An Introduction to the Theory of Quantum Groups

... weirdness surrounding the discoveries of quantum physics. So, just what are these exciting new structures called quantum groups? It’s always good to be honest at the outset of a significant undertaking. With that said, the reader might be disappointed to learn that there is no rigorous, universally ...

... weirdness surrounding the discoveries of quantum physics. So, just what are these exciting new structures called quantum groups? It’s always good to be honest at the outset of a significant undertaking. With that said, the reader might be disappointed to learn that there is no rigorous, universally ...

Quantum groups: A survey of de nitions, motivations, and results

... I advise the reader to begin immediately with Chapter V and nd out what a quantum group is. One can always peek back at the earlier chapters and nd out the de nitions later. This makes it more fun and provides good motivation for learning the earlier background material. It also avoids getting bog ...

... I advise the reader to begin immediately with Chapter V and nd out what a quantum group is. One can always peek back at the earlier chapters and nd out the de nitions later. This makes it more fun and provides good motivation for learning the earlier background material. It also avoids getting bog ...

The Automorphic Universe

... • if µ is ergodic, then f+ (x) is a.e. constant and its value is X f dµ If T is an automorphism, then the limit f− (x) exists asR well and the two limits coincide a.e., f+ (x) = f− (x) (mod 0). The integral X f dµ is the space average of the observable f . The last part of the theorem (that is when ...

... • if µ is ergodic, then f+ (x) is a.e. constant and its value is X f dµ If T is an automorphism, then the limit f− (x) exists asR well and the two limits coincide a.e., f+ (x) = f− (x) (mod 0). The integral X f dµ is the space average of the observable f . The last part of the theorem (that is when ...

Thèse de doctorat - IMJ-PRG

... 6.2 Recollections on quantum shuffle algebras . . . . . . . . . . . . 6.3 Construction of quantum groups . . . . . . . . . . . . . . . . . . 6.4 Main Construction and Rosso’s theorem . . . . . . . . . . . . . 6.5 Coalgebra homology and module structures . . . . . . . . . . . 6.6 A Borel-Weil-Bott ty ...

... 6.2 Recollections on quantum shuffle algebras . . . . . . . . . . . . 6.3 Construction of quantum groups . . . . . . . . . . . . . . . . . . 6.4 Main Construction and Rosso’s theorem . . . . . . . . . . . . . 6.5 Coalgebra homology and module structures . . . . . . . . . . . 6.6 A Borel-Weil-Bott ty ...

For screen - Mathematical Sciences Publishers

... of elements on this basis (see, for example [3], [8]). In the sixth section we adapt a well-known method of triangular splitting to the quantification with constants. The original method appeared in studies of simple finite dimensional Lie algebras. Then it has been extended into the field of quantu ...

... of elements on this basis (see, for example [3], [8]). In the sixth section we adapt a well-known method of triangular splitting to the quantification with constants. The original method appeared in studies of simple finite dimensional Lie algebras. Then it has been extended into the field of quantu ...

A categorification of a quantum Frobenius map

... (p-DG) algebras that will be used later. Then, we define the notion of a “slash-zero formal” p-DG algebra, which is analogous to the definition of formality for the usual differential graded algebras. As in the ordinary DG case, slash-zero formality allows one to compute algebraic invariants of a pD ...

... (p-DG) algebras that will be used later. Then, we define the notion of a “slash-zero formal” p-DG algebra, which is analogous to the definition of formality for the usual differential graded algebras. As in the ordinary DG case, slash-zero formality allows one to compute algebraic invariants of a pD ...

cluster algebras in algebraic lie theory

... studying total positivity in semisimple algebraic groups. A cluster algebra is a commutative ring with a distinguished set of generators and a particular type of relations. Although there can be infinitely many generators and relations, they are all obtained from a finite number of them by means of ...

... studying total positivity in semisimple algebraic groups. A cluster algebra is a commutative ring with a distinguished set of generators and a particular type of relations. Although there can be infinitely many generators and relations, they are all obtained from a finite number of them by means of ...

Endomorphism Bialgebras of Diagrams and of Non

... construct explicit examples of such and check all the necessary properties. This gets even more complicated if we have to verify that something like a comodule algebra over a bialgebra is given. Bialgebras and comodule algebras, however, arise in a very natural way in non-commutative geometry and in ...

... construct explicit examples of such and check all the necessary properties. This gets even more complicated if we have to verify that something like a comodule algebra over a bialgebra is given. Bialgebras and comodule algebras, however, arise in a very natural way in non-commutative geometry and in ...

QUANTUM GROUPS AND DIFFERENTIAL FORMS Contents 1

... SWAPNEEL MAHAJAN Abstract. We construct a quantum semigroup and an algebra of forms appropriate for the generalised homological algebra of N -complexes [8]. This is an analogue to the picture for usual homological algebra, where one has the quantum general linear group [9] and the diﬀerential forms ...

... SWAPNEEL MAHAJAN Abstract. We construct a quantum semigroup and an algebra of forms appropriate for the generalised homological algebra of N -complexes [8]. This is an analogue to the picture for usual homological algebra, where one has the quantum general linear group [9] and the diﬀerential forms ...

The semantics of the canonical commutation relation

... Especially nice finite-dimensional representations have Weyl algebras for which the multipliers e2πiak bl are roots of unity. This is the case when all the ak , bl are rational. We call such algebras rational Weyl algebras. In a certain sense, made precise in the paper, the rational Weyl algebras ap ...

... Especially nice finite-dimensional representations have Weyl algebras for which the multipliers e2πiak bl are roots of unity. This is the case when all the ak , bl are rational. We call such algebras rational Weyl algebras. In a certain sense, made precise in the paper, the rational Weyl algebras ap ...

PARADIGM OF QUASI-LIE AND QUASI-HOM

... Schlessinger. Nowadays deformation-theoretic ideas penetrate most aspects of both mathematics and physics and cut to the core of theoretical and computational problems. In the case of Lie algebras, quantum deformations (or q-deformations) and quantum groups associated to Lie algebras have been inves ...

... Schlessinger. Nowadays deformation-theoretic ideas penetrate most aspects of both mathematics and physics and cut to the core of theoretical and computational problems. In the case of Lie algebras, quantum deformations (or q-deformations) and quantum groups associated to Lie algebras have been inves ...

Light-like -deformations and scalar field theory via Drinfeld twist

... algebra and the corresponding κ-Minkowski space. After proving some useful properties of the star-product, we formulate the free scalar field theory (in section 4) and a model of interacting φ4 field theory on κ-Minkowski space (in section 5). Finally, in section 6 we give our conclusion and future ...

... algebra and the corresponding κ-Minkowski space. After proving some useful properties of the star-product, we formulate the free scalar field theory (in section 4) and a model of interacting φ4 field theory on κ-Minkowski space (in section 5). Finally, in section 6 we give our conclusion and future ...

Classical elliptic current algebras

... for Green distributions consist of the pairs (u, z) such that − Im τ < Im(u − z) < 0 and ¯ = {(u, z) | Im u = Im z}. 0 < Im(u − z) < Im τ respectively and ∆ 2.0.7. Integration contour. The geometric roots of the difference between these two choices can be explained as follows. These choices of test ...

... for Green distributions consist of the pairs (u, z) such that − Im τ < Im(u − z) < 0 and ¯ = {(u, z) | Im u = Im z}. 0 < Im(u − z) < Im τ respectively and ∆ 2.0.7. Integration contour. The geometric roots of the difference between these two choices can be explained as follows. These choices of test ...

DIFFERENTIAL EQUATIONS ON HYPERPLANE COMPLEMENTS II Contents 1 3

... However, as a (H 0 , C[x])-bimodule, H 0 is the space of (algebraic) sections of the vector bundle I over C with fibre at m ∈ C over a given by the (finite-dimensional) induced module Im := H 0 ⊗C[x] Cm , where Cm is endowed with the C[x] module structure given by evaluation at m. By the PBW theorem ...

... However, as a (H 0 , C[x])-bimodule, H 0 is the space of (algebraic) sections of the vector bundle I over C with fibre at m ∈ C over a given by the (finite-dimensional) induced module Im := H 0 ⊗C[x] Cm , where Cm is endowed with the C[x] module structure given by evaluation at m. By the PBW theorem ...

Duncan-Dunne-LINCS-2016-Interacting

... to Span(FinSet) and Cospan(FinSet) respectively. See [28] and [25]. The spider theorem is equivalent to this last fact. For any category T, one can view T + Top as a †-category, hence in all of the above we may assume that δ = µ† and = η † . However it is not always desirable to do so. Whether we ...

... to Span(FinSet) and Cospan(FinSet) respectively. See [28] and [25]. The spider theorem is equivalent to this last fact. For any category T, one can view T + Top as a †-category, hence in all of the above we may assume that δ = µ† and = η † . However it is not always desirable to do so. Whether we ...

M04/16

... attain a deeper understanding of more general measurements, say measurements with a ﬁnite or even inﬁnite number of real values. In this paper we study ways in which the orthosum and sequential product can be extended from eﬀects to more general measurements. Just as a sequential eﬀect algebra descr ...

... attain a deeper understanding of more general measurements, say measurements with a ﬁnite or even inﬁnite number of real values. In this paper we study ways in which the orthosum and sequential product can be extended from eﬀects to more general measurements. Just as a sequential eﬀect algebra descr ...

Conjugation coinvariants of quantum matrices

... induces an action of GL(N, C) on the coordinate algebra C[X ij ] of N × N matrices. The invariant functions with respect to this action are well-known, they are the elements in the C-subalgebra of C[Xij ] that is generated by the trace functions σi , for i = 1, . . . , N, where σi is the sum of the ...

... induces an action of GL(N, C) on the coordinate algebra C[X ij ] of N × N matrices. The invariant functions with respect to this action are well-known, they are the elements in the C-subalgebra of C[Xij ] that is generated by the trace functions σi , for i = 1, . . . , N, where σi is the sum of the ...

Reflection equation algebra in braided geometry 1

... This algebra has a braided bi-algebra structure and can be equipped with the coaction of the RTT algebra (in the standard case it can be also equipped with an action of the QG Uq (sl(n))). The term ”braided” means that ∆(a b) = ∆(a) ∆(b) = (a1 ⊗ a2 ) (b1 ⊗ b2 ) := a1 b̃1 ⊗ ã2 b2 where ∆(a) = a1 ⊗ a ...

... This algebra has a braided bi-algebra structure and can be equipped with the coaction of the RTT algebra (in the standard case it can be also equipped with an action of the QG Uq (sl(n))). The term ”braided” means that ∆(a b) = ∆(a) ∆(b) = (a1 ⊗ a2 ) (b1 ⊗ b2 ) := a1 b̃1 ⊗ ã2 b2 where ∆(a) = a1 ⊗ a ...

Past Research

... I work on the cohomology, structure, and representations of various types of rings, such as Hopf algebras and group-graded algebras. My research program has involved collaborations with many mathematicians, including work with postdocs and graduate students. Below is a summary of some of my past res ...

... I work on the cohomology, structure, and representations of various types of rings, such as Hopf algebras and group-graded algebras. My research program has involved collaborations with many mathematicians, including work with postdocs and graduate students. Below is a summary of some of my past res ...

Titles and Abstracts - The Institute of Mathematical Sciences

... • Masaki Izumi: Toeplitz CAR flows (Joint work with R. Srinivasan) Abstract: In this talk, I will report on our joint work about E0 -semigroups, which are continuous 1-parameter semigroups of unital endomorphisms of B(H). In 1987, Powers constructed the first example of type III E0 -semigroups usin ...

... • Masaki Izumi: Toeplitz CAR flows (Joint work with R. Srinivasan) Abstract: In this talk, I will report on our joint work about E0 -semigroups, which are continuous 1-parameter semigroups of unital endomorphisms of B(H). In 1987, Powers constructed the first example of type III E0 -semigroups usin ...

Shanghai Conference on Representation Theory

... Degeneracy loci formulas in K-theory of the symplectic Grassmannian bundles Takeshi Ikeda (³X ) Let E be a vector bundle of rank 2n over a smooth variety X equipped with a symplectic structure. For 0 ≤ k ≤ n, let SGk (E) denote the Grassmannian bundle that parametrizes isotropic subbundles S ⊂ E of ...

... Degeneracy loci formulas in K-theory of the symplectic Grassmannian bundles Takeshi Ikeda (³X ) Let E be a vector bundle of rank 2n over a smooth variety X equipped with a symplectic structure. For 0 ≤ k ≤ n, let SGk (E) denote the Grassmannian bundle that parametrizes isotropic subbundles S ⊂ E of ...

Whittaker Functions and Quantum Groups

... ) be the states of the top and bottom rows. The partition function then is a row transfer matrix ΘS (δ, ε). The partition function with several rows is the product of the row transfer matrices. Theorem 4. (Baxter) If ∆S = ∆T then ΘS and ΘT commute. Though we will not explain this point, the commuta ...

... ) be the states of the top and bottom rows. The partition function then is a row transfer matrix ΘS (δ, ε). The partition function with several rows is the product of the row transfer matrices. Theorem 4. (Baxter) If ∆S = ∆T then ΘS and ΘT commute. Though we will not explain this point, the commuta ...

A pairing between super Lie-Rinehart and periodic cyclic

... Brillouin zone. Its pairing with an appropriately normalized even dimensional class [e ⊗ e ⊗ e] in HP2(A) computes the Hall conductance σ in noncommutative geometric models of quantum Hall effect ...

... Brillouin zone. Its pairing with an appropriately normalized even dimensional class [e ⊗ e ⊗ e] in HP2(A) computes the Hall conductance σ in noncommutative geometric models of quantum Hall effect ...