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Problem 1. Let A be an associative algebra over a field F. Define a map $[\cdot,\cdot] \colon A \times A \to A$ by $[a, b] := ab - ba$ for $a, b \in A$. Prove that $[\cdot,\cdot]$ is bilinear and A with $[\cdot,\cdot]$ is a Lie algebra.

Name: Problem 1. Let A be an associative algebra over a field F. Define a map [·,·] A × A → A by [a, b] := ab - ba for a, b ∈ A. Prove that [·,·] is bilinear and A with [·,·] is a Lie algebra.
Uploaded: 2022-01-26T05:07:39-08:00
Duration: 4 min 34 s
Channel: Vincenzo Zaccaro
Description: Problem 1. Let A be an associative algebra over a field F. Define a map [·,·] A × A → A by [a, b] := ab - ba for a, b ∈ A. Prove that [·,·] is bilinear and A with [·,·] is a Lie algebra.

          Problem 1. Let A be an associative algebra over a field F. Define a map $[\cdot,\cdot] \colon A \times A \to A$ by $[a, b] := ab - ba$ for $a, b \in A$. Prove that $[\cdot,\cdot]$ is bilinear and A with $[\cdot,\cdot]$ is a Lie algebra.

Problem 1. Let A be an associative algebra over a field F. Define a map [·,·] A × A → A by [a, b] := ab - ba for a, b ∈ A. Prove that [·,·] is bilinear and A with [·,·] is a Lie algebra.

Added by Pedro C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Vincenzo Zaccaro

01/26/2022

Step 1

Step 1: To prove that [,] is bilinear, we need to show that it satisfies the properties of linearity. Show more…

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Problem 1. Let A be an associative algebra over a field F. Define a map [,] : A A -> A by [a, b] := ab -- ba for a,b e A. Prove that [,.] is bilinear and A with [, .] is a Lie algebra.