PBW-type filtration on quantum groups of type $A_n$

We will introduce an $\mathbb{N}$-filtration on the negative part of a quantum group of type $A_n$, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation theory of quivers, by realizing the quantum group as the Hall algebra of a quiver. We show that the induced associated graded module of any simple finite-dimensional module (of type 1) is isomorphic to a quotient of this polynomial algebra by a monomial ideal, and we provide a monomial basis for this associated graded module. This construction can be viewed as a quantum analog of the classical PBW framework, and in fact, by considering the classical limit, this basis is the monomial basis provided by Feigin, Littelmann and the second author in the classical setup.

[1]  Martina Lanini,et al.  Degenerate flag varieties of type A and C are Schubert varieties , 2014, 1403.2889.

[2]  Martina Lanini,et al.  Degenerate flag varieties and Schubert varieties: a characteristic free approach , 2015, 1502.04590.

[3]  George Lusztig,et al.  Introduction to Quantum Groups , 1993 .

[4]  Evgeny Feigin,et al.  PBW filtration and bases for irreducible modules in type An , 2010, 1002.0674.

[5]  Evgeny Feigin,et al.  PBW filtration and bases for symplectic Lie algebras , 2010, 1010.2321.

[6]  Ghislain Fourier,et al.  PBW-degenerated Demazure modules and Schubert varieties for triangular elements , 2014, J. Comb. Theory, Ser. A.

[7]  M. Reineke.  Multiplicative properties of dual canonical bases of quantum groups , 1999 .

[8]  Jin Hong,et al.  Introduction to Quantum Groups and Crystal Bases , 2002 .

[9]  Ghislain Fourier,et al.  The degree of the Hilbert-Poincar\'e polynomial of PBW-graded modules , 2014 .

[10]  Claus Michael Ringel,et al.  Hall algebras and quantum groups , 1990 .

[11]  EVGENY FEIGIN,et al.  FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS , 2013, 1306.1292.

[12]  J. Jantzen.  Lectures on quantum groups , 1995 .

[13]  M. Finkelberg,et al.  Degenerate flag varieties of type A: Frobenius splitting and BW theorem , 2011, 1103.1491.

[14]  Evgeny Feigin,et al.  Nonsymmetric Macdonald polynomials and PBW filtration: Towards the proof of the Cherednik-Orr conjecture , 2015, J. Comb. Theory, Ser. A.

[15]  I. Cherednik,et al.  Extremal part of the PBW-filtration and E-polynomials , 2013, 1306.3146.

[16]  Evgeny Feigin,et al.  ${\mathbb G}_a^M$ degeneration of flag varieties , 2010, 1007.0646.

[17]  Ghislain Fourier,et al.  Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces , 2014, 1410.1126.

[18]  Evgeny Feigin,et al.  PBW-filtration over ℤ and Compatible Bases for V ℤ ( λ ) in Type A n and C n , 2013 .

[19]  Ghislain Fourier,et al.  Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence , 2014, 1410.8744.

[20]  Nonsymmetric Macdonald polynomials and Demazure characters , 2001, math/0105061.

[21]  Daniel Simson,et al.  Elements of the Representation Theory of Associative Algebras: Techniques of Representation Theory , 2006 .

[22]  Feigin Evgeny,et al.  Extremal part of the PBW-filtration and E-polynomials , 2013 .

[23]  Evgeny Feigin,et al.  Nonsymmetric Macdonald polynomials, Demazure modules and PBW filtration , 2014, 1407.6316.

[24]  E. Feigin,et al.  Quiver Grassmannians and degenerate flag varieties , 2011, 1106.2399.

[25]  Teodor Backhaus,et al.  PBW filtration: Feigin-Fourier-Littelmann modules via Hasse diagrams , 2014 .

[26]  Nicolas Bourbaki,et al.  Eléments de mathématique : groupes et algèbres de Lie , 1972 .

[27]  J. Humphreys.  Introduction to Lie Algebras and Representation Theory , 1973 .

[28]  E. Feigin,et al.  Degenerate flag varieties: moment graphs and Schröder numbers , 2012, Journal of Algebraic Combinatorics.

[29]  V. Kac,et al.  Representations of quantum groups at roots of 1 , 1992 .

[30]  Federico Ardila,et al.  Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes , 2010, J. Comb. Theory, Ser. A.