[Paper Review] Cohomology of Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$

Yucai Su, R. B. Zhang|ArXiv.org|Feb 26, 2004

Algebraic structures and combinatorial models20 references3 citations

TL;DR

This paper computes the first and second cohomology groups of the Lie superalgebras $\mathfrak{sl}_{m|n}$ and $\mathfrak{osp}_{2|2n}$ with coefficients in finite-dimensional irreducible modules and Kac modules. It proves that the second cohomology group with coefficients in the universal enveloping algebra under the adjoint action vanishes, implying that $\mathrm{U}(\mathfrak{sl}_{m|n})$ and $\mathrm{U}(\mathfrak{osp}_{2|2n})$ are formally rigid and admit no non-trivial Gerstenhaber-type deformations.

ABSTRACT

We explicitly compute the first and second cohomology groups of the classical Lie superalgebras $sl_{m|n}$ and $osp_{2|2n}$ with coefficients in the finite dimensional irreducible modules and the Kac modules. We also show that the second cohomology groups of these Lie superalgebras with coefficients in the respective universal enveloping algebras (under the adjoint action) vanish. The latter result in particular implies that the universal enveloping algebras $U(sl_{m|n})$ and $U(osp_{2|2n})$ do not admit any non-trivial formal deformations of Gerstenhaber type.

Motivation & Objective

• To compute the first and second Lie superalgebra cohomology groups of $\mathfrak{sl}_{m|n}$ and $\mathfrak{osp}_{2|2n}$ with coefficients in finite-dimensional irreducible modules and Kac modules.
• To resolve Kac's open problem on the first cohomology of basic classical Lie superalgebras with irreducible coefficients.
• To determine the formal deformation theory of the universal enveloping algebras $\mathrm{U}(\mathfrak{sl}_{m|n})$ and $\mathrm{U}(\mathfrak{osp}_{2|2n})$ via cohomological invariants.
• To establish that these universal enveloping algebras are formally rigid, meaning they admit no non-trivial deformations of Gerstenhaber type.
• To provide a comprehensive, elementary computation of cohomology groups using weight diagrams and root system techniques.

Proposed method

• The authors use weight diagrams and root system analysis to classify atypical weights and compute cohomology groups via explicit calculations on Kac modules and irreducible modules.
• They define the northeast chain $NE_\mu$ and the primitive weight graph to analyze the structure of cohomology classes.
• The computation relies on the identification of non-vanishing cohomology classes through the action of positive odd roots and the use of Dynkin labels to track weight shifts.
• The authors employ the adjoint action on the universal enveloping algebra to compute $H^2(\mathfrak{g}, \mathrm{U}(\mathfrak{g}))$, showing it vanishes via structural constraints on the algebra.
• They apply technical lemmas on atypical roots and the $nqc(\mu)$ and $A(\mu)$ matrices to classify cases and determine cohomology dimensions.
• The proof proceeds by case analysis on the highest weights $\mu$, distinguishing cases based on atypicality and root chain structures.

Experimental results

Research questions

• RQ1 What are the first and second cohomology groups of $\mathfrak{sl}_{m|n}$ and $\mathfrak{osp}_{2|2n}$ with coefficients in finite-dimensional irreducible modules and Kac modules?
• RQ2 Does the second cohomology group $H^2(\mathfrak{g}, \mathrm{U}(\mathfrak{g}))$ vanish for $\mathfrak{g} = \mathfrak{sl}_{m|n}$ and $\mathfrak{osp}_{2|2n}$, implying formal rigidity of their universal enveloping algebras?
• RQ3 How do the cohomology groups depend on the atypicality of the highest weight $\mu$ in irreducible modules?
• RQ4 What is the structure of $H^1(\mathfrak{g}, \mathrm{U}(\mathfrak{g}))$ and $H^2(\mathfrak{g}, \mathrm{U}(\mathfrak{g}))$ for these Lie superalgebras?
• RQ5 Can the cohomology groups be computed explicitly using weight diagrams and root system data?

Key findings

• The first cohomology group $H^1(\mathfrak{sl}_{m|n}, V)$ and $H^1(\mathfrak{osp}_{2|2n}, V)$ with coefficients in irreducible modules $V$ are computed explicitly, with non-trivial results depending on the atypicality of the highest weight.
• The second cohomology group $H^2(\mathfrak{sl}_{m|n}, V)$ and $H^2(\mathfrak{osp}_{2|2n}, V)$ with coefficients in irreducible modules vanish for all weights $\mu$ except possibly in specific atypical cases, which are fully classified.
• The second cohomology group $H^2(\mathfrak{g}, \mathrm{U}(\mathfrak{g}))$ vanishes for $\mathfrak{g} = \mathfrak{sl}_{m|n}$ and $\mathfrak{g} = \mathfrak{osp}_{2|2n}$, implying that $\mathrm{U}(\mathfrak{g})$ admits no non-trivial formal deformations of Gerstenhaber type.
• The first cohomology group $H^1(\mathfrak{g}, \mathrm{U}(\mathfrak{g}))$ is non-zero for $\mathfrak{g} = \mathfrak{sl}_{m|n}$ and $\mathfrak{g} = \mathfrak{osp}_{2|2n}$, indicating that the coalgebra structure of $\mathrm{U}(\mathfrak{g})$ admits non-trivial deformations.
• The cohomology computations are fully classified via the $A(\mu)$ and $NE_\mu$ matrices, with explicit formulas for the dimension of cohomology groups in terms of weight data and atypical roots.
• The results resolve Kac's problem on first cohomology with irreducible coefficients for $\mathfrak{sl}_{m|n}$ and $\mathfrak{osp}_{2|2n}$, providing the first complete computation of these groups for non-trivial coefficients in basic classical Lie superalgebras.