Deformation Quantization of Nonassociative Algebras

Deformation Quantization of Nonassociative Algebras

We investigate formal deformations of certain classes of nonassociative algebras including classes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">K</m...

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Bibliographic Details
Main Author: Elisabeth Remm
Format: Article
Language:English
Published: MDPI AG 2024-12-01
Series:Mathematics
Subjects:
nonassociative algebras
deformation quantization
polarization
Online Access:https://www.mdpi.com/2227-7390/13/1/58
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Summary:We investigate formal deformations of certain classes of nonassociative algebras including classes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">K</mi><mo>[</mo><msub><mo>Σ</mo><mn>3</mn></msub><mo>]</mo></mrow></semantics></math></inline-formula>-associative algebras, Lie-admissible algebras and anti-associative algebras. In a process which is similar to Poisson algebra for the associative case, we identify for each type of algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></semantics></math></inline-formula> a type of algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>ψ</mi><mo>)</mo></mrow></semantics></math></inline-formula> such that formal deformations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></semantics></math></inline-formula> appear as quantizations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>ψ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. The process of polarization/depolarization associates to each nonassociative algebra a couple of algebras which products are respectively commutative and skew-symmetric and it is linked with the algebra obtained from the formal deformation. The anti-associative case is developed with a link with the Jacobi–Jordan algebras.
ISSN:2227-7390

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