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Canonical bases and higher representation theory. (English) Zbl 1393.17029
A canonical basis of an algebra or a representation is a basis that has particularly nice properties. Famous examples are the Kazhdan-Lusztig basis of the Hecke algebra of a Weyl group [D. Kazhdan and G. Lusztig, Invent. Math. 53, 165–184 (1979; Zbl 0499.20035)] or Lusztig’s canonical basis for the modified quantized universal enveloping algebra $$\dot U$$ [G. Lusztig, Introduction to quantum groups. Reprint of the 1994 ed. Boston, MA: BirkhĂ¤user (2010; Zbl 1246.17018)]. There was however so far no formal definition of a canonical basis. In the present paper the author suggests a definition of a canonical basis and explains how they typically arise in categorifications. He also shows that his definition incorporates the classical examples of the canonical bases of Kazhdan-Lusztig and Lusztig.
A pre-canonical structure on a free $$\mathbb{Z}[q,q^{-1}]$$-module is a triple consisting of a bar-involution $$\psi$$, a sesquilinear inner product $$<,>$$ and a standard basis $$a_c$$ with a partially ordered index set $$(C,<)$$. A basis $$\{b_c\}$$ of a vector space $$V$$ is called canonical if
(1)
Each $$b_c$$ is invariant under $$\psi$$.
(2)
Each $$b_c$$ is in the set $$a_c + \sum_{c'<c} \mathbb Z[q,q^{-1}] \cdot a_{c'}$$.
(3)
The $$b_c$$ are almost orthonormal: $$<b_c,b_{c'}> \in \delta_{c,c'} + q^{-1} \mathbb Z[[q,q^{-1}]]$$.
A pre-canonical structure can have at most one canonical basis; and in general the dependance on the pre-canonical strucure is weak. It is however difficult to see if a pre-canonical structure has a canonical basis unless the pre-canonical structure comes from a categorification.
The main tool to construct canonical bases is the notion of a humorous category. Such a category $$\mathcal{C}$$ always has an orthodox basis of $$K_0(\mathcal{C})$$ consisting of the classes $$[P_c]$$ of the self-dual indecomposable modules in $$\mathcal{C}$$. If the category moreover satisfies a condition called mixedness, this orthodox basis is a canonical basis in the sense defined above enjoying some additional positivity properties.
The author studies in particular two cases: a categorification $$\dot{\mathcal{U}}$$ of Lusztig’s modified universal enveloping algebra $$\dot{U}$$ and a categorification $$\mathcal{X}^{\underline{\lambda}}$$ of a tensor product sequence of highest and lowest weight integrable representations of a Kac-Moody algebra $$\mathfrak{g}$$. The main theorem for these categorifications is as follows.
Theorem A:
(a)
If $$\mathfrak{g}$$ is a Kac-Moody algebra with symmetric Cartan matrix, the canonical basis of a tensor product of highest-weight integrable representations coincides with the classes of indecomposable objects in the categorification $$\mathcal{X}^{\underline{\lambda}}$$.
(b)
If $$\mathfrak{g}$$ is finite-dimensional and simply laced, the canonical basis of $$\dot{U}$$ coincides with the classes of indecomposable objects in the categorification $$\dot{\mathcal{U}}$$.

MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 18D05 Double categories, $$2$$-categories, bicategories and generalizations (MSC2010)
Keywords:
canonical bases; categorification
Full Text:
References:
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