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Rinat Kedem - One of the best experts on this subject based on the ideXlab platform.

  • ( $${{\mathbf {t}}},{{\mathbf {Q}}}$$ t , Q
    Communications in Mathematical Physics, 2019
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    We introduce the natural ( t ,  Q )-deformation of the Q-System algebra in type A . The Q -Whittaker limit $$t\rightarrow \infty $$ t → ∞ gives the Quantum Q-System algebra of Di Francesco and Kedem (Lett Math Phys 107(2):301–341, [ DFK17 ]), a deformation of the Groethendieck ring of finite dimensional Yangian modules, compatible with graded tensor products (Hatayama et al. in: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Volume 248 of Contemporary Mathematics, Amer. Math. Soc., Providence, [ HKO+99 ]; Feigin and Loktev in: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Volume 194 of Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, [ FL99 ]; Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, [ DFK14 ]). We show that the ( Q ,  t )-deformed algebra is isomorphic to the spherical double affine Hecke algebra of type $${\mathfrak {gl}}_N$$ gl N . Moreover, we describe the kernel of the surjective homomorphism from the Quantum toroidal algebra (Miki in J Math Phys 48(12):123520, [ Mik07 ]) and the elliptic Hall algebra (Schiffmann and Vasserot in Compos Math 147(1):188–234, [ SV11 ]) to this new algebra. It is generated by ( Q ,  t )-determinants, new objects which are a deformation of the Quantum determinant associated with the Quantum Q-System. The functional representation of the algebra is generated by generalized Macdonald operators, obtained from the usual Macdonald operators by the $$SL_2({\mathbb {Z}})$$ S L 2 ( Z ) -action on the spherical Double Affine Hecke Algebra. The generating function for generalized Macdonald operators acts by plethysms on the space of symmetric functions. We give the relation to the plethystic operators from Macdonald theory of Bergeron et al. (J Comb 7(4):671–714, [ BGLX16 ]) in the limit $$N\rightarrow \infty $$ N → ∞ . Thus, the ( Q ,  t )-deformation of the Q-System cluster algebra leads directly to Macdonald theory.

  • (t,Q)-Deformed Q-Systems, DAHA and Quantum Toroidal Algebras via Generalized Macdonald Operators
    Communications in Mathematical Physics, 2019
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    We introduce the natural (t, Q)-deformation of the Q-System algebra in type A. The Q-Whittaker limit \(t\rightarrow \infty \) gives the Quantum Q-System algebra of Di Francesco and Kedem (Lett Math Phys 107(2):301–341, [DFK17]), a deformation of the Groethendieck ring of finite dimensional Yangian modules, compatible with graded tensor products (Hatayama et al. in: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Volume 248 of Contemporary Mathematics, Amer. Math. Soc., Providence, [HKO+99]; Feigin and Loktev in: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Volume 194 of Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, [FL99]; Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, [DFK14]). We show that the (Q, t)-deformed algebra is isomorphic to the spherical double affine Hecke algebra of type \({\mathfrak {gl}}_N\). Moreover, we describe the kernel of the surjective homomorphism from the Quantum toroidal algebra (Miki in J Math Phys 48(12):123520, [Mik07]) and the elliptic Hall algebra (Schiffmann and Vasserot in Compos Math 147(1):188–234, [SV11]) to this new algebra. It is generated by (Q, t)-determinants, new objects which are a deformation of the Quantum determinant associated with the Quantum Q-System. The functional representation of the algebra is generated by generalized Macdonald operators, obtained from the usual Macdonald operators by the \(SL_2({\mathbb {Z}})\)-action on the spherical Double Affine Hecke Algebra. The generating function for generalized Macdonald operators acts by plethysms on the space of symmetric functions. We give the relation to the plethystic operators from Macdonald theory of Bergeron et al. (J Comb 7(4):671–714, [BGLX16]) in the limit \(N\rightarrow \infty \). Thus, the (Q, t)-deformation of the Q-System cluster algebra leads directly to Macdonald theory.

  • (t,Q) Q-Systems, DAHA and Quantum toroidal algebras via generalized Macdonald operators
    arXiv: Mathematical Physics, 2017
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    We introduce difference operators on the space of symmetric functions which are a natural generalization of the $(Q,t)$-Macdonald operators. In the $t\to\infty$ limit, they satisfy the $A_{N-1}$ Quantum $Q$-System. We identify the elements in the spherical $A_{N-1}$ DAHA which are represented by these operators, as well as within the Quantum toroidal algebra of $gl_1$ and the elliptic Hall algebra. We present a plethystic, or bosonic, formulation of the generating functions for the generalized Macdonald operators, which we relate to recent work of Bergeron et al. Finally we derive constant term identities for the current that allow to interpret them in terms of shuffle products. In particular we obtain in the $t\to\infty$ limit a shuffle presentation of the Quantum $Q$-System relations.

  • Quantum Q Systems: from cluster algebras to Quantum current algebras
    Letters in Mathematical Physics, 2017
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    This paper gives a new algebraic interpretation for the algebra generated by the Quantum cluster variables of the $$A_r$$ A r Quantum Q-System (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, 2014 ). We show that the algebra can be described as a Quotient of the localization of the Quantum algebra $$U_{\sQrt{Q}}({\mathfrak {n}}[u,u^{-1}])\subset U_{\sQrt{Q}}(\widehat{{\mathfrak {sl}}}_2)$$ U Q ( n [ u , u - 1 ] ) ⊂ U Q ( sl ^ 2 ) , in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-System, which we call fundamental. The other cluster variables are given by a Quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved Quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97–152, 2011 ) described by Quantum Q-System generate the Cartan currents at level 0, in a non-standard polarization. The rest of the Quantum affine algebra is also described in terms of cluster variables.

  • Difference eQuations for graded characters from Quantum cluster algebra
    2016
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    In this paper, we construct graded tensor product characters in terms of generalized difference Macdonald raising operators which form a representation of the Quantum Q-System. Characters for the graded tensor product of Kirillov-Reshetikhin modules were expressed as the constant term of a non-commutative generating function. This function is written in terms of the generators of a Quantum cluster algebra, subject to recursion relations known as the Quantum Q-System. The latter form a discrete non-commutative integrable System, with a set of commuting conserved Quantities. In type A such conserved Quantities can be treated as analogues of the Casimir elements of $U_Q({\mathfrak sl}_{r+1})$. We show that the graded tensor product character is the analogue of a class I Whittaker function (to which it reduces in the Demazure case), and that the difference eQuations which follow from the action of the conserved Quantities on characters generalize the difference Quantum Toda eQuations of Etingof. Finally, we construct the graded characters as solutions of these eQuations, by introducing a representation of the Quantum Q-System via difference operators which generalize the Macdonald raising difference operators of Kirillov-Noumi in the dual Whittaker limit.

Philippe Di Francesco - One of the best experts on this subject based on the ideXlab platform.

  • ( $${{\mathbf {t}}},{{\mathbf {Q}}}$$ t , Q
    Communications in Mathematical Physics, 2019
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    We introduce the natural ( t ,  Q )-deformation of the Q-System algebra in type A . The Q -Whittaker limit $$t\rightarrow \infty $$ t → ∞ gives the Quantum Q-System algebra of Di Francesco and Kedem (Lett Math Phys 107(2):301–341, [ DFK17 ]), a deformation of the Groethendieck ring of finite dimensional Yangian modules, compatible with graded tensor products (Hatayama et al. in: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Volume 248 of Contemporary Mathematics, Amer. Math. Soc., Providence, [ HKO+99 ]; Feigin and Loktev in: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Volume 194 of Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, [ FL99 ]; Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, [ DFK14 ]). We show that the ( Q ,  t )-deformed algebra is isomorphic to the spherical double affine Hecke algebra of type $${\mathfrak {gl}}_N$$ gl N . Moreover, we describe the kernel of the surjective homomorphism from the Quantum toroidal algebra (Miki in J Math Phys 48(12):123520, [ Mik07 ]) and the elliptic Hall algebra (Schiffmann and Vasserot in Compos Math 147(1):188–234, [ SV11 ]) to this new algebra. It is generated by ( Q ,  t )-determinants, new objects which are a deformation of the Quantum determinant associated with the Quantum Q-System. The functional representation of the algebra is generated by generalized Macdonald operators, obtained from the usual Macdonald operators by the $$SL_2({\mathbb {Z}})$$ S L 2 ( Z ) -action on the spherical Double Affine Hecke Algebra. The generating function for generalized Macdonald operators acts by plethysms on the space of symmetric functions. We give the relation to the plethystic operators from Macdonald theory of Bergeron et al. (J Comb 7(4):671–714, [ BGLX16 ]) in the limit $$N\rightarrow \infty $$ N → ∞ . Thus, the ( Q ,  t )-deformation of the Q-System cluster algebra leads directly to Macdonald theory.

  • (t,Q)-Deformed Q-Systems, DAHA and Quantum Toroidal Algebras via Generalized Macdonald Operators
    Communications in Mathematical Physics, 2019
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    We introduce the natural (t, Q)-deformation of the Q-System algebra in type A. The Q-Whittaker limit \(t\rightarrow \infty \) gives the Quantum Q-System algebra of Di Francesco and Kedem (Lett Math Phys 107(2):301–341, [DFK17]), a deformation of the Groethendieck ring of finite dimensional Yangian modules, compatible with graded tensor products (Hatayama et al. in: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Volume 248 of Contemporary Mathematics, Amer. Math. Soc., Providence, [HKO+99]; Feigin and Loktev in: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Volume 194 of Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, [FL99]; Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, [DFK14]). We show that the (Q, t)-deformed algebra is isomorphic to the spherical double affine Hecke algebra of type \({\mathfrak {gl}}_N\). Moreover, we describe the kernel of the surjective homomorphism from the Quantum toroidal algebra (Miki in J Math Phys 48(12):123520, [Mik07]) and the elliptic Hall algebra (Schiffmann and Vasserot in Compos Math 147(1):188–234, [SV11]) to this new algebra. It is generated by (Q, t)-determinants, new objects which are a deformation of the Quantum determinant associated with the Quantum Q-System. The functional representation of the algebra is generated by generalized Macdonald operators, obtained from the usual Macdonald operators by the \(SL_2({\mathbb {Z}})\)-action on the spherical Double Affine Hecke Algebra. The generating function for generalized Macdonald operators acts by plethysms on the space of symmetric functions. We give the relation to the plethystic operators from Macdonald theory of Bergeron et al. (J Comb 7(4):671–714, [BGLX16]) in the limit \(N\rightarrow \infty \). Thus, the (Q, t)-deformation of the Q-System cluster algebra leads directly to Macdonald theory.

  • (t,Q) Q-Systems, DAHA and Quantum toroidal algebras via generalized Macdonald operators
    arXiv: Mathematical Physics, 2017
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    We introduce difference operators on the space of symmetric functions which are a natural generalization of the $(Q,t)$-Macdonald operators. In the $t\to\infty$ limit, they satisfy the $A_{N-1}$ Quantum $Q$-System. We identify the elements in the spherical $A_{N-1}$ DAHA which are represented by these operators, as well as within the Quantum toroidal algebra of $gl_1$ and the elliptic Hall algebra. We present a plethystic, or bosonic, formulation of the generating functions for the generalized Macdonald operators, which we relate to recent work of Bergeron et al. Finally we derive constant term identities for the current that allow to interpret them in terms of shuffle products. In particular we obtain in the $t\to\infty$ limit a shuffle presentation of the Quantum $Q$-System relations.

  • Quantum Q Systems: from cluster algebras to Quantum current algebras
    Letters in Mathematical Physics, 2017
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    This paper gives a new algebraic interpretation for the algebra generated by the Quantum cluster variables of the $$A_r$$ A r Quantum Q-System (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, 2014 ). We show that the algebra can be described as a Quotient of the localization of the Quantum algebra $$U_{\sQrt{Q}}({\mathfrak {n}}[u,u^{-1}])\subset U_{\sQrt{Q}}(\widehat{{\mathfrak {sl}}}_2)$$ U Q ( n [ u , u - 1 ] ) ⊂ U Q ( sl ^ 2 ) , in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-System, which we call fundamental. The other cluster variables are given by a Quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved Quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97–152, 2011 ) described by Quantum Q-System generate the Cartan currents at level 0, in a non-standard polarization. The rest of the Quantum affine algebra is also described in terms of cluster variables.

  • Difference eQuations for graded characters from Quantum cluster algebra
    2016
    Co-Authors: Philippe Di Francesco, Rinat Kedem
    Abstract:

    In this paper, we construct graded tensor product characters in terms of generalized difference Macdonald raising operators which form a representation of the Quantum Q-System. Characters for the graded tensor product of Kirillov-Reshetikhin modules were expressed as the constant term of a non-commutative generating function. This function is written in terms of the generators of a Quantum cluster algebra, subject to recursion relations known as the Quantum Q-System. The latter form a discrete non-commutative integrable System, with a set of commuting conserved Quantities. In type A such conserved Quantities can be treated as analogues of the Casimir elements of $U_Q({\mathfrak sl}_{r+1})$. We show that the graded tensor product character is the analogue of a class I Whittaker function (to which it reduces in the Demazure case), and that the difference eQuations which follow from the action of the conserved Quantities on characters generalize the difference Quantum Toda eQuations of Etingof. Finally, we construct the graded characters as solutions of these eQuations, by introducing a representation of the Quantum Q-System via difference operators which generalize the Macdonald raising difference operators of Kirillov-Noumi in the dual Whittaker limit.

Christopher J Potter - One of the best experts on this subject based on the ideXlab platform.

  • The Q-System: A Versatile Expression System for Drosophila.
    Methods in molecular biology (Clifton N.J.), 2016
    Co-Authors: Olena Riabinina, Christopher J Potter
    Abstract:

    Binary expression Systems are flexible and versatile genetic tools in Drosophila. The Q-System is a recently developed repressible binary expression System that offers new possibilities for transgene expression and genetic manipulations. In this review chapter, we focus on current state-of-the-art Q-System tools and reagents. We also discuss in vivo applications of the Q-System, together with GAL4/UAS and LexA/LexAop Systems, for simultaneous expression of multiple effectors, intersectional labeling, and clonal analysis.

  • Improved and expanded Q-System reagents for genetic manipulations
    Nature methods, 2015
    Co-Authors: Olena Riabinina, David J. Luginbuhl, Elizabeth Marr, Sha Liu, Liqun Luo, Christopher J Potter
    Abstract:

    The Q System is a repressible binary expression System for transgenic manipulations in living organisms. Through protein engineering and in vivo functional tests, we report here variants of the Q-System transcriptional activator, including QF2, for driving strong and ubiQuitous expression in all Drosophila tissues. Our QF2, Gal4QF and LexAQF chimeric transcriptional activators substantially enrich the toolkit available for transgenic regulation in Drosophila melanogaster.

  • controlling gene expression with the Q repressible binary expression System in caenorhabditis elegans
    Nature Methods, 2012
    Co-Authors: Xing Wei, Christopher J Potter, Liqun Luo, Kang Shen
    Abstract:

    We established a transcription-based binary gene expression System in Caenorhabditis elegans using the recently developed Q System. This System, derived from genes in Neurospora crassa, uses the transcriptional activator QF to induce the expression of target genes. Activation can be efficiently suppressed by the transcriptional repressor QS, and suppression can be relieved by the nontoxic small molecule Quinic acid. We used QF, QS and Quinic acid to achieve temporal and spatial control of transgene expression in various tissues in C. elegans. We also developed a split Q System, in which we separated QF into two parts encoding its DNA-binding and transcription-activation domains. Each domain showed negligible transcriptional activity when expressed alone, but expression of both reconstituted QF activity, providing additional combinatorial power to control gene expression.

  • Using the Q System in Drosophila melanogaster
    Nature Protocols, 2011
    Co-Authors: Christopher J Potter
    Abstract:

    In Drosophila , the GAL4/UAS/GAL80 repressible binary expression System is widely used to manipulate or mark tissues of interest. However, complex biological Systems often reQuire distinct transgenic manipulations of different cell populations. For this purpose, we recently developed the Q System, a second repressible binary expression System. We describe here the basic steps for performing a variety of Q System experiments in vivo . These include how to generate and use Q System reagents to express effector transgenes in tissues of interest, how to use the Q System in conjunction with the GAL4 System to generate intersectional expression patterns that precisely limit which tissues will be experimentally manipulated and how to use the Q System to perform mosaic analysis. The protocol described here can be adapted to a wide range of experimental designs.

  • The Q System: A Repressible Binary System for Transgene Expression, Lineage Tracing, and Mosaic Analysis
    Cell, 2010
    Co-Authors: Christopher J Potter, Bosiljka Tasic, Emilie V. Russler, Liang Liang, Liqun Luo
    Abstract:

    SUMMARY We describe a new repressible binary expression System based on the regulatory genes from the Neurospora Qa gene cluster. This ‘‘Q System’’ offers attractive features for transgene expression in Drosophila and mammalian cells: low basal expression in the absence of the transcriptional activator QF, high QF-induced expression, and QF repression by its repressor QS. Additionally, feeding flies Quinic acid can relieve QS repression. The Q System offers many applications, including (1) intersectional ‘‘logic gates’’ with the GAL4 System for manipulating transgene expression patterns, (2) GAL4-independent MARCM analysis, and (3) coupled MARCM analysis to independently visualize and genetically manipulate siblings from any cell division. We demonstrate the utility of the Q System in determining cell division patterns of a neuronal lineage and gene function in cell growth and proliferation, and in dissecting neurons responsible for olfactory attraction. The Q System can be expanded to other uses in Drosophila and to any organism conducive to transgenesis.

Liqun Luo - One of the best experts on this subject based on the ideXlab platform.

  • Improved and expanded Q-System reagents for genetic manipulations
    Nature methods, 2015
    Co-Authors: Olena Riabinina, David J. Luginbuhl, Elizabeth Marr, Sha Liu, Liqun Luo, Christopher J Potter
    Abstract:

    The Q System is a repressible binary expression System for transgenic manipulations in living organisms. Through protein engineering and in vivo functional tests, we report here variants of the Q-System transcriptional activator, including QF2, for driving strong and ubiQuitous expression in all Drosophila tissues. Our QF2, Gal4QF and LexAQF chimeric transcriptional activators substantially enrich the toolkit available for transgenic regulation in Drosophila melanogaster.

  • controlling gene expression with the Q repressible binary expression System in caenorhabditis elegans
    Nature Methods, 2012
    Co-Authors: Xing Wei, Christopher J Potter, Liqun Luo, Kang Shen
    Abstract:

    We established a transcription-based binary gene expression System in Caenorhabditis elegans using the recently developed Q System. This System, derived from genes in Neurospora crassa, uses the transcriptional activator QF to induce the expression of target genes. Activation can be efficiently suppressed by the transcriptional repressor QS, and suppression can be relieved by the nontoxic small molecule Quinic acid. We used QF, QS and Quinic acid to achieve temporal and spatial control of transgene expression in various tissues in C. elegans. We also developed a split Q System, in which we separated QF into two parts encoding its DNA-binding and transcription-activation domains. Each domain showed negligible transcriptional activity when expressed alone, but expression of both reconstituted QF activity, providing additional combinatorial power to control gene expression.

  • The Q System: A Repressible Binary System for Transgene Expression, Lineage Tracing, and Mosaic Analysis
    Cell, 2010
    Co-Authors: Christopher J Potter, Bosiljka Tasic, Emilie V. Russler, Liang Liang, Liqun Luo
    Abstract:

    SUMMARY We describe a new repressible binary expression System based on the regulatory genes from the Neurospora Qa gene cluster. This ‘‘Q System’’ offers attractive features for transgene expression in Drosophila and mammalian cells: low basal expression in the absence of the transcriptional activator QF, high QF-induced expression, and QF repression by its repressor QS. Additionally, feeding flies Quinic acid can relieve QS repression. The Q System offers many applications, including (1) intersectional ‘‘logic gates’’ with the GAL4 System for manipulating transgene expression patterns, (2) GAL4-independent MARCM analysis, and (3) coupled MARCM analysis to independently visualize and genetically manipulate siblings from any cell division. We demonstrate the utility of the Q System in determining cell division patterns of a neuronal lineage and gene function in cell growth and proliferation, and in dissecting neurons responsible for olfactory attraction. The Q System can be expanded to other uses in Drosophila and to any organism conducive to transgenesis.

Dong Deuk Kwon - One of the best experts on this subject based on the ideXlab platform.