Algebraic Groups
... The proof shows that R∗ is a special open set of R. In particular, R∗ is irreducible of dimension dim R∗ = dim R. 1.2. Isomorphisms and products. It follows from our definition that an algebraic group G is an affine variety with a group structure. These two structures are related in the usual way. N ...
... The proof shows that R∗ is a special open set of R. In particular, R∗ is irreducible of dimension dim R∗ = dim R. 1.2. Isomorphisms and products. It follows from our definition that an algebraic group G is an affine variety with a group structure. These two structures are related in the usual way. N ...
LIE ALGEBRAS OF CHARACTERISTIC 2Q
... the 2 element field, then all the above requirements are met by V, F, and /* as defined on the roots. This follows because //* has a basis of roots, and if ß is any other root, ß is a linear combination of this basis with coefficients from the 2 element field, since/* is nonsingular and its values o ...
... the 2 element field, then all the above requirements are met by V, F, and /* as defined on the roots. This follows because //* has a basis of roots, and if ß is any other root, ß is a linear combination of this basis with coefficients from the 2 element field, since/* is nonsingular and its values o ...
Full text
... its entire domain, proving the lemma for that case. For m even we set k = m/2 and use ...
... its entire domain, proving the lemma for that case. For m even we set k = m/2 and use ...
The structure of reductive groups - UBC Math
... E/F is any finite extension then RE/F Ga has the property that its F -rational points may be canonically identified with E . As an algebraic group it is just isomorphic to G2a , but there is no canonical isomorphism with G2a —in other words, there is no canonical basis of E as an F -vector space. Ex ...
... E/F is any finite extension then RE/F Ga has the property that its F -rational points may be canonically identified with E . As an algebraic group it is just isomorphic to G2a , but there is no canonical isomorphism with G2a —in other words, there is no canonical basis of E as an F -vector space. Ex ...
on h1 of finite dimensional algebras
... The quiver algebra is the vector space of finite linear combinations of paths. The multiplication of two paths is their concatenation if it can be done and 0 otherwise. Each vertex is an idempotent, the sum of all the vertices is the unit element in case Q0 is finite. Some authors are interested in ...
... The quiver algebra is the vector space of finite linear combinations of paths. The multiplication of two paths is their concatenation if it can be done and 0 otherwise. Each vertex is an idempotent, the sum of all the vertices is the unit element in case Q0 is finite. Some authors are interested in ...
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... The group PSL2 (Z=qZ) is central to our work because explicit expander codes can be constructed using the explicit Ramanujan graphs that are known for this group, and our previous work [6] indicates that random Cayley graphs for this group are good expanders. The use of random Cayley graphs|that is, ...
... The group PSL2 (Z=qZ) is central to our work because explicit expander codes can be constructed using the explicit Ramanujan graphs that are known for this group, and our previous work [6] indicates that random Cayley graphs for this group are good expanders. The use of random Cayley graphs|that is, ...
Hochschild cohomology: some methods for computations
... that usually we may reduce the study of indecomposable modules over an algebra to that for the corresponding simply connected algebras, using Galois coverings. Despite this very little is known about computations for particular classes of finite–dimensional algebras, since the computations of these ...
... that usually we may reduce the study of indecomposable modules over an algebra to that for the corresponding simply connected algebras, using Galois coverings. Despite this very little is known about computations for particular classes of finite–dimensional algebras, since the computations of these ...
Article - Archive ouverte UNIGE
... We show that the exponentials in the Clifford and exterior algebras satisfy natural differential equations. In the following Section the dynamical Yang-Baxter equation emerges as a consistency condition for these differential equations. 3.1. Lie algebra consequences of Theorem 2.1. A quadratic Lie a ...
... We show that the exponentials in the Clifford and exterior algebras satisfy natural differential equations. In the following Section the dynamical Yang-Baxter equation emerges as a consistency condition for these differential equations. 3.1. Lie algebra consequences of Theorem 2.1. A quadratic Lie a ...
groups and categories
... A strict monoidal category is exactly the same thing as a monoid in Cat. Examples where the underlying category is a poset P include both the meet x∧y and join x ∨ y operations, with terminal object 1 and initial object 0 as units, respectively (assuming P has these structures), as well as the poset ...
... A strict monoidal category is exactly the same thing as a monoid in Cat. Examples where the underlying category is a poset P include both the meet x∧y and join x ∨ y operations, with terminal object 1 and initial object 0 as units, respectively (assuming P has these structures), as well as the poset ...
splitting in relation algebras - American Mathematical Society
... algebras from representable ones (see [HMT, 3.2.67 and 3.2.69]). In this paper we adapt this method to relation algebras. The conditions for splittability in relation algebras seem to be more complex than in cylindric algebras, where every atom below d(a x a) can be split. We use the terminology and ...
... algebras from representable ones (see [HMT, 3.2.67 and 3.2.69]). In this paper we adapt this method to relation algebras. The conditions for splittability in relation algebras seem to be more complex than in cylindric algebras, where every atom below d(a x a) can be split. We use the terminology and ...
Frobenius algebras and monoidal categories
... An algebra A over a field k is called Frobenius when it is finite dimensional and equipped with a linear function e : A æ æÆ k such that: e(ab) = 0 for all a Œ A implies b = 0. ...
... An algebra A over a field k is called Frobenius when it is finite dimensional and equipped with a linear function e : A æ æÆ k such that: e(ab) = 0 for all a Œ A implies b = 0. ...
Semi-crossed Products of C*-Algebras
... integers, we can define a Banach algebra 1’(Z+, VI, a). The elements will be of the form F = Cn>O 6, @ x,, with X, E ?I, and llFll, = CnZ,, (Ix,I/ < co. In analogy with the above paragraph, the multiplication could be defined by letting the semigroup act on the right or on the left. For technical re ...
... integers, we can define a Banach algebra 1’(Z+, VI, a). The elements will be of the form F = Cn>O 6, @ x,, with X, E ?I, and llFll, = CnZ,, (Ix,I/ < co. In analogy with the above paragraph, the multiplication could be defined by letting the semigroup act on the right or on the left. For technical re ...
Examples of modular annihilator algebras
... of A and Q = R D / by [24, Proposition 3, p. 206]. Then SA C R so that I C R, and therefore / C R . But then / = Ç>, a contradiction. C5. Assume that A is a topological algebra such that every primitive ideal of A is closed. If / is a two-sided ideal of A such that SA C / and Jn C SA for some positi ...
... of A and Q = R D / by [24, Proposition 3, p. 206]. Then SA C R so that I C R, and therefore / C R . But then / = Ç>, a contradiction. C5. Assume that A is a topological algebra such that every primitive ideal of A is closed. If / is a two-sided ideal of A such that SA C / and Jn C SA for some positi ...
Notes for an Introduction to Kontsevich`s quantization theorem B
... of g uniquely extends to a Poisson bracket on the symmetric algebra S(g). The ideal I of S(g) generated by all monomials of degree 2 is a Poisson ideal (i.e. we have {f, I} ⊂ I for all f ∈ S(g)). So the quotient A = S(g)/I becomes a (finite-dimensional) Poisson algebra. Assume that its bracket comes ...
... of g uniquely extends to a Poisson bracket on the symmetric algebra S(g). The ideal I of S(g) generated by all monomials of degree 2 is a Poisson ideal (i.e. we have {f, I} ⊂ I for all f ∈ S(g)). So the quotient A = S(g)/I becomes a (finite-dimensional) Poisson algebra. Assume that its bracket comes ...
Introduction to the Lorentz algebra
... Let now Φ be any commutative unital ring and R any algebra in the category algΦ . Let Max(R) denote the maximal spectrum of R (the set of maximal ideals) and m ∈ Max(R). Let V be any R-algebra (not necessarily associative) with V 2 6= 0 and I / V an ideal. Of course the R-submodule mV is an ideal of ...
... Let now Φ be any commutative unital ring and R any algebra in the category algΦ . Let Max(R) denote the maximal spectrum of R (the set of maximal ideals) and m ∈ Max(R). Let V be any R-algebra (not necessarily associative) with V 2 6= 0 and I / V an ideal. Of course the R-submodule mV is an ideal of ...
LECTURE 12: HOPF ALGEBRA (sl ) Introduction
... the simplest case, Uq (sl2 ). An important feature here is that the tensor product is not commutative in a naive sense. This is a feature and not a bug, this is one of the main reasons why the quantum groups were introduced. 1. Hopf algebras 1.1. Tensor products and duals. Recall that for a group G ...
... the simplest case, Uq (sl2 ). An important feature here is that the tensor product is not commutative in a naive sense. This is a feature and not a bug, this is one of the main reasons why the quantum groups were introduced. 1. Hopf algebras 1.1. Tensor products and duals. Recall that for a group G ...
AN INTRODUCTION TO FLAG MANIFOLDS Notes1 for the Summer
... is a Lie subalgebra (here α and β are two real numbers). The subgroup H prescribed by point (b) above is H = {(eiαt , eiβt ) : t ∈ R}. One can show that if the ratio α/β is irrational, then H is dense in T 2 . So there is no way of making H into a (closed) matrix Lie group — see also the note follow ...
... is a Lie subalgebra (here α and β are two real numbers). The subgroup H prescribed by point (b) above is H = {(eiαt , eiβt ) : t ∈ R}. One can show that if the ratio α/β is irrational, then H is dense in T 2 . So there is no way of making H into a (closed) matrix Lie group — see also the note follow ...
Chapter 6 Orthogonal representations II: Minimal dimension - D-MATH
... on each edge of a triangle (Figure 2). This graph has an orthogonal representation (vi : i = 1, . . . , 6) in R3 : v1 , v2 and v3 are mutually orthogonal unit vectors, and v4 = v1 + v2 , v5 = v1 + v3 , and v6 = v2 + v3 . This representation is not in locally general position, since the nodes non-adj ...
... on each edge of a triangle (Figure 2). This graph has an orthogonal representation (vi : i = 1, . . . , 6) in R3 : v1 , v2 and v3 are mutually orthogonal unit vectors, and v4 = v1 + v2 , v5 = v1 + v3 , and v6 = v2 + v3 . This representation is not in locally general position, since the nodes non-adj ...
One-parameter subgroups and Hilbert`s fifth problem
... The affirmative solution of Hilbert's fifth problem requires that we bridge* thé gap between topologico-algebraic structure and analytic structure. In building this bridge we quite naturally seek an intermediate island on which to rest the piers. Such an island is provided by the one-parameter subgr ...
... The affirmative solution of Hilbert's fifth problem requires that we bridge* thé gap between topologico-algebraic structure and analytic structure. In building this bridge we quite naturally seek an intermediate island on which to rest the piers. Such an island is provided by the one-parameter subgr ...
Subfactors and Modular Tensor Categories
... obtains interesting non-Euclidean geometries, such as hyperbolic geometry. If one describes the properties of an algebra of functions on a (nice) topological space and drops the requirement that multiplication of functions is commutative, one obtains the notion of a C*-algebra. More generally, Conne ...
... obtains interesting non-Euclidean geometries, such as hyperbolic geometry. If one describes the properties of an algebra of functions on a (nice) topological space and drops the requirement that multiplication of functions is commutative, one obtains the notion of a C*-algebra. More generally, Conne ...
Representations on Hessenberg Varieties and Young`s Rule
... Fix G = GLn (C) and let B be the subgroup of upper-triangular matrices. Let the respective Lie algebras be g and b. The flag variety is the homogenous space G/B. It is known to be a smooth complex projective variety [H, Section 21]. Hessenberg varieties are a family of subvarieties of the flag varie ...
... Fix G = GLn (C) and let B be the subgroup of upper-triangular matrices. Let the respective Lie algebras be g and b. The flag variety is the homogenous space G/B. It is known to be a smooth complex projective variety [H, Section 21]. Hessenberg varieties are a family of subvarieties of the flag varie ...
Representations on Hessenberg Varieties and Young`s Rule
... Fix G = GLn (C) and let B be the subgroup of upper-triangular matrices. Let the respective Lie algebras be g and b. The flag variety is the homogenous space G/B. It is known to be a smooth complex projective variety [H, Section 21]. Hessenberg varieties are a family of subvarieties of the flag varie ...
... Fix G = GLn (C) and let B be the subgroup of upper-triangular matrices. Let the respective Lie algebras be g and b. The flag variety is the homogenous space G/B. It is known to be a smooth complex projective variety [H, Section 21]. Hessenberg varieties are a family of subvarieties of the flag varie ...
A brief introduction to pre
... Classical and quantum Yang-Baxter equations: Svinolupov and Sokolov(1994), Etingof and Soloviev(1999), Golubschik and Sokolov(2000), · · · Poisson brackets and infinite-dimensional Lie algebras: Gel’fand and Dorfman(1979), Dubrovin and Novikov(1984), Balinskii and Novikov(1985), · · · Quantum field ...
... Classical and quantum Yang-Baxter equations: Svinolupov and Sokolov(1994), Etingof and Soloviev(1999), Golubschik and Sokolov(2000), · · · Poisson brackets and infinite-dimensional Lie algebras: Gel’fand and Dorfman(1979), Dubrovin and Novikov(1984), Balinskii and Novikov(1985), · · · Quantum field ...
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Completed representation ring spectra of nilpotent groups Algebraic & Geometric Topology [Logo here]
... the stable homotopy category. This process has seen highly useful generalizations, such as the Adams-Novikov spectral sequence exposited in [1], and the E -nilpotent completion spectral sequence of [3]. By using the same method of proof, Carlsson gave a construction for modules over a ring R in [6]. ...
... the stable homotopy category. This process has seen highly useful generalizations, such as the Adams-Novikov spectral sequence exposited in [1], and the E -nilpotent completion spectral sequence of [3]. By using the same method of proof, Carlsson gave a construction for modules over a ring R in [6]. ...
Conf
... associative, commutative and unital ring Φ and consider the category algΦ of associative commutative and unital Φ-algebras. Then for any object R in algΦ we may define the Lorentz type algebra LR := LZ ⊗Z R. This is nothing but the free R-module with basis s12 , s13 , s14 , a23 , a24 and a34 , enric ...
... associative, commutative and unital ring Φ and consider the category algΦ of associative commutative and unital Φ-algebras. Then for any object R in algΦ we may define the Lorentz type algebra LR := LZ ⊗Z R. This is nothing but the free R-module with basis s12 , s13 , s14 , a23 , a24 and a34 , enric ...