[Paper Review] Lusztig isomorphisms for Drinfel'd doubles of Nichols algebras of diagonal type

I. Heckenberger|arXiv (Cornell University)|Oct 24, 2007

Algebraic structures and combinatorial models16 references5 citations

TL;DR

This paper establishes Lusztig isomorphisms for Drinfel'd doubles of Nichols algebras of diagonal type, generalizing Lusztig's original maps from quantized enveloping algebras. It proves these isomorphisms satisfy generalized Coxeter-type relations and uses them to characterize Nichols algebras with finite arithmetic root systems.

ABSTRACT

Abstract. In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel’d double for a class of graded Hopf algebras is investigated. Various quantum algebras, including multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfel’d doubles Lusztig maps are defined. It is shown that these maps induce isomorphisms between doubles of Nichols algebras of diagonal type. Further, the obtained isomorphisms satisfy Coxeter type relations in a generalized sense. As an application, the Lusztig isomorphisms are used to give a characterization of Nichols algebras of diagonal type with finite arithmetic root system.

Motivation & Objective

• To extend Lusztig's isomorphisms from quantized enveloping algebras to Drinfel'd doubles of Nichols algebras of diagonal type.
• To establish a general framework for constructing isomorphisms in a class of quantum algebras, including multiparameter quantizations of semisimple Lie algebras and Lie superalgebras.
• To investigate the algebraic structure of Drinfel'd doubles in the context of Nichols algebras with diagonal braiding.
• To characterize Nichols algebras of diagonal type whose arithmetic root system is finite using the constructed isomorphisms.

Proposed method

• Define Lusztig maps on the Drinfel'd double of a graded Hopf algebra, specifically tailored to Nichols algebras of diagonal type.
• Construct isomorphisms between Drinfel'd doubles using these maps, generalizing Lusztig's original construction for quantized enveloping algebras.
• Demonstrate that the induced isomorphisms satisfy generalized Coxeter-type relations, extending the braid group action structure.
• Utilize the isomorphisms to analyze the arithmetic root system of the Nichols algebra, focusing on finiteness conditions.
• Apply the isomorphisms to relate the structure of the Drinfel'd double to the combinatorics of the root system.
• Use the generalized Coxeter relations to derive constraints on the braiding matrix, leading to a characterization of finiteness.

Experimental results

Research questions

• RQ1 How can Lusztig's isomorphisms be generalized to the setting of Drinfel'd doubles of Nichols algebras of diagonal type?
• RQ2 Do the constructed isomorphisms satisfy generalized Coxeter-type relations in the context of these doubles?
• RQ3 Can the Lusztig isomorphisms be used to characterize Nichols algebras of diagonal type with finite arithmetic root systems?
• RQ4 To what extent do the isomorphisms preserve the quantum group-like structure in multiparameter quantizations?
• RQ5 What structural constraints on the braiding matrix emerge from the interplay between the isomorphisms and the root system finiteness?

Key findings

• Lusztig isomorphisms are successfully defined for Drinfel'd doubles of Nichols algebras of diagonal type, extending their scope beyond quantized enveloping algebras.
• The induced isomorphisms satisfy generalized Coxeter-type relations, generalizing the braid group action structure in quantum groups.
• The isomorphisms provide a structural tool to analyze the arithmetic root system of the Nichols algebra.
• The paper gives a characterization of Nichols algebras of diagonal type with finite arithmetic root systems using the Lusztig isomorphisms.
• The construction applies uniformly to multiparameter quantizations of semisimple Lie algebras and Lie superalgebras.
• The isomorphisms preserve the Drinfel'd double structure and reveal hidden symmetries in the algebraic data.