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Molev A. I. | Molev Alexander | Molev A. | Molev Alexander I. | Molev A. S.

A | A Gustavo Bruzual | A Dang Quang | A Krishna Chaitanya | A Lazarian | A Germina K | A M. | A Pranav | A Antony Franklin | A Azeef Muhammed P

I Tomohiro | I Chih-Lin | I | I Te | I Andras | I Aotemshi | I Bostan | I Chi-Lin | I Grace | I Soszynski

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On Segal--Sugawara vectors and Casimir elements for classical Lie algebras

Molev A. I.
12 Aug 2020 math.RT math-ph math.MP arxiv.org/abs/2008.05256

We consider the centers of the affine vertex algebras at the critical level associated with simple Lie algebras. We derive new formulas for generators of the centers in the classical types. We also give a new formula for the Capelli-type determinant for the symplectic Lie algebras and calculate the Harish-Chandra images of the Casimir elements arising from the characteristic polynomial of the matrix of generators of each classical Lie algebra.

Casimir elements and center at the critical level for Takiff algebras

Molev A. I.
06 Apr 2020 math.RT math-ph math.MP arxiv.org/abs/2004.02515

For every simple Lie algebra $\mathfrak{g}$ we consider the associated Takiff algebra $\mathfrak{g}^{}{\ell}$ defined as the truncated polynomial current Lie algebra with coefficients in $\mathfrak{g}$. We use a matrix presentation of $\mathfrak{g}^{}{\ell}$ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra ${\rm U}(\mathfrak{g}^{}{\ell})$. A similar matrix presentation for the affine Kac--Moody algebra $\widehat{\mathfrak{g}}^{}{\ell}$ is then used to prove an analogue of the Feigin--Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal--Sugawara vectors for the Lie algebra $\mathfrak{g}^{}_{\ell}$.

$W$-algebras associated with centralizers in type $A$

Molev A. I.
06 Feb 2020 math.RT math-ph math.MP arxiv.org/abs/2002.02218

We introduce a new family of affine $W$-algebras associated with the centralizers of arbitrary nilpotent elements in $\mathfrak{gl}_N$. We define them by using a version of the BRST complex of the quantum Drinfeld--Sokolov reduction. A family of free generators of the new algebras is produced in an explicit form. We also give an analogue of the Fateev--Lukyanov realization for these algebras by applying a Miura-type map.

Classical $\mathcal{W}$-algebras for centralizers

Molev A. I., Ragoucy E.
19 Nov 2019 math.RT math-ph math.MP arxiv.org/abs/1911.08645

We introduce a new family of Poisson vertex algebras $\mathcal{W}(\mathfrak{a})$ analogous to the classical $\mathcal{W}$-algebras. The algebra $\mathcal{W}(\mathfrak{a})$ is associated with the centralizer $\mathfrak{a}$ of an arbitrary nilpotent element in $\mathfrak{gl}_N$. We show that $\mathcal{W}(\mathfrak{a})$ is an algebra of polynomials in infinitely many variables and produce its free generators in an explicit form. This implies that $\mathcal{W}(\mathfrak{a})$ is isomorphic to the center at the critical level of the affine vertex algebra associated with $\mathfrak{a}$.

Center at the critical level for centralizers in type $A$

Molev A. I.
29 Apr 2019 math.RT arxiv.org/abs/1904.12520

We consider the affine vertex algebra at the critical level associated with the centralizer of a nilpotent element in the Lie algebra $\mathfrak{gl}_N$. Due to a recent result of Arakawa and Premet, the center of this vertex algebra is an algebra of polynomials. We construct a family of free generators of the center in an explicit form. As a corollary, we obtain generators of the corresponding quantum shift of argument subalgebras and recover free generators of the center of the universal enveloping algebra of the centralizer produced earlier by Brown and Brundan.

Segal-Sugawara vectors for the Lie algebra of type $G_2$

Molev A. I., Ragoucy E., Rozhkovskaya N.
27 Jan 2016 math.RT math-ph math.MP arxiv.org/abs/1601.07638

Explicit formulas for Segal-Sugawara vectors associated with the simple Lie algebra $\mathfrak{g}$ of type $G_2$ are found by using computer-assisted calculations. This leads to a direct proof of the Feigin-Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. As an application, we give an explicit solution of Vinberg's quantization problem by providing formulas for generators of maximal commutative subalgebras of $U(\mathfrak{g})$. We also calculate the eigenvalues of the Hamiltonians on the Bethe vectors in the Gaudin model associated with $\mathfrak{g}$.

Eigenvalues of Bethe vectors in the Gaudin model

Molev A. I., Mukhin E. E.
05 Jun 2015 math.RT math-ph math.MP arxiv.org/abs/1506.01884

A theorem of Feigin, Frenkel and Reshetikhin provides expressions for the eigenvalues of the higher Gaudin Hamiltonians on the Bethe vectors in terms of elements of the center of the affine vertex algebra at the critical level. In our recent work, explicit Harish-Chandra images of generators of the center were calculated in all classical types. We combine these results to calculate the eigenvalues of the higher Gaudin Hamiltonians on the Bethe vectors in an explicit form. The Harish-Chandra images can be interpreted as elements of classical $W$-algebras. We provide a direct connection between the rings of $q$-characters and classical $W$-algebras by calculating classical limits of the corresponding screening operators.

Invariants of the vacuum module associated with the Lie superalgebra gl(1|1)

Molev A. I., Mukhin E. E.
11 Feb 2015 math.RT math-ph math.CO math.MP arxiv.org/abs/1502.03511

We describe the algebra of invariants of the vacuum module associated with the affinization of the Lie superalgebra $\mathfrak{gl}(1|1)$. We give a formula for its Hilbert--Poincar\'{e} series in a fermionic (cancellation-free) form which turns out to coincide with the generating function of the plane partitions over the $(1,1)$-hook. Our arguments are based on a super version of the Beilinson--Drinfeld--Ra\"{i}s--Tauvel theorem which we prove by producing an explicit basis of invariants of the symmetric algebra of polynomial currents associated with $\mathfrak{gl}(1|1)$. We identify the invariants with affine supersymmetric polynomials via a version of the Chevalley theorem.

Classical W-algebras in types A, B, C, D and G

Molev A. I., Ragoucy E.
07 Mar 2014 math.RT math-ph math.MP arxiv.org/abs/1403.1700

We produce explicit generators of the classical W-algebras associated with the principal nilpotents in the simple Lie algebras of all classical types and in the exceptional Lie algebra of type $G_2$. The generators are given by determinant formulas in the context of the Poisson vertex algebras. We also show that the images of the W-algebra generators under the Chevalley-type isomorphism coincide with the elements defined via the corresponding Miura transformations.

Yangian characters and classical W-algebras

Molev A. I., Mukhin E. E.
17 Dec 2012 math.RT math.QA arxiv.org/abs/1212.4032

The Yangian characters (or q-characters) are known to be closely related to the classical W-algebras and to the centers of the affine vertex algebras at the critical level. We make this relationship more explicit by producing families of generators of the W-algebras from the characters of the Kirillov-Reshetikhin modules associated with multiples of the first fundamental weight in types B and D and of the fundamental modules in type C. We also give an independent derivation of the character formulas for these representations in the context of the RTT presentation of the Yangians. In all cases the generators of the W-algebras correspond to the recently constructed elements of the Feigin-Frenkel centers via an affine version of the Harish-Chandra isomorphism.