[Paper Review] An analog of the classical invariant theory for Lie superlagebras

A. N. Sergeev|ArXiv.org|Oct 17, 1998

Advanced Topics in Algebra6 references4 citations

TL;DR

This paper extends classical invariant theory to Lie superalgebras by characterizing G-invariant elements in the tensor algebra of a finite-dimensional superspace V, where G ranges over classical superalgebras such as GL(V), SL(V), OSP(V), and their odd analogs Q(V), SQ(V), Pe(V), SPe(V). The key contribution is a complete description of invariants using superalgebraic structures and supertrace-like invariants, generalizing classical results to the super setting with explicit generators and relations.

ABSTRACT

Let V be a finite-dimensional superspace and G a simple (or a ``close'' to simple) matrix Lie superalgebra, i.e., a Lie subsuperalgebra in GL(V). Under the classical invariant theory for G we mean the description of G-invariant elements of the tensor algebra of V. We give such description for GL(V), SL(V) and OSP(V) and their ``odd'' analogs: Q(V), SQ(V); Pe(V) and SPe(V).

Motivation & Objective

• To generalize classical invariant theory to the setting of Lie superalgebras.
• To describe the structure of G-invariant elements in the tensor algebra of a finite-dimensional superspace V.
• To extend known results for classical Lie algebras to superalgebras such as GL(V), SL(V), OSP(V), and their odd counterparts.
• To provide explicit generators and relations for invariants under these superalgebras.
• To unify the treatment of invariants across both even and odd superalgebras using superalgebraic techniques.

Proposed method

• Use of superalgebraic structures to define and analyze invariants in the tensor algebra of a superspace V.
• Construction of invariants via supertrace and superdeterminant-like operations on tensor powers of V.
• Application of Schur-Weyl duality analogs in the super setting to decompose tensor spaces into irreducible supermodules.
• Identification of fundamental invariants through the use of odd and even superalgebraic generators.
• Use of diagrammatic and combinatorial techniques (e.g., super analogs of Young symmetrizers) to describe relations among invariants.
• Leveraging the structure of matrix Lie superalgebras as subalgebras of GL(V) to classify invariants under their action.

Experimental results

Research questions

• RQ1 What is the structure of the algebra of G-invariant tensors in the tensor algebra of a superspace V for classical Lie superalgebras G?
• RQ2 How do the invariants of odd superalgebras like Q(V) and Pe(V) differ from those of even superalgebras like GL(V) and OSP(V)?
• RQ3 Can classical invariant theory results be extended to superalgebras using superalgebraic tools such as supertraces and superdeterminants?
• RQ4 What are the explicit generators and relations for the algebra of invariants under SL(V) and its odd analog SQ(V)?
• RQ5 How do the symmetries of tensor powers of V decompose under the action of superalgebras like OSP(V) and SPe(V)?

Key findings

• The paper provides a complete description of the algebra of invariants for GL(V), SL(V), and OSP(V), showing that they are generated by supertrace-like invariants.
• For the odd superalgebras Q(V), SQ(V), Pe(V), and SPe(V), the invariants are described via explicit generators, revealing distinct structural behavior compared to even superalgebras.
• The invariants of SL(V) and its odd analog SQ(V) are shown to be related to superdeterminants and supertrace relations.
• The paper establishes that the invariants under OSP(V) and SPe(V) arise from super-symmetric bilinear forms and their contractions.
• A uniform framework is developed that unifies the description of invariants across both even and odd superalgebras using superalgebraic duality principles.
• The results generalize classical invariant theory by extending the role of Schur functors and Young symmetrizers to the super setting.