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Topological Field Theories
Topological Field Theories

... • More importantly, M 7→ C ∗ (M ) − Mod does not respect the tensor structure! We will now concentrate on solving this last issue by using the topological category structures just defined on ] n and Catk . Cob ...
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... G is a basis for F G: φ( ag g) = ag ψ(g). Since ψ is a group homomorphism, it follows immediately that φ is an F -algebra homomorphism. As is our custom, we will reformulate this proposition in two ways: Plain English version: If you want to define an F -algebra homomorphism F G−→R, it is enough (in ...
The Coinvariant Algebra in Positive Characteristic
The Coinvariant Algebra in Positive Characteristic

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Tensor Algebra: A Combinatorial Approach to the Projective Geometry of Figures
Tensor Algebra: A Combinatorial Approach to the Projective Geometry of Figures

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Homework #3 Solutions (due 9/26/06)
Homework #3 Solutions (due 9/26/06)

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Algebra Quals Fall 2012 1. This is an immediate consequence of the
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UNIT-V - IndiaStudyChannel.com
UNIT-V - IndiaStudyChannel.com

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Math 3121 Lecture 9 ppt97
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Module Fundamentals
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solutions to HW#3

... 1.1.5 Multiplication of residue classes in Z/nZ cannot be the operation of a group because cancellation fails: 0 · 1 = 0 · 0 but 1 6= 0. 1.1.6 Determine which of the following sets are groups under addition. 1.1.6(a) The set of rational numbers in lowest terms whose denominators are odd: This set co ...
Math 5594 Homework 2, due Monday September 25, 2006 PJW
Math 5594 Homework 2, due Monday September 25, 2006 PJW

... (b)Give an example of a vector space V with three subspaces U1 , U2 and U3 such that V = U1 + U2 + U3 and U1 ∩ U2 = U1 ∩ U3 = U2 ∩ U3 = {0}, but V 6= U1 ⊕ U2 ⊕ U3 . 16. (page 29, no. 9) Suppose that θ is an endomorphism of the vector space V and θ 2 = 1V . Show that V = U ⊕ W , where U = {v ∈ V : vθ ...
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Lecture 8 - Universal Enveloping Algebras and Related Concepts, II
Lecture 8 - Universal Enveloping Algebras and Related Concepts, II

... The Clifford algebras, on the other hand, are quotients by ideas generated by the nonhomogeneous elements x ⊗ y + y ⊗ x + 2 hx, yi, and so do not inherit a grading in the usual sense. Nevertheless they inherit what is called a Z2 -grading. We can decompose the tensor algebra into two parts T V = T 0 ...
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C - GEOCITIES.ws

... also a right A-module, compatible with the complex algebra structure, equipped with an A -valued inner product  ·, ·: E  E → A which is C- and A -linear in its second variable and satisfies the following relations: 1. ξ , η* =  η , ξ, for every ξ , η  E; 2. ξ , ξ  0, for every ξ  E; 3.  ...
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Tensor product of modules



In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.
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