Illustration of the quantum central limit theorem by
... We want to make these results a bit more transparent by large N. We observe the ...
... We want to make these results a bit more transparent by large N. We observe the ...
Symmetries in Conformal Field Theory
... up so that the exponentiated path integral for the WZW action gives a well-defined section of this line bundle. It turns out that the total space of this line bundle actually yields admits a natural group structure centrally extending the loop group. On the level of (complexified) Lie algebras this ...
... up so that the exponentiated path integral for the WZW action gives a well-defined section of this line bundle. It turns out that the total space of this line bundle actually yields admits a natural group structure centrally extending the loop group. On the level of (complexified) Lie algebras this ...
Postprint
... Let σ be an arbitrary element in grt1 and let Γuσ be a cocycle representing σ in the graph complex (GC2 [[u]], du ). We may assume that Γuσ consists of graphs with at least 4 vertices, see [Wi]. Then the element Φ(Γuσ ) describes an L∞ derivation of the Lie algebra V [1] without linear term. By expo ...
... Let σ be an arbitrary element in grt1 and let Γuσ be a cocycle representing σ in the graph complex (GC2 [[u]], du ). We may assume that Γuσ consists of graphs with at least 4 vertices, see [Wi]. Then the element Φ(Γuσ ) describes an L∞ derivation of the Lie algebra V [1] without linear term. By expo ...
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS Setting. W
... an isomorphism. Then we prove that the natural morphism RepΓ (H0,c , CΓ)// GL(CΓ)Γ → Spec(eH0,c e) is an isomorphism. Let y1 , . . . , yn be the tautological basis in Cn = h and x1 , . . . , xn be the dual basis in h∗ . The elements xn , yn still act on N Sn−1 ∼ = Cn . Show that [xn , yn ] ∈ O = {A| ...
... an isomorphism. Then we prove that the natural morphism RepΓ (H0,c , CΓ)// GL(CΓ)Γ → Spec(eH0,c e) is an isomorphism. Let y1 , . . . , yn be the tautological basis in Cn = h and x1 , . . . , xn be the dual basis in h∗ . The elements xn , yn still act on N Sn−1 ∼ = Cn . Show that [xn , yn ] ∈ O = {A| ...
on line
... group law is polynomial, the product map G × G → G becomes under the correspondence an algebra homomorphism ∆ going the other way. Likewise for the rest of the Hopf algebra structure. Two examples are as follows. The “affine line” is described by the coordinate algebra k[x] (polynomials in one varia ...
... group law is polynomial, the product map G × G → G becomes under the correspondence an algebra homomorphism ∆ going the other way. Likewise for the rest of the Hopf algebra structure. Two examples are as follows. The “affine line” is described by the coordinate algebra k[x] (polynomials in one varia ...
1, 2, 4, 8,... What comes next?
... then it is clear that x · y = 0 implies x = 0 or y = 0. In general, this algebra may not be associative. The identities for the sums of 1, 2 and 4 squares follow immediately from the identity (2) for real numbers R, complex numbers C, and quaternions H. In 1845 Cayley constructed an eight-dimensiona ...
... then it is clear that x · y = 0 implies x = 0 or y = 0. In general, this algebra may not be associative. The identities for the sums of 1, 2 and 4 squares follow immediately from the identity (2) for real numbers R, complex numbers C, and quaternions H. In 1845 Cayley constructed an eight-dimensiona ...
Titles and Abstracts
... constructed as composite operator from simper buildings blocks S1,S2 and S3 which obey Coxeter relations S1S2S1=S2S1S2, S2S3S2=S3S2S3 which are equivalent to the general integral star-triangle relation. The operators S1 and S3 have nice group theoretical meaning–they are intertwining operators for r ...
... constructed as composite operator from simper buildings blocks S1,S2 and S3 which obey Coxeter relations S1S2S1=S2S1S2, S2S3S2=S3S2S3 which are equivalent to the general integral star-triangle relation. The operators S1 and S3 have nice group theoretical meaning–they are intertwining operators for r ...
Section 1-1: Patterns and Expressions
... Expressing a Pattern with Algebra How many toothpicks are in the 20th figure? Figure Number (Input) ...
... Expressing a Pattern with Algebra How many toothpicks are in the 20th figure? Figure Number (Input) ...
Homework 7
... (iv): Let θ denote the 1-form θ := dz − 21 (xdy − ydx). The 2-plane field ξ = ker(θ) is a distribution spanned locally by X and Y . Show that for any points p and q there is a smooth path γ from p to q with θ(γ 0 ) = 0. In fact, show that for any continuous path δ from p to q there is a smooth path ...
... (iv): Let θ denote the 1-form θ := dz − 21 (xdy − ydx). The 2-plane field ξ = ker(θ) is a distribution spanned locally by X and Y . Show that for any points p and q there is a smooth path γ from p to q with θ(γ 0 ) = 0. In fact, show that for any continuous path δ from p to q there is a smooth path ...
Homework 3
... condition obtained in (ii) is satisfied, and hence each subset q∈Uα Tq M is open in T M . 2. Let X, Y, Z be C ∞ manifolds; assume X is a submanifold of Y and Y a submanifold of Z. Show then that X is a submanifold of Z. 3. Let n ∈ N? , and consider S n , the unit sphere of Rn+1 , with its usual topo ...
... condition obtained in (ii) is satisfied, and hence each subset q∈Uα Tq M is open in T M . 2. Let X, Y, Z be C ∞ manifolds; assume X is a submanifold of Y and Y a submanifold of Z. Show then that X is a submanifold of Z. 3. Let n ∈ N? , and consider S n , the unit sphere of Rn+1 , with its usual topo ...
(pdf)
... classical simplicial complexes, finite groups, and categories. For example, we shall show how to describe a space with 2n + 2 points that to the eyes of algebraic topology is “just the same” as the n-sphere S n . For another example, we shall reinterpret an interesting unsolved problem in finite gro ...
... classical simplicial complexes, finite groups, and categories. For example, we shall show how to describe a space with 2n + 2 points that to the eyes of algebraic topology is “just the same” as the n-sphere S n . For another example, we shall reinterpret an interesting unsolved problem in finite gro ...
Symposium Spring 2015 Schedule
... Okounkov and Anatoly Vershik applied the theory of Gelfand-Zetlin bases to the inductive family of symmetric groups, and in doing so recovered the main results of Young's work (most notably the Young graph) in a more natural way. In this talk, I'll give a broad introduction to Gelfand-Zetlin theory ...
... Okounkov and Anatoly Vershik applied the theory of Gelfand-Zetlin bases to the inductive family of symmetric groups, and in doing so recovered the main results of Young's work (most notably the Young graph) in a more natural way. In this talk, I'll give a broad introduction to Gelfand-Zetlin theory ...
PDF
... Moreover, consider the classical configuration space Q = R3 of a classical, mechanical system, or particle whose phase space is the cotangent bundle T ∗ R3 ∼ = R6 , for which the space of (classical) observables is taken to be the real vector space of smooth functions on M , and with T being an elem ...
... Moreover, consider the classical configuration space Q = R3 of a classical, mechanical system, or particle whose phase space is the cotangent bundle T ∗ R3 ∼ = R6 , for which the space of (classical) observables is taken to be the real vector space of smooth functions on M , and with T being an elem ...
Nondegenerate Pairings First let`s straighten out something that was
... Show that g is nondegenerate. Any algebra has a pairing of the above form; if the pairing is nondegenerate the algebra is semisimple. This is either a definition or a theorem depending on your taste: if we define a semisimple algebra to be a direct sum of algebras with no nontrivial two-sided ideals ...
... Show that g is nondegenerate. Any algebra has a pairing of the above form; if the pairing is nondegenerate the algebra is semisimple. This is either a definition or a theorem depending on your taste: if we define a semisimple algebra to be a direct sum of algebras with no nontrivial two-sided ideals ...