The symplectic Verlinde algebras and string K e
... thought that the question may be easier to tackle for quotients of symplectic groups (such as quaternionic Grassmannians), because of their relatively sparse homology. This led us to specialize to the symplectic case (it is, of course, only an example; analogous discussions should exist for other co ...
... thought that the question may be easier to tackle for quotients of symplectic groups (such as quaternionic Grassmannians), because of their relatively sparse homology. This led us to specialize to the symplectic case (it is, of course, only an example; analogous discussions should exist for other co ...
Atom structures of cylindric algebras and relation algebras
... algebra, the notion of a cylindric algebra (or relation algebra) is defined axiomatically. The task is then to show (if possible) that any model of these axioms is isomorphic to a concrete algebra whose elements are n-ary relations on some set and whose operations are defined set-theoretically in t ...
... algebra, the notion of a cylindric algebra (or relation algebra) is defined axiomatically. The task is then to show (if possible) that any model of these axioms is isomorphic to a concrete algebra whose elements are n-ary relations on some set and whose operations are defined set-theoretically in t ...
Central Extensions in Physics
... 3. Except when it denotes the real number 3.1415 . . . , the letter π will always denote a canonical projection between two spaces; it will be clear from the context which projection is meant. 4. Abelian groups will be denoted additively except the group U (1) ⊂ C, which is denoted multiplicatively. ...
... 3. Except when it denotes the real number 3.1415 . . . , the letter π will always denote a canonical projection between two spaces; it will be clear from the context which projection is meant. 4. Abelian groups will be denoted additively except the group U (1) ⊂ C, which is denoted multiplicatively. ...
Set theory and von Neumann algebras
... 2.2. Example. In B(H), p - q iff the range of p has dimension smaller than or equal to the range of q. Thus, the ordering on P (B (H)) /∼ is linear, and isomorphic to {0, 1, ..., n} if dim H = n and to N ∪ {∞} if H is infinite dimensional. The previous example highlights why it must be emphasized th ...
... 2.2. Example. In B(H), p - q iff the range of p has dimension smaller than or equal to the range of q. Thus, the ordering on P (B (H)) /∼ is linear, and isomorphic to {0, 1, ..., n} if dim H = n and to N ∪ {∞} if H is infinite dimensional. The previous example highlights why it must be emphasized th ...
Basics of associative algebras
... Simple and semisimple algebras The above discussion suggests that you can have a wide variety of algebras even in quite small dimension. Not all of them are of equal interest however. Often it suffices to consider certain nice classes of algebras, such as simple algebras. The definition of a simple al ...
... Simple and semisimple algebras The above discussion suggests that you can have a wide variety of algebras even in quite small dimension. Not all of them are of equal interest however. Often it suffices to consider certain nice classes of algebras, such as simple algebras. The definition of a simple al ...
Free Heyting algebras: revisited
... goal of generalizing a method of constructing free algebras for varieties axiomatized by rank 1 axioms to the case of rank 0-1 axioms, we consider the case of Heyting algebras (intuitionistic logic, which is of rank 0-1 for f =→). In particular, we construct free Heyting algebras. For an extension ...
... goal of generalizing a method of constructing free algebras for varieties axiomatized by rank 1 axioms to the case of rank 0-1 axioms, we consider the case of Heyting algebras (intuitionistic logic, which is of rank 0-1 for f =→). In particular, we construct free Heyting algebras. For an extension ...
Effective descent morphisms for Banach modules
... K. Moreover, there is a bifunctor [−, −] : (Ban1 )op × Ban1 → Ban1 (the internal Hom) making the category Ban1 into a symmetric closed monoidal category. For two Banach spaces V and W , [V, W ] is the Banach space whose elements are the bounded linear transformations V → W quipped with the operator ...
... K. Moreover, there is a bifunctor [−, −] : (Ban1 )op × Ban1 → Ban1 (the internal Hom) making the category Ban1 into a symmetric closed monoidal category. For two Banach spaces V and W , [V, W ] is the Banach space whose elements are the bounded linear transformations V → W quipped with the operator ...
Full Text (PDF format)
... n such that mn ◦ (I ⊗ S −2 ⊗ · · · ⊗ S −2n+2 ) ◦ ∆n = ε · 1, where mn , ∆n , S, 1, and ε are the iterated product and coproduct, the antipode, the unit, and the counit. If H is involutive (for example, H is semisimple and cosemisimple), the last formula reduces to mn ◦ ∆n = ε · 1. We give four other ...
... n such that mn ◦ (I ⊗ S −2 ⊗ · · · ⊗ S −2n+2 ) ◦ ∆n = ε · 1, where mn , ∆n , S, 1, and ε are the iterated product and coproduct, the antipode, the unit, and the counit. If H is involutive (for example, H is semisimple and cosemisimple), the last formula reduces to mn ◦ ∆n = ε · 1. We give four other ...
On the classification of 3-dimensional non
... and A is a 3-dimensional non-associative division algebra over k whose left and right determinants factor into linear forms over F , then A is isotopic to a generalized twisted field over k split by F . Remark. In fact, the left determinant of A factors over F if and only if the right determinant do ...
... and A is a 3-dimensional non-associative division algebra over k whose left and right determinants factor into linear forms over F , then A is isotopic to a generalized twisted field over k split by F . Remark. In fact, the left determinant of A factors over F if and only if the right determinant do ...
Composition algebras of degree two
... The quadratic form w(x) = g trace(x2) allows composition for the new product *. A general definition of pseudo-octonions, valid over any field, can be found in [9]. Given 7 the algebraic closure of F, the forms of P8(F) are called Okubo algebras. They have been classified by Elduque and Myung in [6] ...
... The quadratic form w(x) = g trace(x2) allows composition for the new product *. A general definition of pseudo-octonions, valid over any field, can be found in [9]. Given 7 the algebraic closure of F, the forms of P8(F) are called Okubo algebras. They have been classified by Elduque and Myung in [6] ...
Q(xy) = Q(x)Q(y).
... IV. C3($) = C(C2(4>)),a Cayley algebra with basis {1, i, j, ft, /, il, jl, kl} where 1= -I, l2= vl (y^O). Since C3 is not associative the Cayley-Dickson construction ends here. It can be shown [l] that every composition algebra over <£>is isomorphic to one of the above algebras (for suitable choice ...
... IV. C3($) = C(C2(4>)),a Cayley algebra with basis {1, i, j, ft, /, il, jl, kl} where 1= -I, l2= vl (y^O). Since C3 is not associative the Cayley-Dickson construction ends here. It can be shown [l] that every composition algebra over <£>is isomorphic to one of the above algebras (for suitable choice ...
DIALGEBRAS Jean-Louis LODAY There is a notion of
... in a dimonoid. In particular we describe the free dimonoid on a given set. In the second section we introduce the notion of dialgebra and give several examples. We explicitly describe the free dialgebra over a vector space. In the third section we construct the chain complex of a dialgebra D, which ...
... in a dimonoid. In particular we describe the free dimonoid on a given set. In the second section we introduce the notion of dialgebra and give several examples. We explicitly describe the free dialgebra over a vector space. In the third section we construct the chain complex of a dialgebra D, which ...
Noncommutative Lp-spaces of W*-categories and their applications
... We consider vector bundles equipped with a more general type of inner product with values in Dens2d (X) for some d ∈ R. Here it is essential that d is real because we need Dens+ 2d (X) for the positivity property. Such an inner product equips every fiber Vx of V with an inner product with values in ...
... We consider vector bundles equipped with a more general type of inner product with values in Dens2d (X) for some d ∈ R. Here it is essential that d is real because we need Dens+ 2d (X) for the positivity property. Such an inner product equips every fiber Vx of V with an inner product with values in ...
RESULTS ON BANACH IDEALS AND SPACES OF MULTIPLIERS
... A natural generalization of the results of Section 3 can be obtained by replacing G by a noncom pact, locally compact space X and F1 (G) by a Wiener algebra A on X. If A has bounded approximate units essentially the situation of Section 2 is given and the main result applies. There is a great number ...
... A natural generalization of the results of Section 3 can be obtained by replacing G by a noncom pact, locally compact space X and F1 (G) by a Wiener algebra A on X. If A has bounded approximate units essentially the situation of Section 2 is given and the main result applies. There is a great number ...
SYMMETRIC SPACES OF THE NON
... We have a natural extension of Theorem 2.2. Theorem 2.18. — Let H be a closed subgroup of a Lie group G. Then H is an embedded submanifold of G, and equipped with this differential structure it is a Lie group. The Lie algebra of H, which is equal to h = {X ∈ g | expG (tX) ∈ H for all t ∈ R}, is a su ...
... We have a natural extension of Theorem 2.2. Theorem 2.18. — Let H be a closed subgroup of a Lie group G. Then H is an embedded submanifold of G, and equipped with this differential structure it is a Lie group. The Lie algebra of H, which is equal to h = {X ∈ g | expG (tX) ∈ H for all t ∈ R}, is a su ...
§13. Abstract theory of weights
... It is an elementary fact about lattices that Λ/Λr must be a finite group (called the fundal P mental group of Φ). We can see this directly as follows. Write αi = mij λj (mij ∈ Z). j=1 ...
... It is an elementary fact about lattices that Λ/Λr must be a finite group (called the fundal P mental group of Φ). We can see this directly as follows. Write αi = mij λj (mij ∈ Z). j=1 ...
Continuous cohomology of groups and classifying spaces
... primary import of this account. Beyond that, the applications to Lie groupoids, foliations and infinite dimensional Lie algebras are the focus of much current activity; I have tried to make this account as up to date as possible. (An alternate approach to much of this material has appeared in lectur ...
... primary import of this account. Beyond that, the applications to Lie groupoids, foliations and infinite dimensional Lie algebras are the focus of much current activity; I have tried to make this account as up to date as possible. (An alternate approach to much of this material has appeared in lectur ...
AdZ2. bb4l - ESIRC - Emporia State University
... purely algebraic point of view, without reference to Lie groups and differential geometry. Such a view point has the advantage of going immediately into the discussion of Lie algebras without first establishing the topo10g cal machineries for the sake of defining Lie groups from ...
... purely algebraic point of view, without reference to Lie groups and differential geometry. Such a view point has the advantage of going immediately into the discussion of Lie algebras without first establishing the topo10g cal machineries for the sake of defining Lie groups from ...
Closed locally path-connected subspaces of finite
... not homeomorphic to closed subspaces of finite-dimensional topological groups. First we state a Key Lemma treating locally continuum-connected subspaces of finite-dimensional topological groups. By a continuum we understand a connected compact Hausdorff space. We shall say that two points x, y of a top ...
... not homeomorphic to closed subspaces of finite-dimensional topological groups. First we state a Key Lemma treating locally continuum-connected subspaces of finite-dimensional topological groups. By a continuum we understand a connected compact Hausdorff space. We shall say that two points x, y of a top ...
ON NONASSOCIATIVE DIVISION ALGEBRAS^)
... center £7. Then % has characteristic two and ^[x] is an inseparable quadratic field over % for every x of 35 which is not in §. For let the characteristic of $ be different from two. If x is in X) and not in 5, there exists an element u=£-\-x such that u2 =a in $. Since 35 is a central division alge ...
... center £7. Then % has characteristic two and ^[x] is an inseparable quadratic field over % for every x of 35 which is not in §. For let the characteristic of $ be different from two. If x is in X) and not in 5, there exists an element u=£-\-x such that u2 =a in $. Since 35 is a central division alge ...
INFINITESIMAL BIALGEBRAS, PRE
... The main results of this paper establish connections between infinitesimal bialgebras, pre-Lie algebras and dendriform algebras, which were a priori unexpected. An infinitesimal bialgebra (abbreviated ǫ-bialgebra) is a triple (A, µ, ∆) where (A, µ) is an associative algebra, (A, ∆) is a coassociativ ...
... The main results of this paper establish connections between infinitesimal bialgebras, pre-Lie algebras and dendriform algebras, which were a priori unexpected. An infinitesimal bialgebra (abbreviated ǫ-bialgebra) is a triple (A, µ, ∆) where (A, µ) is an associative algebra, (A, ∆) is a coassociativ ...
Morita equivalence for regular algebras
... rings with local units (cf. [1], [2], [3]) and, between them, rings of infinite matrices with a finite number of non-zero entries; left ...
... rings with local units (cf. [1], [2], [3]) and, between them, rings of infinite matrices with a finite number of non-zero entries; left ...
Representation theory and applications in classical quantum
... Pv , it is said that the system is found in the state [v]. The number (p(v), p(w)) is interpreted as the probability to find the system in state [v], when measuring the observable Pv , when before measurement the system is known to be in state [w] (such a state could be prepared through measurement ...
... Pv , it is said that the system is found in the state [v]. The number (p(v), p(w)) is interpreted as the probability to find the system in state [v], when measuring the observable Pv , when before measurement the system is known to be in state [w] (such a state could be prepared through measurement ...
Classical Yang-Baxter Equation and Its Extensions
... It is equivalent to the tensor form (1) of CYBE under the following two conditions: ...
... It is equivalent to the tensor form (1) of CYBE under the following two conditions: ...