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Given $\qalg:= \{ \sum_k A_k \otimes (v\otimes v -q(v))\otimes B_k: v \in V, A_k, B_k \in \talg \} $

 

We have for $A,B \in \clalg$: $AB = A \otimes B + \qalg$, where $AB$ denotes the Clifford (geometric) product and $\otimes$ is of course the tensor product.

Given $\qalg:= \{ \sum_k A_k \otimes (v\otimes v -q(v))\otimes B_k: v \in V, A_k, B_k \in \talg \} $

We have for $A,B \in \clalg$: $AB = A \otimes B + \qalg$, where $AB$ denotes the Clifford (geometric) product and $\otimes$ is of course the tensor product.

Here is a similar question on Math.SE, corroborating my claim that every multivector corresponds to a tensor but not vice versa: What is the relationship of tensor and multivector. These two documents explain how to represent some multivectors as tensors for the special case that $V=\mathbb{R}^3$ and $q=I$ i.e. just the identity, so that the inner product is just the dot product: (1) (2).

Related: Which concepts in differential geometry cannot be represented using geometric algebra? (The answer is essentially: tensors.)

How would one express the result of a tensor product (of two vectors) in the geometric algebra?

Is geometric algebra isomorphic to tensor algebra? (No. Most tensors do not admit of a unique representation in geometric algebra due to the quotient structure.)

What is the hierarchy of algebraic objects meant to capture geometric intuition? (Tensor algebras are essentially the most general possible, even though geometric algebras are in general less unwieldy.)

Here is a similar question on Math.SE, corroborating my claim that every multivector corresponds to a tensor but not vice versa: What is the relationship of tensor and multivector. These two documents explain how to represent some multivectors as tensors for the special case that $V=\mathbb{R}^3$ and $q=I$ i.e. just the identity, so that the inner product is just the dot product: (1) (2).

Related: Which concepts in differential geometry cannot be represented using geometric algebra? (The answer is essentially: tensors.)

How would one express the result of a tensor product (of two vectors) in the geometric algebra?

Is geometric algebra isomorphic to tensor algebra? (No. Most tensors do not admit of a unique representation in geometric algebra due to the quotient structure.)

What is the hierarchy of algebraic objects meant to capture geometric intuition? (Tensor algebras are essentially the most general possible, even though geometric algebras are in general less unwieldy.)

Here is a similar question on Math.SE, corroborating my claim that every multivector corresponds to a tensor but not vice versa: What is the relationship of tensor and multivector. These two documents explain how to represent some multivectors as tensors for the special case that $V=\mathbb{R}^3$ and $q=I$ i.e. just the identity, so that the inner product is just the dot product: (1) (2).