The algebraic and geometric classification of right alternative superalgebras

Technical Summary: Algebraic and Geometric Classification of Right Alternative Superalgebras

Problem Statement
The paper addresses the classification of 3-dimensional right alternative superalgebras over the complex field $\mathbb{C}$. This work extends the classical problem of algebraic classification (up to isomorphism) for non-associative algebras defined by polynomial identities to the super-setting. Specifically, the authors aim to classify all non-trivial 3-dimensional right alternative superalgebras of types $(1,2)$ and $(2,1)$, where the dimension of the even part is 1 or 2, respectively. Furthermore, the study seeks to provide the geometric classification of these varieties, determining their irreducible components and rigid algebras. As a consequence, the authors derive classifications for several subvarieties, including perm, binary perm, associative, binary associative, $(-1,1)$-, and binary $(-1,1)$-superalgebras.

Methodology
The authors employ a two-pronged approach combining algebraic and geometric techniques:

1. Algebraic Classification:

   • Symmetrization: The method relies on the relationship between right alternative superalgebras and Jordan superalgebras. For a right alternative superalgebra $(A, \cdot)$, the symmetrized product $x \bullet y = \frac{1}{2}(xy + (-1)^{|x||y|}yx)$ defines a Jordan superalgebra $A^+$.
   • Cohomological Approach: The authors utilize the space $Z_2(A, A)$ of super-skew-symmetric bilinear maps satisfying a specific identity derived from the right alternative condition. Any right alternative superalgebra structure on a fixed Jordan superalgebra $(A, \bullet)$ can be represented as $x * y = x \bullet y + \theta(x, y)$ for some $\theta \in Z_2(A, A)$.
   • Orbit Analysis: The classification proceeds by computing $Z_2(A, A)$ for each isomorphism class of 3-dimensional Jordan superalgebras of types $(1,2)$ and $(2,1)$. The automorphism group $\text{Aut}(A)$ acts on $Z_2(A, A)$, and the isomorphism classes of right alternative superalgebras correspond to the orbits of this action. The authors explicitly compute these orbits to list all non-isomorphic structures.

2. Geometric Classification:

   • Degenerations: The geometric classification analyzes the variety of superalgebras as a subset of the affine space of structure constants. The authors investigate degenerations (limits of orbits) between algebras. An algebra $A$ degenerates to $B$ ($A \to B$) if $B$ lies in the Zariski closure of the orbit of $A$.
   • Irreducible Components: By identifying the rigid algebras (those whose orbits are open) and the degenerations between them, the authors determine the irreducible components of the varieties.
   • Non-Degeneration Proofs: To establish that a degeneration does not exist, the authors use invariants such as the dimension of the square of the even part, commutativity properties, and specific relations among structure constants, often employing Borel subgroup arguments.

Key Contributions and Results

• Algebraic Classification (Theorems A1 and A2):

  • The paper provides a complete list of all non-trivial 3-dimensional right alternative superalgebras.
  • Type (1,2): There are 28 distinct isomorphism classes (denoted $R_{01}$ through $R_{28}$), including a one-parameter family $R_{\alpha \neq -1}^{01}$ and $R_{\alpha}^{04}$ (where $R_{\alpha}^{04} \cong R_{-\alpha}^{04}$).
  • Type (2,1): There are 39 distinct isomorphism classes (denoted $R_{01}$ through $R_{39}$).
  • The classification is exhaustive, covering all possible underlying Jordan superalgebra structures ($J_{01}$ to $J_{12}$ for type $(1,2)$ and $J_{01}$ to $J_{15}$ for type $(2,1)$).

• Subvariety Classifications:

  • The authors derive specific classifications for subvarieties by filtering the main list based on additional identities.
  • Associative/Commutative: Corollaries 6 and 7 list associative commutative superalgebras.
  • Perm and Binary Perm: Corollaries 10–13 classify perm and binary perm superalgebras. Notably, Corollary 13 states that every 3-dimensional binary perm superalgebra of type $(2,1)$ is a perm superalgebra.
  • Associative and Binary Associative: Corollaries 15–18 classify these, noting that for type $(2,1)$, binary associative superalgebras are either associative or isomorphic to specific cases ($R_{28}, R_{30}$).
  • $(-1,1)$ and Binary $(-1,1)$: Corollaries 21–26 provide classifications for these Lie-admissible varieties. A significant finding is Corollary 24, which states that every 3-dimensional $(-1,1)$-superalgebra of type $(2,1)$ is associative.

• Geometric Classification (Theorems G1 and G2):

  • Type (1,2): The variety has dimension 5 and consists of 13 irreducible components. There are 11 rigid superalgebras. The components are defined by the orbits of specific representatives (e.g., $O(R_{\alpha}^{01})$, $O(R_{\alpha}^{07})$, etc.).
  • Type (2,1): The variety has dimension 5 and consists of 15 irreducible components. There are 15 rigid superalgebras.
  • The paper details the specific degenerations (e.g., $R_{16} \to R_{15}$) and non-degenerations that define the topology of these varieties.

• Geometric Classification of Subvarieties:

  • Theorems 28–40 provide the geometric classification for the subvarieties mentioned above, specifying their dimensions, number of irreducible components, and rigid elements. For instance, the variety of perm superalgebras of type $(1,2)$ has dimension 4 with 8 irreducible components.

Significance and Claims
The paper claims to provide the first complete algebraic and geometric classification of 3-dimensional right alternative superalgebras. The authors position this work as a necessary step in understanding super-generalizations of principal varieties of non-associative algebras.

A specific contribution highlighted is the investigation of the "binary" versions of these algebras (where every 2-generated subalgebra satisfies the identity). The authors address an open question regarding whether every 3-dimensional binary-$\Omega$ algebra is an $\Omega$-algebra in the super-setting. While the super-version is known to be false for Lie algebras, the paper provides a nuanced answer for perm, associative, and $(-1,1)$-superalgebras, showing that in many 3-dimensional cases, the binary condition implies the standard condition, though exceptions exist (e.g., in binary perm and binary associative cases).

The work serves as a foundational reference for the structure theory of low-dimensional non-associative superalgebras, offering explicit multiplication tables and a detailed map of the geometric landscape (degenerations and components) for these varieties.