All Questions
Tagged with quadratic-forms ra.rings-and-algebras
9 questions
2
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2
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Rank of a linear combination of quadratic forms
Suppose we have a set of quadratic forms $Q_i (x_1, \dots, x_n)$ for $1 \leq i \leq k$ in $n$ variables, defined over $\mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that ...
- 50.6k
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Are the following two characterisations of symplectic modules, using the language of form rings, the same?
Page 205 of the book Classical Groups and Algebraic K-Theory defines a symplectic module to be an arbitrary quadratic module $(M,h,q)$ over a form ring $(R,\Lambda)$ with $(J,\varepsilon)$ where $J=\...
- 4,943
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Is there any point in considering Form Rings when 2 admits an inverse?
In the study of quadratic spaces over general rings, there is a type of scalar which people consider called a
Form ring $(R,\Lambda)$ relative to some anti-automorphism denoted $(-)^J:R\to R$ and ...
- 4,943
3
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Bott periodicity in characteristic p via Clifford algebras
I am currently reading Husemoller's wonderful book on fibre bundles, specifically the section on Clifford algebras. He defines these groups $L_k$ as follows. Let $M_k$ denote the free abelian groups ...
- 61
2
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Projective group of Plucker quadric over the reals
A somewhat elementary question but seemingly difficult to find a suitable reference:
Consider the six-dimensional real space $\wedge^2(\mathbb R^4)$ with basis $e_i \wedge e_j \ (i < j)$ where $...
- 356
5
votes
3
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Reference request: correspondence between central simple algebras and quadratic forms
Let $A$ be an algebra over $k$, $\operatorname{tr_A}(x, y):=\operatorname{tr}(m_{xy})$ be a trace form on $A$, and $V_A$ be its restriction on the orthogonal complement to $1$. I wonder why a map $A \...
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- 1
2
votes
1
answer
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Involution on the components of a group algebra
If $G$ is a finite group and $k$ a field, there is a canonical involution (ie an involutive anti-automorphism) $\sigma$ on $k[G]$ induced by $g\mapsto g^{-1}$. Given that the center of $k[G]$ has $(\...
- 2,026
4
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Hermitian forms over quaternion algebra
Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let $\sigma=Int(i)\...
- 41
5
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Canonical form for a pair of quadratic forms
Could anyone recommend a reference to a canonical form for a pair of quadratic forms over R (not necessarily positively definite)? This is probably related to the Weierstrass elementary divisors (but ...
- 263