🧮Non-associative Algebra Unit 5 – Malcev algebras

Key Concepts and Definitions

• Malcev algebras are non-associative algebras that satisfy the Malcev identity: $(xy)(xz) = ((xy)x)z + x((yx)z) + (x(yz))x$
• Malcev algebras are named after Anatoly Malcev, who introduced them in 1955
• The Malcev identity is a weakening of the Jacobi identity for Lie algebras
• Malcev algebras are power-associative, meaning that the subalgebra generated by any element is associative
• The center of a Malcev algebra $A$ is defined as $Z(A) = \{x \in A : xy = yx \text{ for all } y \in A\}$
• A Malcev algebra is called simple if it has no non-trivial ideals and is not a Lie algebra
• The radical of a Malcev algebra $A$ is the largest solvable ideal of $A$

Historical Context and Development

• Malcev algebras were introduced by Anatoly Malcev in 1955 as a generalization of Lie algebras
• The study of Malcev algebras was motivated by the problem of classifying simple non-associative algebras
• In the 1960s and 1970s, the structure theory of Malcev algebras was developed by mathematicians such as Sagle, Kuzmin, and Filippov
• The classification of simple Malcev algebras over algebraically closed fields was completed by Kuzmin in 1968
• The study of Malcev algebras has led to the development of other non-associative structures, such as Bol algebras and Malcev-admissible algebras
• Malcev algebras have found applications in various areas of mathematics, including differential geometry, mathematical physics, and combinatorics

Algebraic Structure and Properties

• Malcev algebras are vector spaces equipped with a bilinear operation $[\cdot,\cdot]$ satisfying the Malcev identity
• The Malcev identity can be written in terms of the Jacobian $J(x,y,z) = [[x,y],z] + [[y,z],x] + [[z,x],y]$ as $J(x,y,[x,z]) = [J(x,y,z),x]$
• Malcev algebras are not necessarily associative or commutative
• The associator of elements $x,y,z$ in a Malcev algebra is defined as $[x,y,z] = (xy)z - x(yz)$
• The Malcev identity implies the flexibility condition: $[x,y,x] = 0$ for all $x,y$ in the algebra
• The center of a Malcev algebra is a subalgebra and an ideal
• The quotient of a Malcev algebra by its radical is a semisimple Malcev algebra

Relationship to Lie Algebras

• Every Lie algebra is a Malcev algebra, but not every Malcev algebra is a Lie algebra
• The Malcev identity reduces to the Jacobi identity when the algebra is anticommutative (i.e., a Lie algebra)
• The universal enveloping algebra of a Malcev algebra is a non-associative algebra that generalizes the universal enveloping algebra of a Lie algebra
• The Poincaré-Birkhoff-Witt theorem for Lie algebras has an analog for Malcev algebras, known as the Pérez-Izquierdo-Shestakov theorem
• The study of Malcev algebras has led to the development of Lie-admissible algebras, which are algebras that give rise to Lie algebras through antisymmetrization of the product

Important Theorems and Proofs

• The Sagle-Kuzmin theorem states that every simple Malcev algebra over an algebraically closed field of characteristic not 2 or 3 is isomorphic to one of the following:
  • A simple Lie algebra
  • The 7-dimensional Malcev algebra $M_7$
  • The 5-dimensional Malcev algebra $M_5$
• The proof of the Sagle-Kuzmin theorem relies on the classification of simple Lie algebras and the study of the Killing form on Malcev algebras
• The Pérez-Izquierdo-Shestakov theorem provides a basis for the universal enveloping algebra of a Malcev algebra, generalizing the Poincaré-Birkhoff-Witt theorem for Lie algebras
• The proof of the Pérez-Izquierdo-Shestakov theorem uses the notion of Ω-algebras and the Diamond Lemma for non-associative algebras
• The Filippov-Kuzmin theorem classifies the finite-dimensional simple Malcev algebras over fields of characteristic 0

Applications in Mathematics and Physics

• Malcev algebras have been used to study the geometry of smooth loops and quasigroups, which are non-associative algebraic structures with applications in differential geometry and topology
• In mathematical physics, Malcev algebras have been used to construct non-associative gauge theories and to study the symmetries of certain quantum systems
• Malcev algebras have been applied to the study of integrable systems and the Yang-Baxter equation, which has connections to quantum groups and statistical mechanics
• In combinatorics, Malcev algebras have been used to construct non-associative analogs of the Hopf algebra of symmetric functions, with applications to the study of free Lie algebras and the descent algebra
• Malcev algebras have been used to construct non-associative deformations of classical and quantum mechanics, providing a framework for studying non-associative physical theories

Computational Techniques and Examples

• The Malcev identity and the flexibility condition can be used to derive identities and properties of Malcev algebras using computer algebra systems (Maple, Mathematica)
• The structure constants of a Malcev algebra can be computed using the Malcev identity and the bilinearity of the product
• Example: The 7-dimensional Malcev algebra $M_7$ has a basis $\{e_1, \ldots, e_7\}$ with non-zero products given by:
  • $e_1 e_2 = e_3$, $e_1 e_4 = e_5$, $e_1 e_6 = e_7$
  • $e_2 e_4 = e_6$, $e_2 e_5 = -e_7$
  • $e_3 e_4 = -e_7$, $e_3 e_5 = e_6$
• The Pérez-Izquierdo-Shestakov theorem can be used to compute a basis for the universal enveloping algebra of a Malcev algebra using Gröbner basis techniques
• Computational methods have been developed for constructing Malcev algebras from Lie algebras and for classifying low-dimensional Malcev algebras

Advanced Topics and Current Research

• The study of infinite-dimensional Malcev algebras and their representations is an active area of research
• Malcev algebras have been generalized to Malcev-admissible algebras, which are algebras that give rise to Malcev algebras through a certain construction
• The cohomology theory of Malcev algebras has been developed, extending the cohomology theory of Lie algebras
• Deformation theory has been applied to the study of Malcev algebras, leading to the notion of Malcev-Poisson algebras and non-associative analogs of quantum groups
• The relationship between Malcev algebras and other non-associative structures, such as Jordan algebras and alternative algebras, is an ongoing area of investigation
• Malcev algebras have been studied in the context of operads and category theory, providing new perspectives on their algebraic structure and representation theory
• The application of Malcev algebras to non-associative geometry and physics is a promising area of current research, with potential implications for the study of quantum gravity and non-commutative spacetime