Jackie Moon
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Apr 13, 2019 · Q3: If Q1, Q2 are wrong, what is the relationship between semi-simple Lie algebras and semi-simple rings? (I would not expect for infinite dimensional representation, above categorical equivalence holds. The "semi-simple" ring should be interpreted as every f.g. module is decomposed into simple modules only.)
Jun 8, 2023 · 237. A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear function with a translation, so while the linear part fixes the origin, the translation can map it somewhere else. Linear functions between vector spaces preserve the vector space structure (so in particular they ...
Jun 14, 2020 · 2. Definition of lower semi-continuous: Give topo space X and mapping f: X → (− ∞, + ∞]. f is lower semi-continuous at x0 if ∀ε> f(x0), ∃V is neighborhood of x0 that if ∀x ∈ V then ε <f(x). Problem: If f is semi-continuous on compact space X then ∃x0 ∈ X that f(x0) = min x ∈ X f(x). My attempt: Set a = inf f(X).
Jun 22, 2020 · Proving a homomorphic image of semi-simple ring is also a semi-simple ring. I have try to search it in the internet but I meet some remark in some book that this problem is not true. In Certain Number-Theoretic Episodes In Algebra, you can see the Observation 14.2.4 page 492, they say that a homomorphic image of semi-simple ring is also a semi-simple ring.
Jul 24, 2014 · Then we have $\{ A | A \succeq 0 \} = \cap_{x} H_x$, and since the intersection of convex sets is again convex, we see that the set of positive semi-definite matrices is convex. Essentially the same reasoning applies to positive definite, negative definite and negative semi-definite.
Warm up. We are trying to integrate the area of a sphere with radius r in spherical coordinates. The angle θ runs from the North pole to South pole in radians. Angle θ equals zero at North pole and π at South pole. The distance on the surface of our sphere between North to South poles is rπ (half the circumference of a circle).
In LaTeX it is coded as \cong. ∼ ∼ is a similarity in geometry and can be used to show that two things are asymptotically equal (they become more equal as you increase a variable like n n). This is a weaker statement than the other two. In LaTeX it is coded as \sim. ≃ ≃ is more of a grab-bag of meaning.
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1. In the direct product of H and K, elements h ∈ H and k ∈ K commute, so kh = hk. In the semidirect product, we have kh = φ(h)k for some automorphism φ of h. Thus, as Qiaochu mentioned, we need a homomorphism from K into Aut(H) in order to make this action make sense.
Nov 23, 2017 · 0. If the matrix is symmetric, positive semi-definiteness (∀z ≠ 0 zTAz ≥ 0) is equivalent to the matrix having non-negative eigenvalues. In general, there is always some computation required. An easy sanity check is to make sure the trace of the matrix is not negative, because that would imply that the matrix has a negative eigenvalue.
