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In this paper, we investigate certain deformations B_q(g) of the negative part U_q~-(g) of quantized enveloping algebras U_q(g). An algorithm is established to determine when a given B_q(g) is a PBW-deformation of U_q~-(g). For g ...
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In this paper, we investigate certain deformations B_q(g) of the negative part U_q~-(g) of quantized enveloping algebras U_q(g). An algorithm is established to determine when a given B_q(g) is a PBW-deformation of U_q~-(g). For g of type A_2 and B_2, we classify PBW-deformations of U_n~-(g). Moreover, we explicitly construct some PBW bases for a class of PBW-deformations B_q(g) of U_n~-(g). As an application, Iorgov-Klimyk's PBW bases for the non-standard quantum deformation U'_q(so(n, C)) of the universal enveloping algebra U(so(n, C)) are recovered.
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A finitely generated quadratic algebra with antisymmetric generating relations is called a weakly symmetric algebra. The automorphism group and Calabi-Yau property of a Poincaré-Birkhoff-Witt (PBW)-deformation of a weakly symmetr...
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A finitely generated quadratic algebra with antisymmetric generating relations is called a weakly symmetric algebra. The automorphism group and Calabi-Yau property of a Poincaré-Birkhoff-Witt (PBW)-deformation of a weakly symmetric algebra are discussed. It is shown that the Calabi-Yau property of a PBW-deformation of a weakly symmetric algebra is equivalent to that of the corresponding augmented PBW-deformation under some mild conditions.
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We introduce a method named homogeneous PBW deformation that preserves the regularity and some other homological properties for multigraded algebras. The method is used to produce Artin-Schelter regular algebras without the hypothesis on grading.
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The paper [9]by Bocklandt, Schedler and Wemyss considers path algebras with relations given by the higher derivations of a superpotential, giving a condition for such an algebra to be Calabi-Yau. In particular they show that the a...
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The paper [9]by Bocklandt, Schedler and Wemyss considers path algebras with relations given by the higher derivations of a superpotential, giving a condition for such an algebra to be Calabi-Yau. In particular they show that the algebra C[V] x G, for Va finite dimensional Cvector space and Ga finite subgroup of GL(V), is Morita equivalent to a path algebra with relations given by a superpotential, and is Calabi-Yau for G 收起
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The Calabi–Yau property of the Poincaré–Birkhoff–Witt deformation of a Koszul Calabi–Yau algebra is characterized. Berger and Taillefer (J Noncommut Geom 1:241–270, 2007, Theorem 3.6) proved that the Poincaré–Birkhoff–Wit...
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The Calabi–Yau property of the Poincaré–Birkhoff–Witt deformation of a Koszul Calabi–Yau algebra is characterized. Berger and Taillefer (J Noncommut Geom 1:241–270, 2007, Theorem 3.6) proved that the Poincaré–Birkhoff–Witt deformation of a Calabi–Yau algebra of dimension 3 is Calabi–Yau under some conditions. The main result in this paper generalizes their result to higher dimensional Koszul Calabi–Yau algebras. As corollaries, the necessary and sufficient condition obtained byHe et al. (J Algebra 324:1921–1939, 2010) for the universal enveloping algebra, respectively, Sridharan enveloping algebra, of a finite-dimensional Lie algebra to be Calabi–Yau, is derived.
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In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painleve algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting...
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In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painleve algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painleve equations. (c) 2020 Elsevier Inc. All rights reserved.
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In this paper, we mainly focus on the Poincare-Birkhoff-Witt (PBW) deformation theory for a class of N-homogeneous algebras; here N >= 2 is an integer, which generalizes the results in [2] and [7]. More precisely, let k be a field...
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In this paper, we mainly focus on the Poincare-Birkhoff-Witt (PBW) deformation theory for a class of N-homogeneous algebras; here N >= 2 is an integer, which generalizes the results in [2] and [7]. More precisely, let k be a field of characteristic zero, V a finite dimensional vector space over k, and A = T(V)/(R) an N-homogeneous algebra (i.e., R subset of V-circle times N) with Tor(A)(3)(k, k) being supported in a single degree d such that d > N. Set F-n := circle plus(0 = 0 and J(n) = 0 for n < N.
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Let be a generalized Koszul algebra over a finite dimensional algebra . We construct a bimodule Koszul resolution of when the projective dimension of equals two. Using this we prove a Poincar,-Birkhoff-Witt (PBW) type theorem for ...
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Let be a generalized Koszul algebra over a finite dimensional algebra . We construct a bimodule Koszul resolution of when the projective dimension of equals two. Using this we prove a Poincar,-Birkhoff-Witt (PBW) type theorem for a deformation of a generalized Koszul algebra. When the projective dimension of is greater than two, we construct bimodule Koszul resolutions for generalized smash product algebras obtained from braidings between finite dimensional algebras and Koszul algebras, and then prove the PBW type theorem. The results obtained can be applied to standard Koszul Artin-Schelter Gorenstein algebras in the sense of Minamoto and Mori (Adv Math 226:4061-4095, 2011).
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We begin the study of PBW deformations of graded algebras relevant to the theory of Hopf algebras. One of our examples is the Fomin-Kirillov algebra E-3. Another one appeared in a paper of Garcia Iglesias and Vay. As a consequence...
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We begin the study of PBW deformations of graded algebras relevant to the theory of Hopf algebras. One of our examples is the Fomin-Kirillov algebra E-3. Another one appeared in a paper of Garcia Iglesias and Vay. As a consequence of our methods, we determine when the deformations are semisimple and we are able to produce PBW bases and polynomial identities for these deformations.
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Let A be a Koszul Artin-Schelter regular algebra with Nakayama automorphism ξ. We show that the Yoneda Ext-algebra of the skew polynomial algebra A[. z; ξ] is a trivial extension of a Frobenius algebra. Then we prove that A[. z;...
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Let A be a Koszul Artin-Schelter regular algebra with Nakayama automorphism ξ. We show that the Yoneda Ext-algebra of the skew polynomial algebra A[. z; ξ] is a trivial extension of a Frobenius algebra. Then we prove that A[. z; ξ] is Calabi-Yau; and hence each Koszul Artin-Schelter regular algebra is a subalgebra of a Koszul Calabi-Yau algebra. A superpotential ? is also constructed so that the Calabi-Yau algebra A[. z; ξ] is isomorphic to the derivation quotient of ?. The Calabi-Yau property of a skew polynomial algebra with coefficients in a PBW-deformation of a Koszul Artin-Schelter regular algebra is also discussed.
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