Session S16 - Quantum symmetries

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The Drinfeld double of the Jordan plane, the super Jordan plane and their restricted versions

Héctor Martín Peña Pollastri

Universidad Nacional de Córdoba, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this work, we consider Hopf algebras naturally associated with the Jordan plane, the super Jordan plane, and their restricted versions in characteristic odd. Specifically, we consider the Drinfeld double of their bosonizations. We describe these algebras in various ways and we compute their simple modules.

The Jordan plane is a well-known quadratic algebra in non-commutative geometry and the theory of quantum groups. Cibils, Lauda, and Witherspoon observed this is a Nichols algebra in characteristic 0 and they compute the corresponding Nichols algebra in positive characteristic. In this case, the resulting algebra is finite-dimensional and we call it the restricted Jordan plane. The super Jordan plane introduced by Andruskiewitsch, Angiono, and Heckenberger in 2016, is an algebra with two generators, one quadratic and one cubic relation. In characteristic 0 it is a Nichols algebra. In 2019 the same authors computed the corresponding Nichols algebra in positive characteristic. This finite-dimensional algebra is called the restricted super Jordan plane. These two algebras are important in the classification of the Nichols algebras with finite Gelfand-Kirillov dimension.

We first consider the restricted Jordan plane, its bosonization $H$ by a cyclic group and its corresponding Drinfeld double $D(H)$. We present the latter with generators and relations. It follows from this presentation that $D(H)$ is an abelian extension (as Hopf algebra) of the restricted enveloping algebra of $\mathfrak{u}(\mathfrak{sl}_2)$ by a local Hopf algebra. This allows us to deduce the simple modules of $D(H)$ are the same as the ones of $\mathfrak{u}(\mathfrak{sl}_2)$. Afterward, we construct an infinite-dimensional covering $\widetilde{D}$ de $D(H)$ and we prove that this is an abelian extension of the enveloping algebra $U(\mathfrak{sl}_2)$ by the algebra of regular functions of a soluble algebraic group. Hence, we obtain a commutative diagram of nine Hopf algebras with three exact rows and three exact columns. The vertical arrows are a (kind of) quantum Frobenius morphisms.

We also obtain similar results with the restricted super Jordan plane but related with the category of super vector spaces. The Drinfeld double constructed $E$ is a bosonization of a Hopf superalgebra $\mathcal E$; that is $E \simeq \mathcal E \# \Bbbk C_2$. Furthermore, $\mathcal E$ is a super abelian extension of the restricted enveloping algebra of the Lie superalgebra $\mathfrak{osp}(1|2)$ by a local Hopf algebra. Hence the simple modules are the same as the ones from $\mathfrak{u}(\mathfrak{osp}(1|2))$. Afterward, we construct an infinite-dimensional covering $\widetilde{\mathcal E}$ of $\mathcal E$. The latter is a super abelian extension of the enveloping algebra $U(\mathfrak{osp}(1|2))$ by the algebra of functions of an algebraic supergroup. We then obtain a similar commutative diagram of nine Hopf superalgebras to the one obtained for the Jordan plane.

Joint work with Nicolás Andruskiewitsch (Universidad Nacional de Córdoba).

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