Let $$ \bigwedge(V) $$ be the exterior algebra of a finite-dimensional vector space $$ V $$. Assuming this algebra has already been constructed and the wedge product defined, briefly, but rigorously give the formulas for:
- The Scalar Product
- The Interior Product
- The Regressive Product
- The Hodge Dual



Given that $\bigwedge(V)$ is the exterior algebra of a finite-dimensional vector space $V$, let's define the requested products and the Hodge dual rigorously:

### 1. Scalar Product
The **scalar product** in $\bigwedge(V)$ is typically the **inner product** induced by an inner product on $V$. If $v_1, v_2, \dots, v_k \in V$ are vectors and $v_1^*, v_2^*, \dots, v_k^* \in V$ are their duals (with respect to the chosen inner product), the scalar product of two basis elements of $\bigwedge^k(V)$ is:
\[
\langle v_1 \wedge v_2 \wedge \dots \wedge v_k, w_1 \wedge w_2 \wedge \dots \wedge w_k \rangle = \det(\langle v_i, w_j \rangle)_{1 \leq i, j \leq k}
\]
where $\langle v_i, w_j \rangle$ denotes the inner product in $V$.

### 2. Interior Product
The **interior product** (also called the contraction) of a vector $v \in V$ with a $k$-form $\omega \in \bigwedge^k(V)$ is denoted by $ \iota_v(\omega) $ and is defined by:
\[
\iota_v(\omega)(v_1, v_2, \dots, v_{k-1}) = \omega(v, v_1, v_2, \dots, v_{k-1})
\]
In terms of the wedge product:
\[
\iota_v(\omega) = \frac{1}{k} \sum_{i=1}^{k} (-1)^{i+1} \langle v, e_i \rangle e_1 \wedge \dots \wedge \hat{e_i} \wedge \dots \wedge e_k
\]
where $e_1, \dots, e_k$ are basis elements, and $\hat{e_i}$ means that $e_i$ is omitted.

### 3. Regressive Product
The **regressive product** (or meet in geometric algebra) is defined on the exterior algebra in terms of the wedge product and duality. For $p$-vector $\alpha$ and $q$-vector $\beta$ in $\bigwedge(V)$, the regressive product is defined as:
\[
\alpha \vee \beta = (\star^{-1} (\star \alpha \wedge \star \beta))
\]
where $\star$ is the Hodge star operator. The regressive product results in a $(p+q-n)$-vector, where $n = \dim(V)$.

### 4. Hodge Dual
The **Hodge dual** of a $k$-form $\omega \in \bigwedge^k(V)$ in an $n$-dimensional space $V$ is a $(n-k)$-form, denoted by $\star \omega$, defined by:
\[
\star \omega = \frac{1}{k!} \langle \omega, \eta \rangle \eta
\]
where $\eta$ is the chosen volume form (a top-dimensional form in $\bigwedge^n(V)$), and $\langle \cdot, \cdot \rangle$ is the inner product on $\bigwedge(V)$.

The explicit formula in a chosen basis $\{e_1, \dots, e_n\}$ is:
\[
\star (e_{i_1} \wedge \dots \wedge e_{i_k}) = \frac{1}{(n-k)!} \epsilon_{i_1 \dots i_k j_{k+1} \dots j_n} e_{j_{k+1}} \wedge \dots \wedge e_{j_n}
\]
where $\epsilon_{i_1 \dots i_n}$ is the Levi-Civita symbol.

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Do you need more details or have any questions about these formulas? Here are some related questions to consider:

1. How does the Hodge dual relate to the Laplacian in differential geometry?
2. What is the significance of the regressive product in geometric algebra?
3. How is the interior product used in the context of differential forms?
4. What is the geometric interpretation of the wedge product?
5. Can the scalar product be generalized to exterior products of different degrees?

**Tip:** When working with the Hodge dual, ensure that the orientation and the metric on your vector space are well-defined, as they significantly influence the duality.

Explore the formulas for scalar product, interior product, regressive product, and Hodge dual in exterior algebra. Learn their applications in vector spaces and algebraic structures. Suitable for advanced mathematics enthusiasts and students.

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