## Quantum Supergroups Of GL(n|m) Type: Differential Forms, Koszul Complexes And Berezinians (1998)

Citations: | 6 - 0 self |

### BibTeX

@MISC{Lyubashenko98quantumsupergroups,

author = {Volodymyr Lyubashenko and Anthony Sudbery},

title = {Quantum Supergroups Of GL(n|m) Type: Differential Forms, Koszul Complexes And Berezinians},

year = {1998}

}

### OpenURL

### Abstract

. We introduce and study the Koszul complex for a Hecke R-matrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke R-matrix. Their behaviour with respect to Hecke sum of R-matrices is studied. Given a Hecke R-matrix in n-dimensional vector space, we construct a Hecke R-matrix in 2n-dimensional vector space commuting with a differential. The notion of a quantum differential supergroup is derived. Its algebra of functions is a differential coquasitriangular Hopf algebra, having the usual algebra of differential forms as a quotient. Examples of superdeterminants related to these algebras are calculated. Several remarks about Woronowicz's theory are made. 0.1. Short description of the paper. 0.1.1. In this paper we will be concerned with differential Hopf algebras generated by sets of matrix elements. We start (Section 1) by giving a construction of such algebras, generalising the construction [31, 27] of a bialgebra generated by a single se...

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12 |
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Citation Context ...], where most results of this section were obtained for symmetric monoidal categories (q = 1). Also the following definition is a braided version of one of the Koszul complexes for quadratic algebras =-=[20]-=-. The Koszul complex for a Hecke symmetry possess more structures, namely, it will have two differentials. Definition 3.1. The Koszul complex of V is the Z�0 × Z�0-graded algebra K •• (V ), quotient o... |

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Citation Context ...coquasitriangular Hopf algebra (Section 2). Shortly, it defines a differential quantum supergroup. A quotient of H is the Z�0-graded differential Hopf algebra Ω of differential forms defined via R in =-=[25, 26, 30, 35]-=-. The classical version (q = 1) of this construction is: take a vector space V , add to it another copy of it with the opposite parity and consider the general linear supergroup of the obtained space.... |

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Citation Context ...reserves the ideal of relations of H and the antipode of H comes as its quotient. 1.4. Differential bialgebra case. Some methods of constructing differential bialgebras were described by Maltsiniotis =-=[18, 19]-=- and Manin [21]. We present here a general framework for such constructions. The results of the previous section are applied to the category V = D of Z-graded differential finite dimensional vector sp... |

9 |
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Citation Context ... ,DIFFERENTIAL FORMS, KOSZUL COMPLEXES AND BEREZINIANS 3 ev : X ⊗ X ∨ → k coev : k → X ∨ ⊗ X by by X X ✚✙ ∨ , ✛✘ X∨ X . 0.3. Preliminaries. We recall some definitions from [10] and some results from =-=[12]-=-. Definition 0.1. Let T : A ⊗ B → B ⊗ A, ai ⊗ bj ↦→ T kl ij bk ⊗ al be a linear map, written in bases (ai), (bj) of finite dimensional vector spaces A, B. Let A ∗ , B ∗ be spaces of linear functionals... |

8 |
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6 |
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(Show Context)
Citation Context ...spaces (it is equipped with a faithful exact monoidal functor C → k-vect), the number n such that Λ n (V ) ̸= 0, Λ n+1 (V ) = 0, is not necessarily dimk V . There are examples constructed by Gurevich =-=[4, 5]-=- in which n < dimk V . Proposition 3.15. Let q be not a root of unity and let C be as in above conjecture. (a) Assume that V ∈ Ob C is even, Λ n+1 (V ) = 0, Λ n (V ) ̸= 0. Then Ber V = Λ n (V ) and sd... |

6 |
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Citation Context ...corresponding supercommutative algebra: specifically, it has a linear basis consisting of all alphabetically ordered monomials in p i j, r i j, s i j, t i j with the powers of p and r not exceeding 1 =-=[27]-=-, Theorem 3. Again Theorem 3.4 tells us that Ber X ≃ Ber ′ X is one dimensional. The basis (ui) = (dvi) ⊂ U = dV has the changed degree p(ui) = (p(vi), 1) ∈ Z/2 × Z, hence, a basic vector of Ber U is ... |

5 |
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- 1991
(Show Context)
Citation Context ...disappears in the final answer because only even elements ¯toi oj ei and t ej contribute to the final formula. When m = 0 the superdeterminant coincides with the usual quantum determinant detq t from =-=[24]-=-. In the formulae c(v ⊗ ωV ) = ωV ⊗ 1 π(ω∨ V , ωV ) (π ⊗ 1)(1 ⊗ c)(˜ω∨ V ⊗ v ⊗ ˜ωV ) (4.4) 1 c(ωV ⊗ v) = ωV ⊗ π(ωV , ωV ∨)(1 ⊗ π)(c ⊗ 1)(˜ωV ⊗ v ⊗ ˜ωV ∨) (4.5) contributions of terms proportional to (... |

5 |
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3 |
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- 1991
(Show Context)
Citation Context ...he equivalence of Theorems 3.4 and 3.8, when the pairings π are not degenerate, that is, when q is not a root of unity. 3.6. 8-dimensions. The number, which is called 8-dimension here, was studied in =-=[17]-=- under the name of rank. We believe that our term is less confusing. Definition 3.3. The 8-dimension of V is the number dim8 V = (k coev −−→ V ∨ ⊗ V c −→ V ⊗ V ∨ ✛✘ ev −→ k) ≡ � � � ❅ V V ❅ ∨ ∈ k ✚✙ T... |

2 |
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- 1986
(Show Context)
Citation Context ...uotient of T • (V ) by relations R(x ⊗ y) = qx ⊗ y, and the external algebra Λ • (V ) has relations x, y ∈ V R(x ⊗ y) = −q −1 x ⊗ y, x, y ∈ V. Our definition of a Koszul complex is a q-deformation of =-=[11]-=-, where most results of this section were obtained for symmetric monoidal categories (q = 1). Also the following definition is a braided version of one of the Koszul complexes for quadratic algebras [... |

2 |
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- 1993
(Show Context)
Citation Context ...coquasitriangular Hopf algebra (Section 2). Shortly, it defines a differential quantum supergroup. A quotient of H is the Z�0-graded differential Hopf algebra Ω of differential forms defined via R in =-=[25, 26, 30, 35]-=-. The classical version (q = 1) of this construction is: take a vector space V , add to it another copy of it with the opposite parity and consider the general linear supergroup of the obtained space.... |

2 |
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(Show Context)
Citation Context ....yndUu ′ 1 . . .u ′ lz1 . . .zk = DV (v ′ 1 . . .v′ m y1 . . .yn)u ′ 1 . . .u′ l z1 . . .zk + (−1) n v ′ 1 . . . v′ m y1 . . .ynDU(u ′ 1 . . .u′ l z1 . . .zk). (b) Follows from (a) by Künneth theorem =-=[8]-=-. 3.4. Dual Koszul complex. Consider now the Koszul complex K •• ( ∨ V ) of the left dual ∨ V , which is generated by ∨ V ⊕ V and has the relations R ♭♭ ( ′ v ⊗ ′ ¯v) = −q −1′ v ⊗ ′ ¯v for ′ v, ′ ¯v ∈... |

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(Show Context)
Citation Context .... Choose graded bases (xa) in spaces X ∈ O. Introduce Z/2 × Z-graded coalgebra C = ⊕X∈O(Endk X) ∗ = k{tX a b} with coaction X → X ⊗ C, xb ↦→ xa ⊗ tX a b on Xi. Then T(C) is a Z/2 × Z-graded bialgebra =-=[9]-=- coacting in Xi. With each map f : X 1 ⊗ . . . ⊗ X k −→ Y 1 ⊗ . . . ⊗ Y m ∈ M, Xi ∈ O, Y j ∈ O, is associated a subspace Rel(f) ⊂ T(C). Choosing graded bases (xi a) ⊂ Xi , (y j b ) ⊂ Y j and writing c... |

1 |
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- 1987
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Citation Context ... ,DIFFERENTIAL FORMS, KOSZUL COMPLEXES AND BEREZINIANS 3 ev : X ⊗ X ∨ → k coev : k → X ∨ ⊗ X by by X X ✚✙ ∨ , ✛✘ X∨ X . 0.3. Preliminaries. We recall some definitions from [10] and some results from =-=[12]-=-. Definition 0.1. Let T : A ⊗ B → B ⊗ A, ai ⊗ bj ↦→ T kl ij bk ⊗ al, be a linear map, written in terms of bases (ai), (bj) of the finite-dimensional vector spaces A, B. Let A ∗ , B ∗ be the spaces of ... |

1 |
Sudbery.The Algebra of Differential Forms on a Full Matric
- unknown authors
- 1993
(Show Context)
Citation Context ...and the antipode of H comes as its quotient. □ 1.4. Differential bialgebra case. Some methods of constructing differential bialgebras were described by Maltsiniotis [18, 19], Manin [21] and one of us =-=[28]-=-. We present here a general framework for such constructions. The results of the previous section are applied to the category V = D of Z-graded differential finite-dimensional vector spaces, (V, d : V... |