Classical Yang-Baxter Equation and Some Related Algebraic
... What is classical Yang-Baxter equation (CYBE)? ◦ From O-operators to CYBE C. Bai, A unified algebraic approach to the classical Yang-Baxter equation, J. Phys. A 40 (2007) 11073-11082. Notation: let ρ : g → gl(V ) be a representation of the Lie algebra g. On the vector space g ⊕ V , there is a natur ...
... What is classical Yang-Baxter equation (CYBE)? ◦ From O-operators to CYBE C. Bai, A unified algebraic approach to the classical Yang-Baxter equation, J. Phys. A 40 (2007) 11073-11082. Notation: let ρ : g → gl(V ) be a representation of the Lie algebra g. On the vector space g ⊕ V , there is a natur ...
Cohomology and K-theory of Compact Lie Groups
... the famous theorem by Borel that H ∗ (G/T, R) is isomorphic to the space harmonic polynomials on t and Solomon’s result on W -invariants of differential forms on t with polynomial coefficients(c.f. [So]), Reeder interpreted the right-hand side of (2) as W -invariant subspace of differential forms wi ...
... the famous theorem by Borel that H ∗ (G/T, R) is isomorphic to the space harmonic polynomials on t and Solomon’s result on W -invariants of differential forms on t with polynomial coefficients(c.f. [So]), Reeder interpreted the right-hand side of (2) as W -invariant subspace of differential forms wi ...
Full text
... Now consider k-th order “generalized Fibonacci sequences” of the form un = ki=1 Ri un−i , starting with k initial values 0, . . . , 0, 1. Zeckendorf representations and arrays exist, for these sequences, as above. That is, the initial row of the array is the sequence ai,j = uj suitable shifted so th ...
... Now consider k-th order “generalized Fibonacci sequences” of the form un = ki=1 Ri un−i , starting with k initial values 0, . . . , 0, 1. Zeckendorf representations and arrays exist, for these sequences, as above. That is, the initial row of the array is the sequence ai,j = uj suitable shifted so th ...
2. Ideals and homomorphisms 2.1. Ideals. Definition 2.1.1. An ideal
... is an automorphism of A. It is easy to see that this is a linear automorphism of A since it has the form 1 + η where η is nilpotent. So, the inverse is 1 − η + η 2 − · · · which is a finite sum. The following lemma shows that exp(−δ) is the inverse of exp δ. Lemma 2.2.1. Suppose that char F = 0 and ...
... is an automorphism of A. It is easy to see that this is a linear automorphism of A since it has the form 1 + η where η is nilpotent. So, the inverse is 1 − η + η 2 − · · · which is a finite sum. The following lemma shows that exp(−δ) is the inverse of exp δ. Lemma 2.2.1. Suppose that char F = 0 and ...
On the Lower Central Series of PI-Algebras
... Cn . Etingof, Kim and Ma [6] gave an explicit description of the quotients A/M4 (A). Dobrovolska, Kim, Ma and Etingof also studied the series Bi [3, 4]. In this paper we are interested in algebras that satisfy polynomial identities or PI-algebras. M. Dehn [2] first considered PI-algebras in 1922. Hi ...
... Cn . Etingof, Kim and Ma [6] gave an explicit description of the quotients A/M4 (A). Dobrovolska, Kim, Ma and Etingof also studied the series Bi [3, 4]. In this paper we are interested in algebras that satisfy polynomial identities or PI-algebras. M. Dehn [2] first considered PI-algebras in 1922. Hi ...
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS -modules. 20. KZ functor, II: image
... Weyl groups and constant functions c we recover the usual Iwahori-Hecke algebras (with the same additional relations) that appear in Lie theory (say in the study of the representations of finite reductive groups). For non-constant c we recover their straightforward generalization (Hecke algebras with ...
... Weyl groups and constant functions c we recover the usual Iwahori-Hecke algebras (with the same additional relations) that appear in Lie theory (say in the study of the representations of finite reductive groups). For non-constant c we recover their straightforward generalization (Hecke algebras with ...
2. Basic notions of algebraic groups Now we are ready to introduce
... and moreover H = Ga1 . . . Gan for some a1 , . . . , an ∈ I. (ii) The groups Sp2n and SOn (in characteristic $= 2) are connected. Incidentally, SOn has index two in On , hence it is the identity component of On . ...
... and moreover H = Ga1 . . . Gan for some a1 , . . . , an ∈ I. (ii) The groups Sp2n and SOn (in characteristic $= 2) are connected. Incidentally, SOn has index two in On , hence it is the identity component of On . ...
HYPERELLIPTIC JACOBIANS AND SIMPLE GROUPS U3 1
... Proof. This is Corollary 7.9 of [16]. 4. Steinberg representation We refer to [7] and [3] for a definition and basic properties of Steinberg representations. Let us fix an algebraic closure of F2 and denote it by F . We write φ : F → F for the Frobenius automorphism x 7→ x2 . Let q = 2m be a positiv ...
... Proof. This is Corollary 7.9 of [16]. 4. Steinberg representation We refer to [7] and [3] for a definition and basic properties of Steinberg representations. Let us fix an algebraic closure of F2 and denote it by F . We write φ : F → F for the Frobenius automorphism x 7→ x2 . Let q = 2m be a positiv ...
Semisimple algebras and Wedderburn`s theorem
... = Mm1 (C) ⊕ · · · ⊕ Mml (C). Since Mn (C)op ∼ = Mn (C) (the operation of transposition yields an isomorphism), we have A∼ = Mm1 (C) ⊕ · · · ⊕ Mml (C). Replacing A with an isomorphic algebra, we may assume that A is actually equal to the direct sums of matrix algebras in question. As we now know the ...
... = Mm1 (C) ⊕ · · · ⊕ Mml (C). Since Mn (C)op ∼ = Mn (C) (the operation of transposition yields an isomorphism), we have A∼ = Mm1 (C) ⊕ · · · ⊕ Mml (C). Replacing A with an isomorphic algebra, we may assume that A is actually equal to the direct sums of matrix algebras in question. As we now know the ...
A NEW PROOF OF E. CARTAN`S THEOREM ON
... PROOF. Consider the one-to-one mapping g—*(f, s) denned by: g—f-s where ƒ G ^ * , s£
... PROOF. Consider the one-to-one mapping g—*(f, s) denned by: g—f-s where ƒ G ^ * , s£
Representations with Iwahori-fixed vectors
... Proof: First, we see that the right-handed versions of the statements follow from the left-handed ones. Let `(wt) > `(w) for w ∈ W and t ∈ S. Since `(wt) > `(w), there is s ∈ S such that `(sw) < `(w). Then `(sw) + 1 = `(w) = `(wt) − 1 ≤ `(swt) + 1 − 1 = `(swt) ≤ `(sw) + 1 Thus, `(w) = `(swt) > `(sw) ...
... Proof: First, we see that the right-handed versions of the statements follow from the left-handed ones. Let `(wt) > `(w) for w ∈ W and t ∈ S. Since `(wt) > `(w), there is s ∈ S such that `(sw) < `(w). Then `(sw) + 1 = `(w) = `(wt) − 1 ≤ `(swt) + 1 − 1 = `(swt) ≤ `(sw) + 1 Thus, `(w) = `(swt) > `(sw) ...
Bochner`s linearization theorem
... The investigations by Cartan [3] of automorphisms and general transformations of a domain in several complex variables into itself by means of complex analytic functions led to the following result: If a compact group of automorphisms has a fixed point then in suitably chosen local coordinates aroun ...
... The investigations by Cartan [3] of automorphisms and general transformations of a domain in several complex variables into itself by means of complex analytic functions led to the following result: If a compact group of automorphisms has a fixed point then in suitably chosen local coordinates aroun ...
INTEGRABILITY CRITERION FOR ABELIAN EXTENSIONS OF LIE
... where Lg∗ denotes the pushforward map induced by the diffeomorphism Lg : G → G, h 7→ gh. By definition, X is completely determined by its value at the identity and g is therefore identified with T1 G as topological vector spaces endowed with the continuous Lie bracket of vector fields. The most stri ...
... where Lg∗ denotes the pushforward map induced by the diffeomorphism Lg : G → G, h 7→ gh. By definition, X is completely determined by its value at the identity and g is therefore identified with T1 G as topological vector spaces endowed with the continuous Lie bracket of vector fields. The most stri ...
Non-standard number representation: computer arithmetic, beta
... is denoted by AN . Let v be a word of A∗ , denote by v n the concatenation of v to itself n times, and by v ω the infinite concatenation vvv · · · . A word is said to be eventually periodic if it is of the form uv ω . An automaton over A, A = (Q, A, E, I, T ), is a directed graph labelled by element ...
... is denoted by AN . Let v be a word of A∗ , denote by v n the concatenation of v to itself n times, and by v ω the infinite concatenation vvv · · · . A word is said to be eventually periodic if it is of the form uv ω . An automaton over A, A = (Q, A, E, I, T ), is a directed graph labelled by element ...
OPERADS, FACTORIZATION ALGEBRAS, AND (TOPOLOGICAL
... Vertical and horizontal compositions of 2-morphisms are defined by choosing collars and gluing. This is well-defined because 2-morphisms are isomorphism classes of 2-bordisms, and thus the composition does not depend on the choice of the collar. However, composition of 1-morphisms requires the use o ...
... Vertical and horizontal compositions of 2-morphisms are defined by choosing collars and gluing. This is well-defined because 2-morphisms are isomorphism classes of 2-bordisms, and thus the composition does not depend on the choice of the collar. However, composition of 1-morphisms requires the use o ...
1 Smooth manifolds and Lie groups
... Given a matrix group G 6 GLn (R) we can differentiate a curve α : (−ε, ε) −→ G and define ...
... Given a matrix group G 6 GLn (R) we can differentiate a curve α : (−ε, ε) −→ G and define ...
An Introduction to Computational Group Theory
... subgroups can be computed, and it is possible to find automorphism groups. For these tasks, if there are algorithms in other representations of groups, they can handle only groups of much smaller order. There are examples of structural exploration of polycyclic groups with log |G| in the thousands; ...
... subgroups can be computed, and it is possible to find automorphism groups. For these tasks, if there are algorithms in other representations of groups, they can handle only groups of much smaller order. There are examples of structural exploration of polycyclic groups with log |G| in the thousands; ...
lecture 3
... • an action transition with label p is enabled if p is the next action symbol in x; advance the head past p • a test transition with label b is enabled if b, where is the current atom in x; do not advance the head • accept if occupying an accept state while scanning n ...
... • an action transition with label p is enabled if p is the next action symbol in x; advance the head past p • a test transition with label b is enabled if b, where is the current atom in x; do not advance the head • accept if occupying an accept state while scanning n ...
A NOTE ON NORMAL VARIETIES OF MONOUNARY ALGEBRAS 1
... A variety is called normal if no laws of the form s = t are valid in it where s is a variable and t is not a variable. For every variety V let N (V ) denote the smallest normal variety (of the same type as V ) containing V . Remark. From the results in Section 2 it follows that the non-normal elemen ...
... A variety is called normal if no laws of the form s = t are valid in it where s is a variable and t is not a variable. For every variety V let N (V ) denote the smallest normal variety (of the same type as V ) containing V . Remark. From the results in Section 2 it follows that the non-normal elemen ...
AN EXTENSION OF YAMAMOTO`S THEOREM
... Loesener [6] rediscovered (1.3). We remark that (1.1) remains true for Hilbert space operators [1, p.45]. Also see [3, 4, 8] for some generalizations of Yamamoto’s theorem. Equation (1.3) relates the two important sets of scalars of X in (1.2) in a very nice asymptotic way. It may be interpreted as ...
... Loesener [6] rediscovered (1.3). We remark that (1.1) remains true for Hilbert space operators [1, p.45]. Also see [3, 4, 8] for some generalizations of Yamamoto’s theorem. Equation (1.3) relates the two important sets of scalars of X in (1.2) in a very nice asymptotic way. It may be interpreted as ...
Introduction to linear Lie groups
... group member a group member T-1 such that T·T-1=E. • Associative property: T·(T’·T’’)= (T·T’)·T’’ 2. Elements of group form a “topological space” 3. Elements also constitute an “analytic manifold” Non countable number elements lying in a region “near” its identity ...
... group member a group member T-1 such that T·T-1=E. • Associative property: T·(T’·T’’)= (T·T’)·T’’ 2. Elements of group form a “topological space” 3. Elements also constitute an “analytic manifold” Non countable number elements lying in a region “near” its identity ...
Locally compact quantum groups 1. Locally compact groups from an
... and at measured spaces (X , µ) where it’s natural to look at L∞ (X ). As the other talks in this series have looked at Banach algebras, I’ll start instead there. ...
... and at measured spaces (X , µ) where it’s natural to look at L∞ (X ). As the other talks in this series have looked at Banach algebras, I’ll start instead there. ...
Separation of Variables and the Computation of Fourier
... David Maslen1 , Daniel N. Rockmore2†, and Sarah Wolff2‡ ...
... David Maslen1 , Daniel N. Rockmore2†, and Sarah Wolff2‡ ...
When is a group homomorphism a covering homomorphism?
... Let us extract another consequence of theorem 3. If ϕ is an analytic homomorphism from a Lie group G to a Lie group H, let ϕ∗ denote the differential of ϕ at eG . Note that, since every connected Lie group is locally compact, Lindelöf and a Baire space, theorem 3 implies that an analytic homomorphis ...
... Let us extract another consequence of theorem 3. If ϕ is an analytic homomorphism from a Lie group G to a Lie group H, let ϕ∗ denote the differential of ϕ at eG . Note that, since every connected Lie group is locally compact, Lindelöf and a Baire space, theorem 3 implies that an analytic homomorphis ...