On Isotropy Groups of Quantum Weyl Algebras and Jordanian Plane

Technical Summary: Isotropy Groups of Quantum Weyl Algebra and Jordanian Plane

Problem Statement
This paper investigates the isotropy groups of derivations on two specific non-commutative algebras: the quantum Weyl algebra $A_1^q(k)$ and the Jordanian plane $\Lambda_2(k)$. The central object of study is the isotropy group of a derivation $\delta$, defined as the subgroup of automorphisms $\rho$ that commute with $\delta$ (i.e., $\rho \delta \rho^{-1} = \delta$), subject to the constraint that $\rho$ must also commute with the underlying automorphism $\sigma$ if $\delta$ is a $\sigma$-derivation. The authors aim to determine the structure of these groups explicitly and utilize them to distinguish between the structural properties of the two algebras, which, despite both being two-generated quadratic algebras, are known to be non-isomorphic.

Methodology
The authors employ a combination of classification theorems, direct computation, and structural decomposition:

1. Quantum Weyl Algebra ($A_1^q(k)$):

   • Classification: The study relies on Brzeziński's classification of $\sigma$-derivations for $A_1^q(k)$ where $q$ is not a root of unity. The automorphism group of this algebra is restricted to diagonal automorphisms $\rho_\beta(x) = \beta^{-1}x, \rho_\beta(y) = \beta y$.
   • Decomposition: The authors decompose the space of $\sigma$-derivations into a direct sum of subspaces: inner derivations ($Inn_\sigma$), a specific family of non-inner derivations associated with polynomials in $h=1-yx$($H_\mu$), and a family of non-inner derivations associated with monomials in $x$ and $y$ ($P_\mu$).
   • Intersection Formula: Utilizing a general lemma, they prove that the isotropy group of an arbitrary $\sigma$-derivation is the intersection of the isotropy groups of its non-zero components in this decomposition.
   • Arithmetic Analysis: For each component, they derive explicit conditions on the parameter $\beta$ of the automorphism $\rho_\beta$, often reducing the problem to arithmetic conditions involving roots of unity and greatest common divisors (gcd) of exponents.

2. Jordanian Plane ($\Lambda_2(k)$):

   • Automorphism and Derivation Structure: The authors utilize known descriptions of the automorphism group (triangular type: $\phi(x) = \gamma x + g(y), \phi(y) = \gamma y$) and derivations (including inner and locally nilpotent types).
   • Commutativity Criteria: They establish a necessary and sufficient condition for an automorphism to belong to the isotropy group of an inner derivation: $\rho(w) - w \in k$.
   • Explicit Computation: They apply this criterion to monomial inner derivations and locally nilpotent derivations (defined by $\delta(y)=0, \delta(x)=p(y)$) to compute their specific isotropy groups.

Key Contributions and Results

• Quantum Weyl Algebra:

  • Explicit Isotropy Groups: The paper provides arithmetic descriptions of isotropy groups for all families of $\sigma$-derivations.
    • For non-inner derivations of type 2a (involving $f(h)$), the isotropy group is the full automorphism group $Aut(R)$ if the derivation is inner, or trivial/restricted based on specific root-of-unity conditions if non-inner.
    • For type 2b derivations (involving $h^d b(y)$ and $h^d a(x)$), the isotropy group is isomorphic to a cyclic group $Z_d$, where $d$ is the greatest common divisor of the exponents in the polynomials $a(x)$ and $b(y)$ plus one.
    • For inner derivations $ad_w$, the isotropy group is determined by the gcd of the set $S = \{j-i \mid c_{i,j} \neq 0\}$. If $\gcd(S) = 0$, the group is $k^*$; if $\gcd(S) = d > 1$, it is $Z_d$; if $\gcd(S) = 1$, it is trivial.
  • Intersection Theorem: The authors prove that for any $\sigma$-derivation $\delta$, its isotropy group is the intersection of the isotropy groups of its non-zero components in the decomposition $Der_\sigma(R) = Inn_\sigma(R) \oplus H_\mu \oplus P_\mu$.

• Jordanian Plane:

  • Monomial Inner Derivations: The isotropy group of $ad_{x^m y^n}$ is characterized. For $m \ge 1$ and $(m,n) \neq (1,0)$, the group is isomorphic to $Z_{m+n}$ (consisting of automorphisms with $\gamma^{m+n}=1$ and $g(y)=0$).
  • Locally Nilpotent Derivations: For derivations defined by $\delta_p(y)=0$ and $\delta_p(x)=p(y)$, the isotropy group consists of automorphisms $\phi_{\gamma, g}$ satisfying $p(\gamma y) = \gamma p(y)$.
  • Triangular Subgroups: A significant finding is that the isotropy group of the locally nilpotent derivation $\delta_1$ (where $p(y)=1$) is isomorphic to the additive group of polynomials $(k[y], +)$, which forms an infinite-dimensional triangular subgroup.

Significance and Structural Distinction
The paper concludes that the collection of isotropy groups serves as a natural invariant reflecting the structural differences between the two algebras:

• In the quantum Weyl algebra (with $q$ not a root of unity), the automorphism group is one-dimensional (a torus $k^*$). Consequently, all isotropy groups of derivations are constrained to be subgroups of this torus, typically finite cyclic groups or the torus itself.
• In the Jordanian plane, the automorphism group is larger, allowing for "large triangular subgroups." The authors demonstrate that the isotropy group of a specific locally nilpotent derivation contains an infinite-dimensional triangular subgroup isomorphic to $(k[y], +)$.

This contrast provides a new perspective on the non-isomorphism of $A_1^q(k)$ and $\Lambda_2(k)$. While their non-isomorphism is a known result (Shirikov), this work suggests that the behavior of isotropy groups—specifically the presence of infinite-dimensional triangular subgroups in the Jordanian plane versus their absence in the quantum Weyl algebra—offers a distinct invariant for distinguishing non-commutative algebras. The paper also poses a broader question regarding the characterization of isotropy groups for Ore extensions of $k[x]$.