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... The category of commutative Hopf algebras is anti-equivalent to the category of affine group schemes. The prime spectrum of a commutative Hopf algebra is an affine group scheme of multiplicative units. And going in the opposite direction, the algebra of natural transformations from an affine group s ...
... The category of commutative Hopf algebras is anti-equivalent to the category of affine group schemes. The prime spectrum of a commutative Hopf algebra is an affine group scheme of multiplicative units. And going in the opposite direction, the algebra of natural transformations from an affine group s ...
1. Prove that the following are all equal to the radical • The union of
... of V and any two elements w1 , w2 of V , there exists an element r of R such that rv1 = w1 and rv2 = w2 . Show that R equals Endk (V ). A proof from first principles should be easy to give, but here is a “high level” proof. Since R is 2-transitive, it follows from an observation in §3 of the notes ...
... of V and any two elements w1 , w2 of V , there exists an element r of R such that rv1 = w1 and rv2 = w2 . Show that R equals Endk (V ). A proof from first principles should be easy to give, but here is a “high level” proof. Since R is 2-transitive, it follows from an observation in §3 of the notes ...
Solutions
... Therefore the kernel is a subgroup. We need to check normality. Let x be any element of G. The element xax−1 is mapped to Φ(x)e0 Φ(x)−1 , where e0 is the identity element in H. We get that Φ(x)e0 Φ(x)−1 = Φ(x) · Φ(x)−1 = e0 , and the kernel is normal. 2. The image is a subgroup: Its non-empty and if ...
... Therefore the kernel is a subgroup. We need to check normality. Let x be any element of G. The element xax−1 is mapped to Φ(x)e0 Φ(x)−1 , where e0 is the identity element in H. We get that Φ(x)e0 Φ(x)−1 = Φ(x) · Φ(x)−1 = e0 , and the kernel is normal. 2. The image is a subgroup: Its non-empty and if ...
Solutions 8 - D-MATH
... 1. We start by eliminating non-simple groups of order < 60. First, recall that pgroups (pn , n > 1) have non-trivial center and that the center of a group is a normal subgroup. Recall from exercise sheet 7, that there are no simple groups of order pq or p2 q. The candidates left are then the groups ...
... 1. We start by eliminating non-simple groups of order < 60. First, recall that pgroups (pn , n > 1) have non-trivial center and that the center of a group is a normal subgroup. Recall from exercise sheet 7, that there are no simple groups of order pq or p2 q. The candidates left are then the groups ...
Order (group theory)
... The following partial converse is true for finite groups: if d divides the order of a group G and d is a prime number, then there exists an element of order d in G (this is sometimes called Cauchy's theorem). The statement does not hold for composite orders, e.g. the Klein four-group does not have a ...
... The following partial converse is true for finite groups: if d divides the order of a group G and d is a prime number, then there exists an element of order d in G (this is sometimes called Cauchy's theorem). The statement does not hold for composite orders, e.g. the Klein four-group does not have a ...
Factorization of unitary representations of adele groups
... series representations of reductive linear groups over local fields. The fact that in some sense there is just one irreducible representation of a reductive linear p-adic group will have to be pursued later. References and historical notes will be added later, maybe. Many of the statements made here ...
... series representations of reductive linear groups over local fields. The fact that in some sense there is just one irreducible representation of a reductive linear p-adic group will have to be pursued later. References and historical notes will be added later, maybe. Many of the statements made here ...
Solutions — Ark 1
... k[X1 , . . . , Xn ] over the field k is prime if and only if P (X1 , . . . , Xn ) is irreducible. (Hint: Use that k[X1 , . . . , Xn ] is UFD.) Solution: In fact, we are going to show that in any ring A being a UFD an element f is irreducible if and only if the principal ideal (f ) is prime. The easy ...
... k[X1 , . . . , Xn ] over the field k is prime if and only if P (X1 , . . . , Xn ) is irreducible. (Hint: Use that k[X1 , . . . , Xn ] is UFD.) Solution: In fact, we are going to show that in any ring A being a UFD an element f is irreducible if and only if the principal ideal (f ) is prime. The easy ...
THE BASICS OF EXT AND TOR In lecture, we have omitted the
... In lecture, we have omitted the proofs about some basic facts of Ext and Tor, including that they are well defined. Our reasoning is that the true interest of homological algebra is the application of these tools, and these proofs do not contribute much towards this aim. However, being mathematician ...
... In lecture, we have omitted the proofs about some basic facts of Ext and Tor, including that they are well defined. Our reasoning is that the true interest of homological algebra is the application of these tools, and these proofs do not contribute much towards this aim. However, being mathematician ...
Chapter 1 (as PDF)
... • the integers (Z, +, ∗) form an integral domain but not a field; • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) form fields; • the set of 2 × 2 matrices with real entries forms a non-commutative ring with identity w.r.t. matrix addition and multiplication. • the group Zn w ...
... • the integers (Z, +, ∗) form an integral domain but not a field; • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) form fields; • the set of 2 × 2 matrices with real entries forms a non-commutative ring with identity w.r.t. matrix addition and multiplication. • the group Zn w ...
EXAMPLE SHEET 3 1. Let A be a k-linear category, for a
... satisfies ei pej q “ δij . Prove that i“1 ei b ei P V b V is independent of the choice of the basis of V . 3. Let k be a field and Mn pkq the algebra of n ˆ n matrices with entries in k, and denote by OpMn pkqq be the free commutative algebra on the variables tXij : 1 ď i, j ď nu (ie the plynomial a ...
... satisfies ei pej q “ δij . Prove that i“1 ei b ei P V b V is independent of the choice of the basis of V . 3. Let k be a field and Mn pkq the algebra of n ˆ n matrices with entries in k, and denote by OpMn pkqq be the free commutative algebra on the variables tXij : 1 ď i, j ď nu (ie the plynomial a ...
Applications of Logic to Field Theory
... Theorem. ACF∪Ψ0 is uncountably categorical. That is, if κ is uncountable, then any two algebraically closed fields of characteristic zero and cardinality κ are isomorphic. Similarly, for each prime p, ACF ∪ {ψp } is uncountably categorical. Thus, for example, C is the unique (up to isomorphism) alg ...
... Theorem. ACF∪Ψ0 is uncountably categorical. That is, if κ is uncountable, then any two algebraically closed fields of characteristic zero and cardinality κ are isomorphic. Similarly, for each prime p, ACF ∪ {ψp } is uncountably categorical. Thus, for example, C is the unique (up to isomorphism) alg ...
A note on some properties of the least common multiple of
... for some prime q. Let H = V G. If there is a regular orbit and H does not have full index then q is quasi-central in H. G Proof. Since there is a regular orbit, there is an element v ∈ V so that v = |G|. Then CH (v) is q-group. So the only possible quasi-prime is q. ...
... for some prime q. Let H = V G. If there is a regular orbit and H does not have full index then q is quasi-central in H. G Proof. Since there is a regular orbit, there is an element v ∈ V so that v = |G|. Then CH (v) is q-group. So the only possible quasi-prime is q. ...
2 Incidence algebras of pre-orders - Rutcor
... algebra contains all diagonal matrices, which constitute a larger but still commutative subring, and thus the incidence algebra is a matrix algebra over the algebra of ...
... algebra contains all diagonal matrices, which constitute a larger but still commutative subring, and thus the incidence algebra is a matrix algebra over the algebra of ...
O I A
... belongs to A( ρ ) as claimed, concluding the proof that A( ρ ) = A . The two claims above being proved, the isomorphism stated in the theorem is now established. As for the second statement of the theorem, the equivalence of (ii) and (iii) follows from the preliminary discussion about conjugation by ...
... belongs to A( ρ ) as claimed, concluding the proof that A( ρ ) = A . The two claims above being proved, the isomorphism stated in the theorem is now established. As for the second statement of the theorem, the equivalence of (ii) and (iii) follows from the preliminary discussion about conjugation by ...
18. Cyclotomic polynomials II
... where d is summed over some set of integers all strictly smaller than n. Let Φn (x) be the nth cyclotomic polynomial. Show that, on one hand, Φn (q) divides q n − q, but, on the other hand, this is impossible unless n = 1. Thus D = k. ) First, the center k of D is defined to be k = center D = {α ∈ D ...
... where d is summed over some set of integers all strictly smaller than n. Let Φn (x) be the nth cyclotomic polynomial. Show that, on one hand, Φn (q) divides q n − q, but, on the other hand, this is impossible unless n = 1. Thus D = k. ) First, the center k of D is defined to be k = center D = {α ∈ D ...
HW2 Solutions Section 16 13.) Let G be the additive group of real
... There are only two elements of order 3, namely (1 2 3) and (1 3 2). So we think these guys might be in the same conjugacy class. So we just have to find an element σ ∈ S3 such that σ(1 2 3)σ −1 = (1 3 2). Note that all the cycles of length 2 are self-inverse. ...
... There are only two elements of order 3, namely (1 2 3) and (1 3 2). So we think these guys might be in the same conjugacy class. So we just have to find an element σ ∈ S3 such that σ(1 2 3)σ −1 = (1 3 2). Note that all the cycles of length 2 are self-inverse. ...
570 SOME PROPERTIES OF THE DISCRIMINANT MATRICES OF A
... where ti(eres) and fa{erea) are the first and second traces, respectively, of eres. The first forms in terms of the constants of multiplication arise from the isomorphism between the first and second matrices of the elements of A and the elements themselves. The second forms result from direct calcu ...
... where ti(eres) and fa{erea) are the first and second traces, respectively, of eres. The first forms in terms of the constants of multiplication arise from the isomorphism between the first and second matrices of the elements of A and the elements themselves. The second forms result from direct calcu ...
