The standard algebras

Abstract

We discuss the tensor, symmetric, and exterior algebras of a vector space. -- Chapter 0 contains algebraic preliminaries. -- In Chapter I we define the tensor product VⓧW of two vector spaces and then the tensor product of a finite number of vector spaces. A theorem concerning the existence and uniqueness of the tensor product is proved. Let Vr denote Vⓧ...ⓧV (r times) . We define an operation called multiplication of tensors which pairs an element of Vr and an element of Vs with an element of Vr+s . This defines a multiplicative structure on the (weak) direct sum ⓧV=R+V+V* + VⓧV+V*+... We call ⓧV the tensor algebra of the vector space V and prove a theorem concerning its existence and uniqueness. Let V* denote the dual space of V and (V*)r denote Vⓧ...ⓧV (r times) We show that (V*)r can be identified with (Vr)* , the dual space of Vr . This identification establishes the pseudo product for the pair Vr, (Vr)* : -- [special characters omitted] -- In the final section we discuss the induced covariant and contravariant homomorphisms. -- We give parallel discussions for the symmetric and exterior algebras. In Chapter II we give constructual and conceptual definitions of V(r) , the space of symmetric contravariant tensors of degree r , and show the existence and uniqueness of V(r) . We define an operation called symmetric multiplication which pairs an element of V(r) and an element of V(s) with an element of V(r+s) . We then have a multiplicative structure on the direct sum -- [special characters omitted] -- and we call OV the symmetric algebra of V . We prove its existence and uniqueness. We discuss the duality in the symmetric algebra and show that (V(r))* can be identified with (V*)(r) . This establishes the pseudo product for the pair V(r), (V(r))* . In fact, we prove the formula -- [special characters omitted] -- and show the relationship between this pseudo product and the permanent function. -- In Chapter III we define V[r] , the space of antisymmetric (alternate) tensors of degree r . We proceed as in Chapter II. Having defined exterior multiplication, we have a multiplicative structure on the direct sum -- [special characters omitted] -- and we call ∧V the exterior algebra of V . We show that (V[r])* can be identified with (V*)[r] We prove that -- [special characters omitted] -- and show the relationship between this pseudo product and the determinant.