82 research outputs found
Cluster algebras and representation theory
We apply the new theory of cluster algebras of Fomin and Zelevinsky to study
some combinatorial problems arising in Lie theory. This is joint work with
Geiss and Schr\"oer (3, 4, 5, 6), and with Hernandez (8, 9)
Cluster algebras in algebraic Lie theory
We survey some recent constructions of cluster algebra structures on
coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody
groups. We also review a quantized version of these results.Comment: Invited survey; to appear in Transformation Group
Kac-Moody groups and cluster algebras
Let Q be a finite quiver without oriented cycles, let \Lambda be the
associated preprojective algebra, let g be the associated Kac-Moody Lie algebra
with Weyl group W, and let n be the positive part of g. For each Weyl group
element w, a subcategory C_w of mod(\Lambda) was introduced by Buan, Iyama,
Reiten and Scott. It is known that C_w is a Frobenius category and that its
stable category is a Calabi-Yau category of dimension two. We show that C_w
yields a cluster algebra structure on the coordinate ring \CC[N(w)] of the
unipotent group N(w) := N \cap (w^{-1}N_-w). Here N is the pro-unipotent
pro-group with Lie algebra the completion of n. One can identify \CC[N(w)] with
a subalgebra of the graded dual of the universal enveloping algebra U(n) of n.
Let S^* be the dual of Lusztig's semicanonical basis S of U(n). We show that
all cluster monomials of \CC[N(w)] belong to S^*, and that S^* \cap \CC[N(w)]
is a basis of \CC[N(w)]. Moreover, we show that the cluster algebra obtained
from \CC[N(w)] by formally inverting the generators of the coefficient ring is
isomorphic to the algebra \CC[N^w] of regular functions on the unipotent cell
N^w := N \cap (B_-wB_-) of the Kac-Moody group G with Lie algebra g. We obtain
a corresponding dual semicanonical basis of \CC[N^w]. As one application we
obtain a basis for each acyclic cluster algebra, which contains all cluster
monomials in a natural way.Comment: 85 pages. This paper removes the assumption of adaptability of a Weyl
group element, which was needed in our preprint "Cluster algebra structures
and semicanoncial bases for unipotent groups", arXiv:math/0703039. v2: We
corrected several typos and reorganized some Sections. v3: New section 2.8,
section 13.1 improved, several small corrections. To appear in Adv. Mat
A quantum cluster algebra of Kronecker type and the dual canonical basis
The article concerns the dual of Lusztig's canonical basis of a subalgebra of
the positive part U_q(n) of the universal enveloping algebra of a Kac-Moody Lie
algebra of type A_1^{(1)}. The examined subalgebra is associated with a
terminal module M over the path algebra of the Kronecker quiver via an Weyl
group element w of length four.
Geiss-Leclerc-Schroeer attached to M a category C_M of nilpotent modules over
the preprojective algebra of the Kronecker quiver together with an acyclic
cluster algebra A(C_M). The dual semicanonical basis contains all cluster
monomials. By construction, the cluster algebra A(C_M) is a subalgebra of the
graded dual of the (non-quantized) universal enveloping algebra U(n).
We transfer to the quantized setup. Following Lusztig we attach to w a
subalgebra U_q^+(w) of U_q(n). The subalgebra is generated by four elements
that satisfy straightening relations; it degenerates to a commutative algebra
in the classical limit q=1. The algebra U_q^+(w) possesses four bases, a PBW
basis, a canonical basis, and their duals. We prove recursions for dual
canonical basis elements. The recursions imply that every cluster variable in
A(C_M) is the specialization of the dual of an appropriate canonical basis
element. Therefore, U_q^+(w) is a quantum cluster algebra in the sense of
Berenstein-Zelevinsky. Furthermore, we give explicit formulae for the quantized
cluster variables and for expansions of products of dual canonical basis
elements.Comment: 32 page
Quantum Unipotent Subgroup and dual canonical basis
Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate
ring of the unipotent subgroup, associated with a Weyl group element
and they proved cluster monomials are contained in Lusztig's dual
semicanonical basis . We give a set up for the quantization of their
results and propose a conjecture which relates the quantum cluster algebras to
the dual canonical basis . In particular, we prove that the quantum
analogue of has the induced basis from ,
which contains quantum flag minors and satisfies a factorization property with
respect to the `-center' of . This generalizes Caldero's
results from ADE cases to an arbitary symmetrizable Kac-Moody Lie algebra.Comment: 41 page
Partial flag varieties and preprojective algebras
Let L be a preprojective algebra of Dynkin type, and let G be the
corresponding complex semisimple simply connected algebraic group. We study
rigid modules in subcategories sub(Q) for Q an injective L-module, and we
introduce a mutation operation between complete rigid modules in sub(Q). This
yields cluster algebra structures on the coordinate rings of the partial flag
varieties attached to G.Comment: 42 pages, 12 figures, 4 tables. Version 3 : minor corrections and one
reference added. Final version to appear in Annales de l'Institut Fourie
Semicanonical bases and preprojective algebras
We study the multiplicative properties of the dual of Lusztig's semicanonical
basis.The elements of this basis are naturally indexed by theirreducible
components of Lusztig's nilpotent varieties, whichcan be interpreted as
varieties of modules over preprojective algebras.We prove that the product of
two dual semicanonical basis vectorsis again a dual semicanonical basis vector
provided the closure ofthe direct sum of thecorresponding two irreducible
components is again an irreducible component.It follows that the semicanonical
basis and the canonical basiscoincide if and only if we are in Dynkin type
with .Finally, we provide a detailed study of the varieties of
modules over the preprojectivealgebra of type .We show that in this case
the multiplicative properties ofthe dual semicanonical basis are controlled by
the Ringel form of a certain tubular algebra of type (6,3,2) and by
thecorresponding elliptic root system of type .Comment: Minor corrections. Final version to appear in Annales Scientifiques
de l'EN
Auslander algebras and initial seeds for cluster algebras
Let be a Dynkin quiver and the corresponding set of positive roots.
For the preprojective algebra associated to we produce a rigid
-module with pairwise non-isomorphic indecomposable
direct summands by pushing the injective modules of the Auslander algebra of
to .
If is a maximal unipotent subgroup of a complex simply connected simple
Lie group of type , then the coordinate ring is an upper cluster
algebra. We show that the elements of the dual semicanonical basis which
correspond to the indecomposable direct summands of coincide with certain
generalized minors which form an initial cluster for , and that the
corresponding exchange matrix of this cluster can be read from the Gabriel
quiver of .
Finally, we exploit the fact that the categories of injective modules over
and over its covering are triangulated in order to
show several interesting identities in the respective stable module categories.Comment: 23 pages, Version 2: Reference [7] corrected+update
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