Notes on Tensor Products and the Exterior Algebra
ثبت نشده
چکیده
Note that the three vector spaces involved aren’t necessarily the same. What these examples have in common is that in each case, the product is a bilinear map. The tensor product is just another example of a product like this. If V1 and V2 are any two vector spaces over a field F, the tensor product is a bilinear map: V1 × V2 → V1 ⊗ V2 , where V1 ⊗ V2 is a vector space over F. The tricky part is that in order to define this map, we first need to construct this vector space V1 ⊗ V2. We give two definitions. The first is an axiomatic definition, in which we specify the properties that V1 ⊗ V2 and the bilinear map must have. In some sense, this is all we need to work with tensor products in a practical way. Later we’ll show that such a space actually exists, by constructing it.
منابع مشابه
The non-abelian tensor product of normal crossed submodules of groups
In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the...
متن کاملA few classical results on tensor, symmetric and exterior powers
0.1. Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.3. Basic conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.4. Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.5. Tensor powers of k-modules . . . . . . . . . . . ...
متن کاملWedge Products of Alternating Forms
Let V be a vector space of dimension d < ∞. In these notes, we discuss how to define a wedge product on the space of all alternating forms on V × · · · × V ’s so as to make it isomorphic to the exterior algebra Λ(V ). We start by reviewing the equivalence class definition of the exterior algebra over the dual space V ∗ of V . Define ◦ C(V ) = ⊕∞ k=0 V ∗⊗k (with V ∗⊗k being the tensor product of...
متن کاملOn p-minimal homological models of twisted tensor products of elementary complexes localised over a prime
In this paper, working over Z(p) and using algebra perturbation results from [18], p-minimal homological models of twisted tensor products (TTPs) of Cartan’s elementary complexes are obtained. Moreover, making use of the notion of indecomposability of a TTP, we deduce that a homological model of a indecomposable p-minimal TTP of length ` (` ≥ 2) of exterior and divided power algebras is a tenso...
متن کاملIrreducibility of the tensor product of Albeverio's representations of the Braid groups $B_3$ and $B_4$
We consider Albeverio's linear representations of the braid groups $B_3$ and $B_4$. We specialize the indeterminates used in defining these representations to non zero complex numbers. We then consider the tensor products of the representations of $B_3$ and the tensor products of those of $B_4$. We then determine necessary and sufficient conditions that guarantee the irreducibility of th...
متن کاملAdjunctions between Hom and Tensor as endofunctors of (bi-) module category of comodule algebras over a quasi-Hopf algebra.
For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor endofunctors V k - and - kV are left adjoint to some kinds of Hom-endofunctors of _HM. The units and counits of these adjunctions are formally trivial as in the classical case.The category of (bi-) modules over a quasi-Hopf algebra is monoidal and some generalized versions of Hom-tensor relations have been st...
متن کامل