Centralizers of Nilpotent Elements in Basic Classical Lie Superalgebras in Good Characteristic

Area of Science:

• Lie superalgebra theory
• Representation theory
• Abstract algebra

Background:

• Basic classical Lie superalgebras g = g0 ⊕ g1 over an algebraically closed field K with characteristic p > 0.
• Reductive algebraic group G0 with Lie(G0) = g0.
• Nilpotent elements e in g0.

Purpose of the Study:

• Calculate a basis for the centralizer g^e of e in g.
• Determine a basis for the center z(g^e) of g^e.
• Classify nilpotent elements in exceptional Lie superalgebras D(2,1;α), G(3), and F(4) based on reachability and Panyushev property.

Main Methods:

• Utilizing associated cocharacters τ: K× → G0 of nilpotent elements.
• Employing algebraic group theory and representation theory techniques.
• Analyzing the structure of centralizers and centers of nilpotent elements.

Main Results:

• A method for computing bases of g^e and z(g^e) is established.
• Classification of specific types of nilpotent elements (reachable, strongly reachable, Panyushev property) is provided for D(2,1;α), G(3), and F(4).

Conclusions:

• The study provides computational tools for understanding the structure of Lie superalgebras via nilpotent elements.
• The classification advances the study of exceptional Lie superalgebras and their properties.
• Findings contribute to the broader understanding of algebraic structures in characteristic p > 0.