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Tensor Products Read chapter Read first chapter

2017 | OriginalPaper | Chapter

2. Tensor Algebras

Author : Alexey L. Gorodentsev

Published in: Algebra II

Publisher: Springer International Publishing

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Abstract

Let V be a vector space over an arbitrary field \(\mathbb{k}\). We write \(V ^{\otimes n}\stackrel{\mathrm{def}}{=}V \otimes V \otimes \,\cdots \, \otimes V\) for the tensor product of n copies of V and call it the n th tensor power of V.

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previous chapter Tensor Products
next chapter Symmetric Functions
Footnotes
1
See Sect. 7.​2 of Algebra I.
 
2
See Sect. 6.6.1 of Algebra I.
 
3
See Proposition 2.1 of Algebra I.
 
4
See Example 15.3 in Algebra I.
 
5
See Proposition 12.​2 of Algebra I.
 
6
See Sect. 7.1.4 of Algebra I.
 
7
See Example 2.1 on p. 22.
 
8
Recall that the zero set of this form in \(\mathbb{P}(V )\) is the hyperplane intersecting the quadric \(Z(f) \subset \mathbb{P}(V )\) along its apparent contour viewed from v.
 
9
That is, of first degree.
 
10
With respect to inclusions.
 
11
See Sect. 2.2.3 on p. 23.
 
12
Here we use that \(\mathbb{k}\) is algebraically closed.
 
13
See Sect. 11.3.3 of Algebra I.
 
14
Compare with Sect. 2.5.6 on p. 41.
 
15
Compare with Problem 17.20 of Algebra I.
 
16
Though the last two curves are given by their affine equations within the standard chart \(U_{0} \subset \mathbb{P}_{2}\), the points at infinity should also be taken into account.
 
17
That is, a triple of rational functions \(x_{0}(t),x_{1}(t),x_{2}(t) \in \mathbb{k}(t)\) such that f(x 0(t), x 1(t), x 2(t)) = 0 in \(\mathbb{k}(t)\), where \(f \in \mathbb{k}[x_{0},x_{1},x_{2}]\) is the equation of the curve.
 
18
Compare with Example 11.7 and the proof of Proposition 17.6 in Algebra I.
 
19
See Sect. 15.3.1 of Algebra I.
 
20
This is clear if the identity in question expresses some basis-independent properties of the linear operator but not its matrix in some specific basis.
 
21
Even for the diagonal matrices with distinct eigenvalues, because the conjugation classes of these matrices are dense in \(\mathrm{Mat}_{n}(\mathbb{C})\) as well.
 
22
See Example 1.​3 on p. 8 and Example 17.6 from Algebra I.
 
23
Note that the decomposition of a Grassmannian polynomial into a sum of decomposable monomials is highly nonunique.
 
24
See Sect. 2.3.1 on p. 27 and Sect. 2.3.3 on p. 27.
 
Literature
[DK]
Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.MathSciNetCrossRef
[Fu]
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.MATH
[FH]
Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.MATH
[Mo]
Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
Metadata
Title
Tensor Algebras
Author
Alexey L. Gorodentsev
Copyright Year
2017
Publisher
Springer International Publishing
Book
Algebra II

Print ISBN: 978-3-319-50852-8

Electronic ISBN: 978-3-319-50853-5

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-50853-5_2

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