Affine group schemes over symmetric monoidal categories

... category and let G be an affine commutative group scheme over C free and of finite rank r ≥ 1. Then, for any algebra B in C and any element u in the group G(B), we have u r = 1 B , where 1 B denotes the identity element of G(B). (For the definition of G(B), see (3-5).) The relative algebraic geometr ...

... category and let G be an affine commutative group scheme over C free and of finite rank r ≥ 1. Then, for any algebra B in C and any element u in the group G(B), we have u r = 1 B , where 1 B denotes the identity element of G(B). (For the definition of G(B), see (3-5).) The relative algebraic geometr ...

algebra boolean circuit outline schaums switching

... be conveniently and unambiguously omitted. For that purpose, we adopt the following conventions for omission of parentheses. (1) Every statement form other than a statement letter has an outer pair of parentheses. We may omit this outer pair without any danger of ambiguity. Thus instead of ((A v B) ...

... be conveniently and unambiguously omitted. For that purpose, we adopt the following conventions for omission of parentheses. (1) Every statement form other than a statement letter has an outer pair of parentheses. We may omit this outer pair without any danger of ambiguity. Thus instead of ((A v B) ...

arXiv:1510.01797v3 [math.CT] 21 Apr 2016 - Mathematik, Uni

... In the case where R = k is a field, Sweedler’s finite dual coalgebra construction A• [22] of a kalgebra A provides an extension of the above construction to arbitrary algebras. The underlying vector space A◦ of A• is the subspace of A∗ consisting of those linear forms whose kernel contains a cofinit ...

... In the case where R = k is a field, Sweedler’s finite dual coalgebra construction A• [22] of a kalgebra A provides an extension of the above construction to arbitrary algebras. The underlying vector space A◦ of A• is the subspace of A∗ consisting of those linear forms whose kernel contains a cofinit ...

Chapter 9 Lie Groups, Lie Algebras and the Exponential Map

... The topological structure of certain linear groups determined by equations among the real and the imaginary parts of their entries can be determined by refining the polar form of matrices. Such groups are called pseudo-algebraic groups. For example, the groups SO(p, q) ands SU(p, q) are pseudoalgebr ...

... The topological structure of certain linear groups determined by equations among the real and the imaginary parts of their entries can be determined by refining the polar form of matrices. Such groups are called pseudo-algebraic groups. For example, the groups SO(p, q) ands SU(p, q) are pseudoalgebr ...

On fusion categories - Annals of Mathematics

... results which are partially or fully due to other authors, and our own results are blended in at appropriate places. Sections 3-7 are mostly devoted to review of the technical tools and to proofs of the results of Section 2. Namely, in Section 3, we prove one of the main theorems of this paper, sayi ...

... results which are partially or fully due to other authors, and our own results are blended in at appropriate places. Sections 3-7 are mostly devoted to review of the technical tools and to proofs of the results of Section 2. Namely, in Section 3, we prove one of the main theorems of this paper, sayi ...

Varieties of cost functions

... Example 3. Consider the stabilisation monoid S1 defined in Example 2 and let I = {0}. Let h : {a, b} → M be the labelling map defined by h(a) = s and h(b) = 1. Then S1 is a stabilisation monoid that can make a distinction between products with no s (that are 1), products containing “few” s (that are ...

... Example 3. Consider the stabilisation monoid S1 defined in Example 2 and let I = {0}. Let h : {a, b} → M be the labelling map defined by h(a) = s and h(b) = 1. Then S1 is a stabilisation monoid that can make a distinction between products with no s (that are 1), products containing “few” s (that are ...

slides

... Let M be a locally finite monoid. Then, its canonical filtration induces (functorially) a structure of an exhausted and separated filtered algebra on R[[M]]. It is given by In = { f ∈ R M : ∀x(`(x) < n ⇒ f (x) = 0) }. The associated (linear) topology is always stronger than the product topology (i.e ...

... Let M be a locally finite monoid. Then, its canonical filtration induces (functorially) a structure of an exhausted and separated filtered algebra on R[[M]]. It is given by In = { f ∈ R M : ∀x(`(x) < n ⇒ f (x) = 0) }. The associated (linear) topology is always stronger than the product topology (i.e ...

The Lambda Calculus is Algebraic - Department of Mathematics and

... point of view. We suggest another way of looking at the problem, which yields a sense in which the lambda calculus is equivalent to an algebraic theory. The basic observation is that the failure of the ξ-rule is not a deficiency of the lambda calculus itself, nor of combinatory algebras, but rather ...

... point of view. We suggest another way of looking at the problem, which yields a sense in which the lambda calculus is equivalent to an algebraic theory. The basic observation is that the failure of the ξ-rule is not a deficiency of the lambda calculus itself, nor of combinatory algebras, but rather ...

The Brauer group of a field - Mathematisch Instituut Leiden

... (iii) The group Br(k) is abelian. (iv) All elements of Br(k) have ﬁnite order. (v) If k is perfect of characteristic p with p a prime number, then every element of Br(k) has order not divisible by p. For a proof of (i), (ii) and (iii) see section 1 of Chapter 2, and for a proof of (iv) see section ...

... (iii) The group Br(k) is abelian. (iv) All elements of Br(k) have ﬁnite order. (v) If k is perfect of characteristic p with p a prime number, then every element of Br(k) has order not divisible by p. For a proof of (i), (ii) and (iii) see section 1 of Chapter 2, and for a proof of (iv) see section ...

Classification of Semisimple Lie Algebras

... spread over hundreds of articles written by many individual authors. This fragmentation raised considerable doubts about the validity of the proof, since there is no way any one person could check the proof from beginning to end. Since then, considerable effort has been devoted to the simplification ...

... spread over hundreds of articles written by many individual authors. This fragmentation raised considerable doubts about the validity of the proof, since there is no way any one person could check the proof from beginning to end. Since then, considerable effort has been devoted to the simplification ...

CLUSTER ALGEBRAS AND CLUSTER CATEGORIES

... • commutative and non commutative algebraic geometry and in particular the study of stability conditions in the sense of Bridgeland [13], Calabi-Yau algebras [66, 57], Donaldson-Thomas invariants [106, 68, 82, 81, 95, 47, 46, 21, 48] . . . ; • and in the representation theory of quivers and finite-d ...

... • commutative and non commutative algebraic geometry and in particular the study of stability conditions in the sense of Bridgeland [13], Calabi-Yau algebras [66, 57], Donaldson-Thomas invariants [106, 68, 82, 81, 95, 47, 46, 21, 48] . . . ; • and in the representation theory of quivers and finite-d ...