[Paper Review] Chevalley Supergroups of type D(2,1;a)

Giovanna Fabio|arXiv (Cornell University)|Jun 2, 2010

Advanced Topics in Algebra10 references3 citations

TL;DR

This paper completes the construction of Chevalley supergroups for all simple Lie superalgebras by extending the classical Chevalley method to Lie superalgebras of type $D(2,1;a)$ for non-integer $a$. It introduces a new class of multiplicative one-parameter subgroups modeled on the group functor $A \mapsto P_a(A)$, which allows the realization of the supergroup structure even when $a \notin \mathbb{Z}$, and proves that the resulting functor is representable and has the original Lie superalgebra as its tangent algebra.

ABSTRACT

I present a construction "a` la Chevalley" of affine supergroups associated with simple Lie superalgebras of (classical) type D(2,1;a), for any possible value of the parameter a - in particular, including non-integral values of a. This extends the similar work performed in [R. Fioresi, F. Gavarini, "Chevalley Supergroups", Memoirs of the AMS 215 (2012), no. 1014 - arXiv:0808.0785v8 [math.RA]], where all other simple Lie superalgebras of classical type were considered. The case of simple Lie superalgebras of Cartan type is dealt with in [F. Gavarini, "Algebraic supergroups of Cartan type", Forum Mathematicum (to appear), 92 pages - arXiv:1109.0626v5 [math.RA], so this work completes the program of constructing connected affine supergroups associated with any simple Lie superalgebra.

Motivation & Objective

• To complete the program of constructing connected affine supergroups for all simple Lie superalgebras by addressing the missing case of $D(2,1;a)$ with $a \notin \mathbb{Z}$.
• To extend the Chevalley construction to supergroups associated with Lie superalgebras of type $D(2,1;a)$, where the standard method fails due to non-integral parameter $a$.
• To define a new class of multiplicative one-parameter subgroups using the group functor $A \mapsto P_a(A)$, enabling the construction when $a$ is not an integer.
• To prove that the resulting group functor is representable, thus forming an affine supergroup, and that its tangent Lie superalgebra is isomorphic to the original $\mathfrak{g} = D(2,1;a)$.

Proposed method

• Construct a Chevalley basis for $\mathfrak{g} = D(2,1;a)$ with integral structure, ensuring integrality in the universal enveloping superalgebra $U(\mathfrak{g})$.
• Fix a faithful finite-dimensional $\mathfrak{g}$-module $V$ and construct a $\mathbb{Z}$-lattice $M \subset V$ stable under the Kostant $\mathbb{Z}$-form of $U(\mathfrak{g})$.
• Define additive one-parameter subgroups for each root $\alpha \in \Delta$, modeled on $1 + \vartheta X_\alpha$ for $\vartheta \in A_1$.
• Introduce multiplicative one-parameter subgroups of $a$-type using the group functor $P_a(A) = \{ t \in A_0 \mid t^{a^k} \text{ exists for all } k \}$, which are essential for non-integer $a$.
• Define the group functor $\mathbf{G}_V$ as the subgroup of $\mathrm{GL}(V(A))$ generated by all homogeneous one-parameter subgroups, and sheafify it to obtain a representable affine supergroup.
• Establish a factorization $\mathbf{G}_V = \mathbf{G}_0 \times \mathbf{G}_1^{-,<} \times \mathbf{G}_1^{+,<}$, proving representability via the representability of $\mathbf{G}_0$ and $\mathbf{G}_1$.

Experimental results

Research questions

• RQ1 Can the Chevalley construction be extended to Lie superalgebras of type $D(2,1;a)$ when $a$ is not an integer?
• RQ2 What new group functors are required to realize the supergroup structure in the non-integer $a$ case?
• RQ3 Is the resulting group functor representable, and does it yield an affine supergroup?
• RQ4 Does the tangent Lie superalgebra of the constructed supergroup match the original $\mathfrak{g} = D(2,1;a)$?
• RQ5 How do different choices of $\mathfrak{g}$-modules affect the resulting supergroup up to isomorphism?

Key findings

• The construction of Chevalley supergroups for $D(2,1;a)$ with $a \notin \mathbb{Z}$ is completed by introducing a new class of multiplicative one-parameter subgroups based on the group functor $A \mapsto P_a(A)$.
• The resulting group functor $\mathbf{G}_V$ is sheafified and proven to be representable, hence forming an affine supergroup.
• The tangent Lie superalgebra of $\mathbf{G}_V$ is isomorphic to $\mathfrak{g} = D(2,1;a)$, as required by Lie's Third Theorem for supergroups.
• The construction is functorial in the choice of the $\mathfrak{g}$-module $V$, and the resulting supergroup is independent of the choice of admissible lattice in $V$ up to isomorphism.
• A factorization $\mathbf{G}_V = \mathbf{G}_0 \times \mathbf{G}_1^{-,<} \times \mathbf{G}_1^{+,<}$ confirms the supergroup structure and enables the representability proof.
• The result completes the program of constructing connected affine supergroups for all simple Lie superalgebras, including all classical and Cartan-type cases.