# The standard algebras

Pritchett, Hazel A. M. (1965) The standard algebras. Masters thesis, Memorial University of Newfoundland. [English] PDF (Migrated (PDF/A Conversion) from original format: (application/pdf)) - Accepted Version
Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission.

• [English] PDF - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. (Original Version)

## 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.

Item Type: Thesis (Masters) http://research.library.mun.ca/id/eprint/7123 7123 Bibliography: leaf 51. Science, Faculty of > Mathematics and Statistics March 1965 Submission Algebra; Vector analysis View Item