The Nil Hecke Ring and Cohomology of G/P for a Kac
... the minimal parabolic containing r;). Kac and Peterson have extended the definition of the ring of operators & on H(G/B) to the general case and they have used these operators to study the topology of G (as well as G/B). The problems, we wish to deal with, are to describe H(G/B): (I ) as a ring, in ...
... the minimal parabolic containing r;). Kac and Peterson have extended the definition of the ring of operators & on H(G/B) to the general case and they have used these operators to study the topology of G (as well as G/B). The problems, we wish to deal with, are to describe H(G/B): (I ) as a ring, in ...
THE ZEN OF ∞-CATEGORIES Contents 1. Derived categories
... localization functor R → R[W−1 ], but (since we are speaking strictly) this induces an isomorphism on sets of objects and so there is no real ambiguity. 4As an example of the equivalence relation on zigzags, if one of the backwards-pointing weak equivalences happens to be an isomorphism, then the di ...
... localization functor R → R[W−1 ], but (since we are speaking strictly) this induces an isomorphism on sets of objects and so there is no real ambiguity. 4As an example of the equivalence relation on zigzags, if one of the backwards-pointing weak equivalences happens to be an isomorphism, then the di ...
Automorphisms of 2--dimensional right
... A right-angled Artin group is a group given by a finite presentation whose only relations are commutators of the generators. These groups have nice algorithmic properties and act naturally on CAT(0) cube complexes. Also known as graph groups, they occur in many different mathematical contexts; for s ...
... A right-angled Artin group is a group given by a finite presentation whose only relations are commutators of the generators. These groups have nice algorithmic properties and act naturally on CAT(0) cube complexes. Also known as graph groups, they occur in many different mathematical contexts; for s ...
Strong isomorphism reductions in complexity theory
... considered by the fourth author (see [8]) inspired by the analogous notion from descriptive set theory (see [9]). In descriptive set theory, C and D denote classes of structures with universe N and the function f satisfying (1) is required to be Borel (in the topology generated by the first-order de ...
... considered by the fourth author (see [8]) inspired by the analogous notion from descriptive set theory (see [9]). In descriptive set theory, C and D denote classes of structures with universe N and the function f satisfying (1) is required to be Borel (in the topology generated by the first-order de ...
ALGEBRA HANDOUT 2: IDEALS AND
... (IR1) I is a subgroup of the additive group of R. (IR2) For any r ∈ R and any i ∈ I, ri ∈ I. We often employ notation like rI = {ri | i ∈ I} and then (IR2) can be stated more succinctly as: for all r ∈ R, rI ⊂ I. In other words, an ideal is a subset of a ring R which is a subgroup under addition (in ...
... (IR1) I is a subgroup of the additive group of R. (IR2) For any r ∈ R and any i ∈ I, ri ∈ I. We often employ notation like rI = {ri | i ∈ I} and then (IR2) can be stated more succinctly as: for all r ∈ R, rI ⊂ I. In other words, an ideal is a subset of a ring R which is a subgroup under addition (in ...
THE JACOBSON DENSITY THEOREM AND APPLICATIONS We
... This makes sense even when A does not have an identity. It is associative (proof: embed A into a larger ring with identity and pull back) and 0 acts as the identity for it. An element z of A is right quasi-regular if it has a right inverse11 with respect to this composition, i.e., if there exists z ...
... This makes sense even when A does not have an identity. It is associative (proof: embed A into a larger ring with identity and pull back) and 0 acts as the identity for it. An element z of A is right quasi-regular if it has a right inverse11 with respect to this composition, i.e., if there exists z ...
Fell bundles associated to groupoid morphisms
... π : G → H the canonical morphism. Then we get a Fell bundle E over H with the fiber Cr∗ (K) over the identity element, such that Cr∗ (G) ∼ = Cr∗ (E). It is equivalent to think of this structure as a grading of Cr∗ (G) by H. Since the groups are discrete the bundle “transverse” bundle structure is tr ...
... π : G → H the canonical morphism. Then we get a Fell bundle E over H with the fiber Cr∗ (K) over the identity element, such that Cr∗ (G) ∼ = Cr∗ (E). It is equivalent to think of this structure as a grading of Cr∗ (G) by H. Since the groups are discrete the bundle “transverse” bundle structure is tr ...
Mixed Tate motives over Z
... where GU is a prounipotent group whose Lie algebra is free, generated by one element σ2n+1 in degree −2n − 1 for all n ≥ 1. Let AMT be the graded affine ring of GU over Q. It is a cofree commutative graded Hopf algebra cogenerated by one element f2n+1 in degree 2n+1 for all n ≥ 1. Consider the free ...
... where GU is a prounipotent group whose Lie algebra is free, generated by one element σ2n+1 in degree −2n − 1 for all n ≥ 1. Let AMT be the graded affine ring of GU over Q. It is a cofree commutative graded Hopf algebra cogenerated by one element f2n+1 in degree 2n+1 for all n ≥ 1. Consider the free ...
On Rule Formats for Zero and Unit Elements
... ensure the validity of algebraic properties by design, using the so called SOS rule formats [5]. Results in this research area roughly state that if the specification of (parts of) the operational semantics of a language has a certain form then some semantic property is guaranteed to hold. The liter ...
... ensure the validity of algebraic properties by design, using the so called SOS rule formats [5]. Results in this research area roughly state that if the specification of (parts of) the operational semantics of a language has a certain form then some semantic property is guaranteed to hold. The liter ...
Base change for unit elements of Hecke algebras
... What we now know about the correspondence -y 8 can be summarized as follows. The correspondence sets up a bijection from the set of conjugacy classes in G ( F ) of elements y E G ( F ) such that (A’ ) can be solved to the set of 0-conjugacy classes in G ( E ) of elements 8 E G(E) such that (D’) can ...
... What we now know about the correspondence -y 8 can be summarized as follows. The correspondence sets up a bijection from the set of conjugacy classes in G ( F ) of elements y E G ( F ) such that (A’ ) can be solved to the set of 0-conjugacy classes in G ( E ) of elements 8 E G(E) such that (D’) can ...
Coxeter functors and Gabriel's theorem
... We hope that this technique is useful not only for the solution of Gabriel's problem and the classification of quadruples of subspaces, but also for the solution of many other problems (possibly, not only problems of linear algebra). Some arguments on Gabriel's problem, similar to those used in this ...
... We hope that this technique is useful not only for the solution of Gabriel's problem and the classification of quadruples of subspaces, but also for the solution of many other problems (possibly, not only problems of linear algebra). Some arguments on Gabriel's problem, similar to those used in this ...
AN INTRODUCTION TO KK-THEORY These are the lecture notes of
... E ⊗B F or E ⊗φ F . LEMMA 32. Let E1B , E2B and FC be Hilbert B, C module respectively, and let φ : B → L(F ) be a ∗-homomorphism. Let T ∈ L(E1 , E2 ). Then e1 ⊗ f → T (e1 ) ⊗ f defines a map T ⊗1 ∈ L(E1 ⊗B F, E2 ⊗B F ) such that (T ⊗1)∗ = T ∗ ⊗1 and kT ⊗ 1k ≤ kT k. If φ(B) ⊂ K(F ), then T ∈ K(E1 , E ...
... E ⊗B F or E ⊗φ F . LEMMA 32. Let E1B , E2B and FC be Hilbert B, C module respectively, and let φ : B → L(F ) be a ∗-homomorphism. Let T ∈ L(E1 , E2 ). Then e1 ⊗ f → T (e1 ) ⊗ f defines a map T ⊗1 ∈ L(E1 ⊗B F, E2 ⊗B F ) such that (T ⊗1)∗ = T ∗ ⊗1 and kT ⊗ 1k ≤ kT k. If φ(B) ⊂ K(F ), then T ∈ K(E1 , E ...
Introducing Algebraic Number Theory
... It is natural to conjecture that we can identify those primes that√can be represented as x2 + |d|y 2 , where d is a negative integer, by working in the ring Z[ d]. But the Gaussian integers (d = −1) form a Euclidean domain, in particular a unique factorization domain. On the other hand, unique facto ...
... It is natural to conjecture that we can identify those primes that√can be represented as x2 + |d|y 2 , where d is a negative integer, by working in the ring Z[ d]. But the Gaussian integers (d = −1) form a Euclidean domain, in particular a unique factorization domain. On the other hand, unique facto ...
MAXIMAL AND NON-MAXIMAL ORDERS 1. Introduction Let K be a
... a fractional ideal, a finitely generated nonzero R-module contained in F rac(R). In class we defined fractional ideals only for Dedekind domains such as OK , but the definition can be applied to arbitrary orders. The reason we looked only at fractional ideals in Dedekind domains is in that case we c ...
... a fractional ideal, a finitely generated nonzero R-module contained in F rac(R). In class we defined fractional ideals only for Dedekind domains such as OK , but the definition can be applied to arbitrary orders. The reason we looked only at fractional ideals in Dedekind domains is in that case we c ...