Noncommutative geometry of homogenized
quantum 𝔰𝔩(2, ℂ)

Chirvăsitu, Alexandru; Smith, S. Paul; Wong, Liang Ze

doi:10.2140/pjm.2018.292.305

Cited by 7 publications

(5 citation statements)

References 22 publications

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“…He also observed that "degenerate" versions of the 4-dimensional Sklyanin algebras lead to the quantized enveloping algebras U q (sl 2 ). A detailed examination of this degeneration procedure is carried out in [11].…”

Section: Introductionmentioning

confidence: 99%

Exotic elliptic algebras of dimension 4

Chirvăsitu

Smith

2017

Advances in Mathematics

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Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A = A(E, τ ) that depends on this data:where S(E, τ ) is the 4-dimensional Sklyanin algebra associated to (E, τ ), M2(k) is the ring of 2 × 2 matrices over k, and Γ is (Z/2) × (Z/2) acting in a particular way as automorphisms of S and M2(k). The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E[2] on E. Following the ideas and results in papers of Artin-Tate-Van den Bergh, Smith-Stafford, and Levasseur-Smith, this paper examines the line modules, point modules, and fat point modules, over A, and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P 3 that we denote by Proj nc (A). Point modules and fat point modules determine "points" in Proj nc (A). Line modules determine "lines" in Proj nc (A). Line modules for A are in bijection with certain lines in P(A * 1 ) ∼ = P 3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G(1, 3). Shelton-Vancliff call L the line scheme for A. We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plücker P 5 , and that deg(L) = 20. The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P(A * 1 ). Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A.2010 Mathematics Subject Classification. 16E65, 16S38, 16T05, 16W50. Key words and phrases. Sklyanin algebras, elliptic algebras, line modules.2. The 4-dimensional Sklyanin algebras S(E, τ ) 2.1. Definition of the Sklyanin algebra. Always, α 1 , α 2 , α 3 are fixed elements in k − {0, ±1} such that α 1 + α 2 + α 3 + α 1 α 2 α 3 = 0. We often write α = α 1 , β = α 2 , and γ = α 3 . As in [9, §6], we fix a, b, c, i ∈ k such that a 2 = α, b 2 = β, c 2 = γ, and i 2 = −1.

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Section: Introductionmentioning

confidence: 99%

Exotic elliptic algebras of dimension 4

Chirvăsitu

Smith

2017

Advances in Mathematics

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“…Or, conversely, the U q (sl n )'s are "degenerations" of Q n 2 ,n−1 (E, τ ). A detailed examination of this degeneration process for n = 2 is carried out in [CSW18].…”

Section: The Artin-van Den Bergh Theorem and The Functor γmentioning

confidence: 99%

Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

Chirvasitu,

Kanda,

Smith

2019

Preprint

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The elliptic algebras in the title are connected graded C-algebras, denoted Q n,k (E, τ ), depending on a pair of relatively prime integers n > k ≥ 1, an elliptic curve E, and a point τ ∈ E. For fixed n and k, they form a flat family of deformations of the polynomial ring C[x 0 , . . . , x n−1 ]. This paper examines a canonical homomorphism from Q n,k (E, τ ) to the twisted homogeneous coordinate ring) are generated in degrees ≤ 3, and the non-commutative scheme Proj nc (Q n,k (E, τ )) has a closed subvariety that is isomorphic to E g or S g E, respectively. When X n/k = E g and τ = 0, the results about B(X n/k , σ ′ , L ′ n/k ) show that the morphism |L n/k | : E g → P n−1 embeds E g as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.ALEX CHIRVASITU, RYO KANDA, AND S. PAUL SMITH 6. Relations for B(S g E, σ, L) 28 6.1. Preparations 28 6.2. Surjectivity of multiplication maps 28 7. The mapk ) 34 7.4. The algebras Q 2k+1,k (E, τ ) 34 8. The rings B(E g , σ, L n/k ) 35 8.1. The main result in this section 36 8.2. Notation 36 8.3. Preliminary results 36 8.4. Proof of Proposition 8.1 38 9. Relations for B(E g , σ, L) 39 9.1. Notation 39 9.2. The surjectivity and kernel of the map Q n,k (E, τ ) → B(E g , σ, L n/k ) when X n/k = E g 40 References 43

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“…Since dim k (R) = 6 = dim(P 3 × P 3 ), Γ = ∅. ) and the class of the zero locus of a non-zero element in V ⊗ V is equal to s + t. If dim(Γ) = 0, then the class of Γ is (s + t) 6 since dim(R) = 6. But (s + t) 6 = 20s 3 t 3 so the cardinality of Γ is 20 when its points are counted with multiplicity.…”

Section: The Zero Locus Of the Relations For A(α β γ)mentioning

confidence: 99%

“…In conclusion, we have Proof. Simply compose ψ and its analogue for ℓ 2 (which are double covers by Proposition 5.4) with the two-fold covers of the form (5)(6). and (α j + α k ){x j , a i } + (α j α k + 1)[x j , c i ] − (α k + 1) α j {x i , a j } + [x i , c j ] + (α j − 1) [x 0 , a k ] − {x 0 , c k } = (α 1 + α 2 + α 3 + α 1 α 2 α 3 )[x k , x 2 j ], and (α i + α k ){x i , a j } − α i (α j α k + 1)[x i , c j ] + (α i + 1) {x 0 , c k } − [x 0 , a k ] − (α k − 1) α i {x j , a i } − [x j , c i ] = − (α 1 + α 2 + α 3 + α 1 α 2 α 3 )[x k , x 2 i ] and − (α j + α k ){x 0 , c i } − α i (α j α k + 1)[x 0 , a i ] + α i (α k + 1) α j {x k , a j } − [x k , c j ] + α i (α j − 1) α k {x j , a k } + [x j , c k ] = − (α 1 + α 2 + α 3 + α 1 α 2 α 3 )[x i , x 2 0 ].…”

Section: Atmentioning

confidence: 99%

Some algebras having relations like those for the 4-dimensional Sklyanin algebras

Chirvăsitu¹,

Smith²

2017

Preprint

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The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of noncommutative graded algebras, often denoted A(E, τ ), that depend on a quartic elliptic curve E ⊆ P 3 and a translation automorphism τ of E. They are graded algebras generated by four degreeone elements subject to six quadratic relations and in many important ways they behave like the polynomial ring on four indeterminates apart from the minor difference that they are not commutative. They can be seen as "elliptic analogues" of the enveloping algebra of gl(2, C) and the quantized enveloping algebras Uq(gl2).Recently, Cho, Hong, and Lau, conjectured that a certain 2-parameter family of algebras arising in their work on homological mirror symmetry consists of 4-dimensional Sklyanin algebras. This paper shows their conjecture is false in the generality they make it. On the positive side, we show their algebras exhibit features that are similar to, and differ from, analogous features of the 4-dimensional Sklyanin algebras in interesting ways. We show that most of the Cho-Hong-Lau algebras determine, and are determined by the graph of a bijection between two 20-point subsets of the projective space P 3 .The paper also examines a class of 4-generator 6-relator algebras admitting presentations analogous to those of the 4-dimensional Sklyanin algebras. This class includes the 4-dimensional Sklyanin algebras and most of the Cho-Hong-Lau algebras.

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Noncommutative geometry of homogenized quantum 𝔰𝔩(2, ℂ)

Cited by 7 publications

References 22 publications

Exotic elliptic algebras of dimension 4

Exotic elliptic algebras of dimension 4

Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

Some algebras having relations like those for the 4-dimensional Sklyanin algebras

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