insight - Algebra - # Algebra Embedding Problem

The Embedding Problem for Simple Algebras with Involution Over Algebraically Closed Fields of Characteristic Not 2

Q: How does the characteristic of the field impact the embedding problem for algebras with involution?

The characteristic of the field plays a crucial role in the embedding problem for algebras with involution. The provided excerpt focuses on fields of characteristic different from 2, and this is not merely a technical assumption. Here's why: Decomposition into symmetric and skew-symmetric parts: In characteristic different from 2, we can decompose an algebra with involution A into the direct sum A = A+ ⊕ A−, where A+ consists of symmetric elements (a* = a) and A− consists of skew-symmetric elements (a* = -a). This decomposition is fundamental to many arguments related to involutions. In characteristic 2, this decomposition fails, significantly altering the behavior of ∗-identities. Form of the symplectic involution: The definition of the symplectic involution on M₂ₖ(K) relies on the matrix T, which involves subtracting matrix units. In characteristic 2, addition and subtraction coincide, making this definition problematic. Different ∗-identities: The ∗-identities satisfied by matrix algebras with involution can differ depending on the characteristic. For instance, the minimal degree of standard ∗-identities can change. In essence, characteristic 2 introduces complications that require different techniques and often lead to different outcomes compared to the case of characteristic not equal to 2.

Q: Could there be examples of algebras with involution that satisfy each other's *-identities but are not embeddable, perhaps in the case of infinite-dimensional algebras?

Yes, it is indeed possible to have algebras with involution that satisfy each other's *-identities but are not embeddable, particularly in the realm of infinite-dimensional algebras. Here's why: Finite-dimensional vs. Infinite-dimensional behavior: The positive embedding results discussed in the excerpt heavily rely on the finite-dimensionality of the algebras involved. Finite-dimensional simple algebras over an algebraically closed field are well-understood and classified. This classification, along with properties of standard polynomials, enables the embedding arguments. Infinite-dimensional algebras exhibit far more complexity and are not as easily classified. Loss of structural constraints: In the infinite-dimensional setting, we lose some of the structural constraints present in the finite-dimensional case. For example, an infinite-dimensional algebra satisfying the identities of a matrix algebra might not have a bound on the dimensions of its irreducible representations, unlike its finite-dimensional counterpart. This lack of boundedness can obstruct embeddings. Constructing explicit examples of such non-embeddable algebras can be quite technical. However, the key takeaway is that the techniques used for finite-dimensional algebras do not directly transfer to the infinite-dimensional case, and counterexamples to embeddability are more likely to arise.

Q: What are the implications of these findings for the representation theory of algebras with involution, and how can they be applied to other areas of mathematics or physics?

The findings about the embedding problem for algebras with involution have significant implications for representation theory and connections to other fields: Representation Theory: Understanding representations: Embedding theorems provide a way to study representations of an algebra with involution by relating them to representations of matrix algebras, which are well-understood. If we can embed an algebra A into a matrix algebra, representations of the matrix algebra restrict to representations of A. Classifying involutions: The embedding problem is closely tied to the classification of involutions on algebras. Different types of involutions lead to different embedding results, shedding light on the structure of algebras with involutions. Applications in Other Areas: Physics: Algebras with involution appear naturally in quantum mechanics and quantum field theory. For instance, the *-algebras of observables in quantum mechanics are algebras with involution. Embedding theorems can provide insights into the structure of these algebras and their representations, potentially leading to a deeper understanding of physical systems. Geometry: Involutions often arise in geometric contexts, such as in the study of symmetric spaces. The embedding problem for algebras with involution can have implications for the classification and properties of these spaces. Invariant Theory: Invariant theory studies actions of groups on algebraic structures. Involutions can be viewed as actions of the cyclic group of order 2. Embedding theorems can be helpful in understanding invariants of these actions. In summary, the embedding problem for algebras with involution is not merely an algebraic curiosity. It has profound connections to representation theory and applications in various areas of mathematics and physics, providing tools and insights into the structure and behavior of these fundamental algebraic objects.

Core Concepts

Under the assumption of an algebraically closed field with characteristic not 2, finite-dimensional central simple algebras with involution can be embedded into each other, preserving the involution, if the target algebra satisfies the *-identities of the source algebra.

Abstract

Bibliographic Information:

Gomez Parada, J. A. (2024). The Embedding Problem in Algebras with Involution [Preprint]. arXiv:2411.06952v1

Research Objective:

This research investigates the embedding problem for finite-dimensional central simple algebras with involution over an algebraically closed field of characteristic not equal to 2. The author aims to determine if an algebra with involution A can be embedded into another algebra with involution B, preserving the involutions, given that A satisfies the *-polynomial identities of B.

Methodology:

The author utilizes the properties of *-polynomial identities, standard polynomials, and the classification of finite-dimensional simple algebras with involution. The analysis focuses on the minimal degree of standard polynomials that become *-identities for matrix algebras with transpose and symplectic involutions. Explicit embedding constructions are provided for different cases based on the types of involutions involved.

Key Findings:

The paper demonstrates that if A and B are finite-dimensional central simple algebras with involution over an algebraically closed field of characteristic not 2, and A satisfies the *-identities of B, then there exists an embedding of A into B that preserves the involutions. This holds true for various combinations of involution types (transpose, symplectic, and exchange) on A and B.

Main Conclusions:

The research concludes that the embedding problem has a positive solution for finite-dimensional central simple algebras with involution over algebraically closed fields of characteristic not 2, under the condition that the target algebra satisfies the *-identities of the source algebra.

Significance:

This work contributes to the understanding of the embedding problem in the context of algebras with involution, extending previous results on simple algebras and algebras with trace. It provides valuable insights into the structural relationships between different types of involutions and their impact on the embeddability of algebras.

Limitations and Future Research:

The study focuses specifically on finite-dimensional central simple algebras over algebraically closed fields of characteristic not 2. Further research could explore the embedding problem for more general classes of algebras with involution, such as those with different dimensions, non-central simple algebras, or algebras over fields with characteristic 2. Investigating the embedding problem for algebras with additional structures beyond involutions could also be a promising direction for future work.

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The Embedding Problem in Algebras with Involution

by Jonatan Andr... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06952.pdf

The Embedding Problem in Algebras with Involution

Deeper Inquiries

How does the characteristic of the field impact the embedding problem for algebras with involution?

The characteristic of the field plays a crucial role in the embedding problem for algebras with involution. The provided excerpt focuses on fields of characteristic different from 2, and this is not merely a technical assumption.
Here's why:

Decomposition into symmetric and skew-symmetric parts:  In characteristic different from 2, we can decompose an algebra with involution A into the direct sum A = A+ ⊕ A−, where A+ consists of symmetric elements (a* = a) and A− consists of skew-symmetric elements (a* = -a). This decomposition is fundamental to many arguments related to involutions. In characteristic 2, this decomposition fails, significantly altering the behavior of ∗-identities.

Form of the symplectic involution: The definition of the symplectic involution on M₂ₖ(K) relies on the matrix T, which involves subtracting matrix units. In characteristic 2, addition and subtraction coincide, making this definition problematic.

Different ∗-identities: The ∗-identities satisfied by matrix algebras with involution can differ depending on the characteristic. For instance, the minimal degree of standard ∗-identities can change.
In essence, characteristic 2 introduces complications that require different techniques and often lead to different outcomes compared to the case of characteristic not equal to 2.

Could there be examples of algebras with involution that satisfy each other's *-identities but are not embeddable, perhaps in the case of infinite-dimensional algebras?

Yes, it is indeed possible to have algebras with involution that satisfy each other's *-identities but are not embeddable, particularly in the realm of infinite-dimensional algebras.
Here's why:

Finite-dimensional vs. Infinite-dimensional behavior: The positive embedding results discussed in the excerpt heavily rely on the finite-dimensionality of the algebras involved.  Finite-dimensional simple algebras over an algebraically closed field are well-understood and classified. This classification, along with properties of standard polynomials, enables the embedding arguments. Infinite-dimensional algebras exhibit far more complexity and are not as easily classified.

Loss of structural constraints:  In the infinite-dimensional setting, we lose some of the structural constraints present in the finite-dimensional case. For example, an infinite-dimensional algebra satisfying the identities of a matrix algebra might not have a bound on the dimensions of its irreducible representations, unlike its finite-dimensional counterpart. This lack of boundedness can obstruct embeddings.
Constructing explicit examples of such non-embeddable algebras can be quite technical. However, the key takeaway is that the techniques used for finite-dimensional algebras do not directly transfer to the infinite-dimensional case, and counterexamples to embeddability are more likely to arise.

What are the implications of these findings for the representation theory of algebras with involution, and how can they be applied to other areas of mathematics or physics?

The findings about the embedding problem for algebras with involution have significant implications for representation theory and connections to other fields:
Representation Theory:

Understanding representations:  Embedding theorems provide a way to study representations of an algebra with involution by relating them to representations of matrix algebras, which are well-understood. If we can embed an algebra A into a matrix algebra, representations of the matrix algebra restrict to representations of A.

Classifying involutions:  The embedding problem is closely tied to the classification of involutions on algebras.  Different types of involutions lead to different embedding results, shedding light on the structure of algebras with involutions.
Applications in Other Areas:

Physics: Algebras with involution appear naturally in quantum mechanics and quantum field theory. For instance, the *-algebras of observables in quantum mechanics are algebras with involution.  Embedding theorems can provide insights into the structure of these algebras and their representations, potentially leading to a deeper understanding of physical systems.

Geometry:  Involutions often arise in geometric contexts, such as in the study of symmetric spaces. The embedding problem for algebras with involution can have implications for the classification and properties of these spaces.

Invariant Theory:  Invariant theory studies actions of groups on algebraic structures. Involutions can be viewed as actions of the cyclic group of order 2.  Embedding theorems can be helpful in understanding invariants of these actions.
In summary, the embedding problem for algebras with involution is not merely an algebraic curiosity. It has profound connections to representation theory and applications in various areas of mathematics and physics, providing tools and insights into the structure and behavior of these fundamental algebraic objects.