Linear and multilinear algebra; matrix theory Definition

Exterior algebra

Builds on Vector space Tensor product of vector spaces

Definition

Let $F$ be a field and let $V$ be a vector space over $F$ . Let $T(V) = \bigoplus_{k \ge 0} V^{\otimes k}$ be the tensor algebra of $V$ , where $V^{\otimes 0} = F$ and the product is the tensor product. Let $I$ be the two-sided ideal of $T(V)$ generated by the elements $v \otimes v$ for $v \in V$ . The exterior algebra of $V$ is the quotient algebra $\Lambda(V) = T(V) / I$ . The product in $\Lambda(V)$ is written $\wedge$ .

Remarks

For all $u, v \in V$ , $v \wedge v = 0$ and $u \wedge v = -v \wedge u$ . The image of $V^{\otimes k}$ is the $k$ -th exterior power $\Lambda^k(V)$ ; if $\dim V = n$ , then $\dim \Lambda^k(V) = \binom{n}{k}$ and $\dim \Lambda(V) = 2^n$ . If $(e_1, \dots, e_n)$ is a basis of $V$ , then $\Lambda^n(V)$ is spanned by $e_1 \wedge \cdots \wedge e_n$ . For $v_j = \sum_i a_{ij} e_i$ , $v_1 \wedge \cdots \wedge v_n = \det(a_{ij})\, e_1 \wedge \cdots \wedge e_n$ .

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Vector space definition
Bilinear form definition
Tensor product of vector spaces definition
Exterior algebra definition

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Sullivan minimal model

Tag	`08HK38GCSM`
URI	`m2://08HK38GCSM`
Kind	definition
Writing	W1 stated
MSC	15A75
Defines	Exterior algebra
Uses	Vector space, Tensor product of vector spaces

Exterior algebra

Definition

Remarks

Reading path

Where this leads

Further reading

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