bases for unipotent groups (2007), 121 pp |
| [37] | Geiß, C.; Leclerc, B.; Schröer, J., Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (Grenoble), 58, 825-876 (2008) · Zbl 1151.16009 |
| [38] | Happel, D., Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Math. Soc. Lecture Note Ser., vol. 119 (1988), Cambridge University Press: Cambridge University Press Cambridge, x+208 pp · Zbl 0635.16017 |
| [39] | Happel, D., Partial tilting modules and recollement, (Proceedings of the International Conference on Algebra, Part 2. Proceedings of the International Conference on Algebra, Part 2, Novosibirsk, 1989. Proceedings of the International Conference on Algebra, Part 2. Proceedings of the International Conference on Algebra, Part 2, Novosibirsk, 1989, Contemp. Math., vol. 131 (1992), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 345-361 · Zbl 0760.18010 |
| [40] | Iyama, O., Auslander correspondence, Adv. Math., 210, 1, 51-82 (2007) · Zbl 1115.16006 |
| [41] | Iyama, O.; Reiten, I., 2-Auslander algebras associated with reduced words in Coxeter groups (2010), 14 pp |
| [42] | Joseph, A., Quantum Groups and Their Primitive Ideals, Ergeb. Math. Grenzgeb. (3), vol. 29 (1995), Springer-Verlag: Springer-Verlag Berlin, x+383 pp · Zbl 0808.17004 |
| [43] | Kac, V., Infinite-Dimensional Lie Algebras (1990), Cambridge University Press: Cambridge University Press Cambridge, xxii+400 pp · Zbl 0716.17022 |
| [44] | Kac, V.; Peterson, D., Regular functions on certain infinite-dimensional groups, (Arithmetic and Geometry, vol. II. Arithmetic and Geometry, vol. II, Progr. Math., vol. 36 (1983), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 141-166 · Zbl 0578.17014 |
| [45] | Kashiwara, M.; Saito, Y., Geometric construction of crystal bases, Duke Math. J., 89, 9-36 (1997) · Zbl 0901.17006 |
| [46] | Keller, B., On triangulated orbit categories, Doc. Math., 10, 551-581 (2005), (electronic) · Zbl 1086.18006 |
| [47] | Keller, B.; Reiten, I., Cluster tilted algebras are Gorenstein and stably Calabi-Yau, Adv. Math., 211, 123-151 (2007) · Zbl 1128.18007 |
| [48] | Kumar, S., Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progr. Math., vol. 204 (2002), Birkhäuser Boston: Birkhäuser Boston Boston, MA · Zbl 1026.17030 |
| [49] | Leclerc, B., Dual canonical bases, quantum shuffles and \(q\)-characters, Math. Z., 246, 691-732 (2004) · Zbl 1052.17008 |
| [50] | Lusztig, G., Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc., 4, 2, 365-421 (1991) · Zbl 0738.17011 |
| [51] | Lusztig, G., Semicanonical bases arising from enveloping algebras, Adv. Math., 151, 2, 129-139 (2000) · Zbl 0983.17009 |
| [52] | Palu, Y., Cluster characters for 2-Calabi-Yau triangulated categories, Ann. Inst. Fourier (Grenoble), 58, 6, 2221-2248 (2008) · Zbl 1154.16008 |
| [53] | Reutenauer, C., Free Lie Algebras, London Mathematical Society Monographs, New Series, vol. 7 (1993), Oxford Science Publications, The Clarendon Press, Oxford University Press: Oxford Science Publications, The Clarendon Press, Oxford University Press New York, xviii+269 pp · Zbl 0798.17001 |
| [54] | Richmond, N., A stratification for varieties of modules, Bull. Lond. Math. Soc., 33, 5, 565-577 (2001) · Zbl 1054.16007 |
| [55] | Ringel, C. M., Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., vol. 1099 (1984), Springer-Verlag: Springer-Verlag Berlin, xiii+376 pp · Zbl 0546.16013 |
| [56] | Ringel, C. M., The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z., 208, 2, 209-223 (1991) · Zbl 0725.16011 |
| [57] | Ringel, C. M., The category of good modules over a quasi-hereditary algebra, (Proceedings of the Sixth International Conference on Representations of Algebras. Proceedings of the Sixth International Conference on Representations of Algebras, Ottawa, ON, 1992. Proceedings of the Sixth International Conference on Representations of Algebras. Proceedings of the Sixth International Conference on Representations of Algebras, Ottawa, ON, 1992, Carleton-Ottawa Math. Lecture Note Ser., vol. 14 (1992), Carleton Univ.: Carleton Univ. Ottawa, ON), 17 pp · Zbl 0824.16007 |
| [58] | Ringel, C. M., PBW-bases of quantum groups, J. Reine Angew. Math., 470, 51-88 (1996) · Zbl 0840.17010 |
| [59] | Ringel, C. M., Iyamaʼs finiteness theorem via strongly quasi-hereditary algebras (2009), 5 pp |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
