Semicanonical Bases Arising From Enveloping Algebras

By G. Lusztig. 2000

168 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d6414abe32ad129b6adc5ba85e8749b5d2c27e95

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Bernard Leclerc
2. B. Keller
3. Osamu Iyama
4. C. Geiss
5. J. Schröer
6. I. Reiten
7. Andrzej Skowronski
8. K. Yamagata

In Geometry and Topology

1. B. Keller
2. Bernard Leclerc
3. Osamu Iyama
4. C. Geiss
5. I. Reiten
6. J. Schröer
7. Andrzej Skowronski
8. K. Yamagata

In Statistical and Nonlinear Physics

1. Bernard Leclerc
2. C. Geiss
3. B. Keller
4. J. Schröer
5. Andrzej Skowronski
6. K. Yamagata
7. Laurent Demonet
8. H. Nakajima

Citations in the corpus

• Cluster algebra structures and semicanonical bases for unipotent groups (2007, background): “f irreducible components of Λd. For each Z ∈Irr(Λd) there exists a dense open subset Uin Zsuch that δX = δY for all X,Y ∈U. If X∈Uwe say that Xis a generic point of Z. Define ρZ := δX for some X∈U. By [L1, L3], S∗ := {ρZ |Z∈Irr(Λd),d∈Nn} 12 CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHROER¨ is a basis”
• Classifying $$\tau $$τ-tilting modules over preprojective algebras of Dynkin type (2013, background): “For example, they play central roles in Lusztig’s Lagrangian construction of semicanonical basis [28,29].”
• Preprojective algebras and MV polytopes (2010, background): “On the algebraic side, crystals are implemented by special bases in representations, for instance Lusztig’s canonical and semicanonical bases [15, 17, 20].”
• Triangular bases in quantum cluster algebras and monoidal categorification conjectures (2015, background): “gebra, which contains the cluster monomials. As an important consequence, they identified the generic basis with Lusztig’s dual semicanonical basis, which is a variant of the dual canonical basis, cf. [Lus00]. By [Pla13], this basis is parametrized by tropical points as expected by [FG09]. (ii) For c”
• Kac–Moody groups and cluster algebras (2010, background): “ follows that M= M d∈Nn Md is an Nn-graded C-algebra. Let U(n) be the enveloping algebra of the positive part nof the Kac-Moody Lie algebra gassociated to Q, compare Section 4.1. Theorem 2.4 (Lusztig [L2]). There is an isomorphism of Nn-graded C-algebras U(n) →M defined by Ei →1i for 1 ≤i≤n. Let Irr(”
• Quiver varieties and cluster algebras (2009, background): “Therefore (a) it has the structure of the monoidal abelian category, where the tensor product is given by the convolution diagram coming from exact sequences of quiver representations; (b) it inherit various combinatorial structure from the module category repQ, and probably also from the cluster…”
• Semicanonical bases and preprojective algebras (2004, methodology): “Following [41] define (f ′ ∗ f ′′)(x) = ∫”
• Cluster structures on strata of flag varieties (2014, background): “ic module M on Z. Let Z denote the union of all irreducible components of all varieties Λd (d ∈NI). The collection S∗ ∶={ϕZ SZ ∈Z} is dual to the semicanonical basis of U(n) constructed by Lusztig in [L3]. Hence S∗ is a C-basis ofC[N] which we call the dual semicanonical basis, Note that if M is a r”
• Absolutely indecomposable representations and Kac-Moody Lie algebras (2001, background): “By [ 28 ] we have nα = dim U(g + )α. Theorem 3.2 yields the formula”
• Cluster algebras and derived categories (2012, background): “A proof of the corollary based on the study of stability conditions on Dfd(Γ) is given in section 4.2 of [106].”
• Tensor diagrams and cluster algebras (2012, background): “Zelevinsky [50], while the square lies in the dual semicanonical basis, see [17,40].”
• Representations of Algebras and Related Topics (2011, background): “ initial motivation of Fomin and Zelevinsky to define cluster algebras was to give a combinatorial and algebraic setup to understand canonical basis and total positivity in algebraic groups (see [77], [78], [50]), these algebras have led to many applications in very different domains of mathematics: •”
• Cluster algebras and representation theory (2010, methodology): “is dual to the basis S = {fZ | Z ∈ I} ofM∼= U(n) constructed by Lusztig in [47], and called by him the semicanonical basis.”
• Rigid modules over preprojective algebras (2005, background): “In [29], [31], Lusztig has given a geometric construction of U(n) as an algebra (M, ∗) of constructible functions on varieties of finite-dimensional Λ-modules, where ∗ is a convolution product inspired by Ringel’s multiplication for Hall algebras [36].”