Pseudo-Euclidean Novikov Superalgebras: Structure and Properties

Technical Summary: Pseudo-Euclidean Novikov Superalgebras: Structure and Properties

Problem Statement
This paper investigates the structure and classification of pseudo-Euclidean Novikov superalgebras. A pseudo-Euclidean Novikov superalgebra is defined as a Novikov superalgebra $(A, \bullet)$ endowed with a non-degenerate symmetric bilinear form $\langle \cdot, \cdot \rangle$ such that all left multiplication operators $L_u$ are $\langle \cdot, \cdot \rangle$-antisymmetric. The study is motivated by the geometric context of flat pseudo-Euclidean Lie superalgebras, where the Levi-Civita product induces a left-symmetric structure. While the relationship between flat pseudo-Euclidean Lie algebras and Novikov algebras has been explored in the non-super setting (e.g., in the Lorentzian case), this work extends the theory to the super setting, addressing the structural decomposition and construction of these objects, particularly when the underlying two-sided ideal is degenerate.

Methodology
The authors employ a structural approach combining algebraic identities, representation theory, and inductive construction methods:

1. Algebraic Characterization: The paper establishes that a pseudo-Euclidean Novikov superalgebra is equivalent to a flat pseudo-Euclidean Lie superalgebra $(A^-, \langle \cdot, \cdot \rangle)$ where the right multiplication operators commute ($[R_u, R_v] = 0$). This equivalence allows the reduction of the problem to the study of flat Lie superalgebras with commuting right multiplications.
2. Structural Decomposition: The authors prove that for such superalgebras, the product satisfies $(u \bullet v) \bullet w = 0$ for all $u, v, w$. This leads to the introduction of a distinguished subclass called Milnor superalgebras, characterized by the existence of a non-degenerate two-sided ideal $I$ such that $A \bullet A \subseteq I$ and $L_u = 0$ for all $u \in I$.
3. Representation and Bimodules: The paper introduces the notion of representations for left-Leibniz $L$-superalgebras. It establishes a characterization of pseudo-Euclidean Novikov superalgebras via bimodule isomorphisms, specifically showing that the existence of the required bilinear form is equivalent to the isomorphism between the regular bimodule $(A, R_\bullet, L_\bullet)$ and a dual bimodule involving the product $\star$ defined by $\langle u \bullet v, w \rangle = \langle u, v \star w \rangle$.
4. Double Extension Procedure: Inspired by the double extension of quadratic Lie algebras (Medina and Revoy), the authors develop a "double extension" procedure for pseudo-Euclidean Novikov superalgebras. This involves central extensions and almost semi-direct products to construct new algebras from existing ones, handling both even and odd parity cases.
5. Inductive Classification: Using the double extension method, the authors prove that any pseudo-Euclidean Novikov superalgebra with a degenerate two-sided ideal can be obtained via a sequence of double extensions starting from a Milnor superalgebra.

Key Contributions and Results

• Equivalence and Structural Properties:

  • It is proven that a pseudo-Euclidean superalgebra is Novikov if and only if it is an LR-superalgebra (both left and right symmetric) if and only if it is a left-Leibniz $L$-superalgebra.
  • A key lemma establishes that if left multiplications are antisymmetric and right multiplications commute, the algebra satisfies $(u \bullet v) \bullet w = 0$.
  • The nilpotency of the superalgebra is shown to be equivalent to the nilpotency of the associated Lie superalgebra $A^-$ and the nilpotency of all left multiplication operators.

• Milnor Superalgebras:

  • The paper defines Milnor superalgebras as a subclass where the algebra decomposes into a non-degenerate ideal and its orthogonal complement, with specific vanishing conditions on the product.
  • Theorem: Any pseudo-Euclidean Novikov superalgebra with a non-degenerate two-sided ideal is necessarily a Milnor superalgebra.

• Construction via Double Extensions:

  • The authors introduce four types of double extensions: even/odd extensions of even/odd pseudo-Euclidean Novikov superalgebras.
  • Theorem: Every pseudo-Euclidean Novikov superalgebra with a degenerate two-sided ideal is obtained via a double extension of a lower-dimensional pseudo-Euclidean Novikov superalgebra.
  • Main Structural Theorem: Any pseudo-Euclidean Novikov superalgebra is either a Milnor superalgebra or can be obtained by a finite sequence of double extensions starting from a Milnor superalgebra.

• Classification in Low Dimensions:

  • The paper provides a complete classification of pseudo-Euclidean Novikov superalgebras of total dimension at most four over an algebraically closed field of characteristic zero.
  • The classification distinguishes between cases where the product space $A \bullet A$ is non-degenerate (Milnor type) and where it is degenerate (constructed via double extensions).
  • Specific isomorphism classes are listed for dimensions 2, 3, and 4, covering both even and odd parity cases, including trivial, solvable, and nilpotent examples.

Significance
The paper claims to provide a systematic structural theory for pseudo-Euclidean Novikov superalgebras, bridging the gap between flat pseudo-Euclidean Lie superalgebras and Novikov superalgebras. By introducing the concept of Milnor superalgebras and the double extension procedure, the authors reduce the classification problem to a manageable inductive process. The complete classification up to dimension four serves as a foundational step for understanding higher-dimensional structures. The work generalizes previous results on pseudo-Euclidean Novikov algebras (non-super) and flat Lie algebras to the super setting, offering a unified framework that handles both non-degenerate and degenerate metric cases. The results are presented as a direct extension of known theorems in Lie theory and Novikov algebra theory, without claiming new external applications beyond the structural classification itself.