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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: Birkhäuser (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)
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