An Abelian algebra is nilpotent. If $ V $
is Abelian. The unique non-Abelian three-dimensional nilpotent Lie algebra $ \mathfrak g $
10 KB (1,457 words) - 18:48, 13 January 2024
It is clear from Stokes' theorem (cf. [[Stokes theorem|Stokes theorem]]) that the integral $ \int _ {C} A $
extends to the non-Abelian or Yang–Mills case.
8 KB (1,229 words) - 08:30, 26 March 2023
...of the group $ G $ of automorphisms which preserve the structure of the Abelian variety, by the group $ A(k) $ of translations in the points of $ A $
...Grothendieck gave a proof of this fact for projective varieties, and this theorem has been extended to the case of proper flat schemes of morphisms. The sche
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...tryagin [[#References|[2]]] on the theory of characters of locally compact Abelian groups (cf. [[Character of a group|Character of a group]]), posed the probl
...veloped mainly on the basis of the theory of characters of locally compact Abelian groups established by Pontryagin ([[#References|[2]]], see also [[#Referenc
66 KB (9,085 words) - 17:28, 31 March 2020
Using this theorem, it can be proved that every $ l $-
group is large. E.g., it contains the classes of Abelian torsion-free groups, locally nilpotent torsion-free groups, and many others
9 KB (1,403 words) - 22:15, 5 June 2020
$#C+1 = 45 : ~/encyclopedia/old_files/data/P072/P.0702770 Plancherel theorem
denotes the inverse, then Plancherel's theorem can be rephrased as follows: $ {\mathcal F} $
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...f bad reduction (see [[#References|[4]]], and also [[Siegel theorem|Siegel theorem]] on integer points).
...famous conjectures, namely the Tate conjecture concerning endomorphisms of Abelian varieties over number fields (cf. [[Tate conjectures|Tate conjectures]]) an
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...e group. Every subgroup of a free group is also free (the Nielsen–Schreier theorem, see [[#References|[1]]], [[#References|[2]]]).
Free groups of certain varieties have special names, for example, free Abelian, free nilpotent, free solvable, free Burnside; they are free groups of the
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...elian group]] are of the form $\pm g$, $g \in G$. Of course, if $G$ is non-Abelian, then any conjugate of $\pm g$ is also of finite order; however, these are
...r, it was proved in [[#References|[a6]]] that if $n = 2$ and $G$ is finite Abelian, then $U$ is conjugate in $\mathbf{Q}G_{2\times 2}$ to $\operatorname{diag}
9 KB (1,457 words) - 17:05, 26 January 2021
$#C+1 = 29 : ~/encyclopedia/old_files/data/P071/P.0701120 Paley\ANDWiener theorem
...ner theorem; the most frequently encountered analogues of the Paley–Wiener theorem are a description of the image of the space $ C _ {0} ^ \infty ( G) $
4 KB (591 words) - 08:04, 6 June 2020
3) In the category of Abelian groups, the functor $ \mathop{\rm Hom} ( A , Y ) $
...roups is the left adjoint of the functor of taking the torsion part of any Abelian group.
8 KB (1,301 words) - 20:14, 4 April 2020
...assigns the sum in the usual sense to any convergent series (an [[Abelian theorem]]). The series
2 KB (345 words) - 21:12, 23 November 2023
there is associated a sequence of Abelian groups $ H ^ { n } ( G, A) $,
is an Abelian group and $ G $
16 KB (2,427 words) - 09:48, 26 March 2023
are Abelian for all $ 0 \leq i < n $ );
is a one-dimensional (Abelian) Lie algebra for $ 0 \leq i < m $ .
9 KB (1,348 words) - 08:49, 8 April 2023
Given an arbitrary Abelian group $ \pi $
one can define a simplicial set (in fact, a simplicial Abelian group) $ E ( \pi , n) $.
32 KB (4,905 words) - 09:31, 13 February 2024
...his 1963 notes [[#References|[a7]]]. This facilitated a proof of a duality theorem of A. Grothendieck (cf. [[#References|[a5]]]). Let $ {\mathcal C} $
...seful to keep the example below in mind: the category of complexes over an Abelian category (and algebraic mapping cones, the corresponding long exact sequenc
16 KB (2,338 words) - 06:56, 10 May 2022
...orem of local [[Class field theory|class field theory]] there is for every Abelian extension of local fields $ K / k $
3 KB (398 words) - 08:27, 6 June 2020
...among all algebraic number fields is illustrated by the [[Kronecker–Weber theorem]], which states that a finite extension $ K/ \mathbf Q $
is Abelian if and only if $ K \subset K _ {n} $
12 KB (1,769 words) - 11:19, 26 March 2023
In the special case of the group algebra of a locally compact Abelian group (with convolution taken as multiplication in the algebra, cf. also [[
...[[#References|[a2]]], Sect. 11.13). A well-known such theorem is Wiener's theorem (cf. also [[#References|[a1]]], Chapt. XI, Sect. 2): If $ f ( t) = \sum _
3 KB (474 words) - 19:41, 5 June 2020
...ty|Rational variety]], [[Unirational variety|Unirational variety]]). Since Abelian varieties can never be rational, the main interest is in rationality theore
...ective). Fairly complete results are obtained for tori which split over an Abelian extension of the ground field (see [[#References|[5]]]). The first example
8 KB (1,072 words) - 20:22, 21 December 2019