Abstract
We describe an explicit formula for the second-order quantum argument shifts of an arbitrary central element of the universal enveloping algebra of a general linear Lie algebra. We identify the generators of the subalgebra generated by the quantum argument shifts up to the second order.
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1. Introduction
Let \(g\) be a complex Lie algebra. The Lie–Poisson bracket on the symmetric algebra \(S(g)\) is the unique Poisson bracket extending the Lie bracket,
We suppose that \(\xi\) is an arbitrary element of the dual space \(g^*\) and let \(\bar\partial_\xi\) denote the constant vector field in the direction \(\xi\). We write \( \kern1.1pt\overline{\vphantom{C}\kern6.3pt}\kern-7.4pt C \) for the Poisson center of the symmetric algebra \(S(g)\). We define \( \kern1.1pt\overline{\vphantom{C}\kern6.3pt}\kern-7.4pt C _\xi\) as the algebra generated by the set \(\bigcup_{n=0}^\infty\bar\partial_\xi^n \kern1.1pt\overline{\vphantom{C}\kern6.3pt}\kern-7.4pt C \). Mishchenko and Fomenko [1] showed the following theorem.
Theorem 1.
The algebra \( \kern1.1pt\overline{\vphantom{C}\kern6.3pt}\kern-7.4pt C _\xi\) is Poisson commutative.
Vinberg [2] inquired whether the argument shift algebra \( \kern1.1pt\overline{\vphantom{C}\kern6.3pt}\kern-7.4pt C _\xi\) could be extended to a commutative subalgebra \(C_\xi\) of the universal enveloping algebra \(U(g)\). Nazarov and Olshanski [3] constructed the quantum argument shift algebra \(C_\xi\) for any regular semisimple \(\xi\) in terms of (i) the Yangian in the case \(g=gl_d(\mathbb C)\) and (ii) the twisted Yangians in the orthogonal and symplectic cases. Tarasov [4] constructed the same quantum argument shift algebra for \(g=gl_d(\mathbb C)\) via the symmetrization map. The quantum argument shift algebra \(C_\xi\) is also constructed via the Feigin–Frenkel center for (i) any simple complex Lie algebra \(g\) and any regular \(\xi\) [5], [6], and (ii) any simple complex Lie algebra of type \(A\) or \(C\) and any \(\xi\) [7], [8].
So far, the argument shift operator \(\bar\partial_\xi\) had not been quantized. Gurevich, Pyatov, and Saponov [9] defined the quantum derivations \(\partial^i_j\) on the universal enveloping algebra \(Ugl_d(\mathbb C)\). We found an explicit formula for the quantum derivations of appropriate elements [10] and showed a quantum analogue of the Mishchenko and Fomenko theorem [11].
In the following, we present an explicit formula for the quantum argument shifts of an arbitrary central element up to the second order (see Proposition 1). We also identify a reduced set of generators of the algebra generated by the quantum argument shifts up to the second order (see Corollary 1 and Theorem 5). This reduced set of generators provides an alternative to those given by Futorny and Molev [7]. Complex combinatorial formulas play an essential role here (see Theorem 4 and Proposition 4).
2. Preliminaries
We write \(\delta\) for the identity matrix and let \(x^{\mathrm T}\) be the transpose of a matrix \(x\). We suppose that \(d\) is a nonnegative integer and let \(M(d,A)\) denote the algebra of \(d\times d\) matrices with entries in an algebra \(A\). We write \(x^i_j\) for the \((i,j)\) element of a \(d\times d\) matrix \(x\) and
for the \(i\)th row vector and the \(j\)th column vector of the matrix \(x\).
We define the generating matrix of the Lie algebra \(gl_d=gl_d(\mathbb C)\) as the \(d\times d\) matrix \(e\) composed of the indeterminates \(e^i_j\) (generators of the Lie algebra \(gl_d\)). The universal enveloping algebra of the Lie algebra \(gl_d\) is the quotient algebra
where \(\mathbb{C}\langle e^i_j\rangle\) denotes the free unital algebra on the indeterminates \(e^i_j\) and the denominator in the right-hand side denotes the ideal generated by the elements
The following relation holds in the universal enveloping algebra \(Ugl_d\):
This can be proved by induction.
Quantum derivations on the universal enveloping algebra \(Ugl_d\) were defined in [9]. We give a slightly modified definition of these operators as follows.
Definition 1.
The quantum derivations on the universal enveloping algebra \(Ugl_d\) are the matrix elements of a unique homomorphism of unital complex algebras
such that \(\partial \operatorname{tr} (\xi e)= \operatorname{tr} (\xi e)+\xi\) for any numerical matrix \(\xi\).
We define the polynomials
The following theorem is proved in [10].
Theorem 2.
The quantum derivations of the matrix elements \((e^n)^i_j\) are given by
We write \(C\) for the center of the universal enveloping algebra \(Ugl_d\). The center \(C\) is generated by the elements \( \operatorname{tr} e, \operatorname{tr} e^2,\ldots{}\,\).
We suppose that \(\xi\) is an arbitrary numerical matrix. The map \(\partial_\xi= \operatorname{tr} (\xi\partial)\) is called the quantum argument shift operator in the direction \(\xi\). We define \(C_\xi\) as the algebra generated by the set \(\bigcup_{n=0}^\infty\partial_\xi^nC\). The following theorem is proved in [11], [12].
Theorem 3.
The algebra \(C_\xi\) is a quantum argument shift algebra in the direction \(\xi\).
3. Formulas for second-order quantum argument shifts
We present formulas for the second-order quantum argument shifts of central elements. Theorem 2 suffices for this purpose. We adopt the convention that \( \operatorname{tr} e^{-1}=1\) for simplicity of notation. The following formulas give the quantum argument shifts of an arbitrary central element up to the second order.
Proposition 1.
and
for a finite product \( \operatorname{tr} e^{n_1} \operatorname{tr} e^{n_2}\ldots{}\,\).
Proof
is by direct computation. We have
by Theorem 2 and the identity \(\sum_{m=0}^nf^{(n-m-1)}_{+}(x)x^m=f^{(n)}_{-}(x)\). We obtain
We proceed to calculate the second-order quantum argument shifts
and
by formula (3). Because
we arrive at formula (2). \(\blacksquare\)
We write \(A[S]\) for the algebra generated by an algebra \(A\) and a set \(S\) contained in the quantum argument shift algebra \(C_\xi\). We define
Formula (4) implies the following assertion.
Corollary 1.
\(C_\xi^{(1)}=C[ \operatorname{tr} \bigl(\xi e^n\bigr)\colon n=1,2,\ldots\,]\).
We have
by formula (2). Formula (5) implies the corollary.
Corollary 2.
The algebra \(C_\xi^{(2)}\) is contained in the algebra generated by the algebra \(C_\xi^{(1)}\) and the elements
Proof.
The elements of the form
belong to the additive monoid generated by the elements
Any element of \(C_\xi^{(2)}\) is contained in the algebra generated by the algebra \(C_\xi^{(1)}\) and the elements
We suppose that \(m\) and \(n\) are nonnegative integers. We have
by Theorem 2 and thus
by Corollary 1.
Definition 2.
We define the \((m+n)\times n\) integer matrix \(P^{(m)}_n\) as the coefficients of the polynomials
and let \(P_n=P^{(0)}_n\).
The matrix \(P_n\) is the submatrix of the matrix \(P_{n+1}\) in the top right corner, \(P_{n+1}=\bigl(\begin{smallmatrix} * & P_n \\ 1 & 0 \end{smallmatrix}\bigr)\) and \(P^{(m)}_n=\binom{0}{P_n}\) (the first \(m\) row vectors are null). For instance, because
we have \(P_4=\biggl(\begin{smallmatrix} 0 & 1 & 0 & 1 \\ 3 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{smallmatrix}\biggr)\).
Definition 3.
We define
for any \(m\times n\) numerical matrix \(x\).
By formula (7), we now have
4. Generators of the algebra \(C_\xi^{(2)}\)
We give the reduced set of generators of the algebra \(C_\xi^{(2)}\). The generators given in Corollary 2 can be expressed in terms of lower triangular matrices.
Definition 4.
Let \(n\) be a nonnegative integer and \(x\) an \(n\times n\) numerical matrix. We define the \(n\times n\) lower triangular numerical matrix \(\sigma(x)\) by the formula
Proposition 2.
\((\tau_\xi \mathbin{\stackrel{\scriptscriptstyle{\circ}}{{}_{\vphantom{.}}}} \sigma)(x)=\tau_\xi(x)\) for any numerical square matrix \(x\).
Proof.
We suppose that \(m\) and \(n\) are nonnegative integers and let \((\zeta_1,\ldots,\zeta_n)\) be a finite sequence of elements of the set \(M(d,\mathbb{C})\sqcup\{e\}\). We have
by the commutation relation (1), and thus
because the algebra \(C_\xi^{(1)}=C\bigl[ \operatorname{tr} (\xi e^n)\colon n=1,2,\ldots\,\bigr]\) (see Corollary 1) is commutative by Theorem 3. We have
for any \(n\times n\) numerical matrix \(x\) by formula (9). \(\blacksquare\)
Proposition 3.
For any nonnegative integers \(m\) and \(n\), we have
Proof.
We have
by formula (8) and Proposition 2. \(\blacksquare\)
The following theorem plays an essential role in reducing the number of the generators given in Corollary 2 and Proposition 3. The proof is given in the Appendix.
Theorem 4.
For any nonnegative integers \(m\) and \(n\), we have
The following theorem is the main result in this paper.
Theorem 5.
The algebra \(C_\xi^{(2)}\) is given by
Proof.
The algebra \(C_\xi^{(2)}\) is contained in the algebra
by Proposition 3 and Theorem 4. We prove that the elements \(\tau_\xi(P^{(n)}_n)\) and \(\tau_\xi(P^{(n)}_{n+1})+\tau_\xi(P^{(n+1)}_n)\) belong to the algebra
by induction on the nonnegative integer \(n\). Suppose that the integer \(n\) is positive and the elements \(\tau_\xi(P^{(m)}_m)\), \( \tau_\xi(P^{(m)}_{m+1})+\tau_\xi(P^{(m+1)}_m)\) belong to algebra (12) for any nonnegative integer \(m<n\). The element \(\tau_\xi(P^{(n)}_n)\) belongs to algebra (12) because the element \(\partial_\xi^2 \operatorname{tr} e^{2n+1}-(4n+2)\tau_\xi(P^{(n)}_n)\) belongs to the submodule
modulo \(C_\xi^{(1)}\) by Theorem 4. Similarly, the element \(\tau_\xi(P^{(n)}_{n+1})+\tau_\xi(P^{(n+1)}_n)\) belongs to algebra (12). \(\blacksquare\)
We compute the first several elements of the generators:
They form a commutative family together with the elements \(\bigl\{ \operatorname{tr} (\xi e^n)\colon n=1,2,\dots\bigr\}\) (see Theorem 3 and Corollary 1).
References
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2024, Vol. 220, pp. 275–285 https://doi.org/10.4213/tmf10578.
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Appendix: Proof of Theorem 4
We note that relation (10) for \(m+1\) implies the same relation for \(m\) and is therefore equivalent to the relation
Relation (13) is equivalent to the combinatorial relation
Relation (14) is equivalent to the polynomial relation
Similar arguments apply to the case in (11). We thus arrive at the following proposition.
Proposition 4.
-
1.
Theorem 4 is equivalent to the following conditions.
For any nonnegative integers \(n_1\), \(n_2\), and \(n_3\), we have
$$\begin{aligned} \, &\binom{2n_1+n_2+2n_3+1}{2n_3}+\binom{n_2+2n_3}{2n_3}= \nonumber\\ &\qquad=\sum_{n_4=0}^{n_3}\biggl(\binom{n_1+n_2+n_3+n_4+1}{2n_4}+\binom{n_1+n_2+n_3+n_4}{2n_4}\biggr)\binom{n_1+n_3-n_4}{2(n_3-n_4)}, \end{aligned}$$(15)$$\begin{aligned} \, &\binom{2n_1+n_2+2n_3+2}{2n_3}+\binom{n_2+2n_3}{2n_3}= \nonumber\\ &\qquad =\sum_{n_4=0}^{n_3}\binom{n_1+n_2+n_3+n_4+1}{2n_4}\biggl(\binom{n_1+n_3-n_4+1}{2(n_3-n_4)}+\binom{n_1+n_3-n_4}{2(n_3-n_4)}\biggr). \end{aligned}$$(16)For any nonnegative integers \(m\) and \(n\), we have
$$\begin{aligned} \, &f^{(m+2n)}_{+}(x)+f^{(m)}_{+}(x)x^{2n}=\sum_{k=0}^n\biggl(\binom{2n-k}k+\binom{2n-k-1}{k-1}\biggr)f^{(m+k)}_{+}(x)x^k, \\ &f^{(m+2n+1)}_{+}(x)+f^{(m)}_{+}(x)x^{2n+1}=\sum_{k=0}^n\binom{2n-k}k\bigl(f^{(m+k+1)}_{+}(x)x^k+f^{(m+k)}_{+}(x)x^{k+1}\bigr). \end{aligned}$$ -
2.
Relation (15) is equivalent to the relation
$$\sigma(P_{2n})=\sum_{m=1}^n\biggl(\binom{2n-m}m+\binom{2n-m-1}{m-1}\biggr)P^{(m)}_m.$$ -
3.
Relation (16) is equivalent to the relation
$$\sigma(P_{2n+1})=\sum_{m=0}^n\binom{2n-m}m\bigl(P^{(m)}_{m+1}+P^{(m+1)}_m\bigr).$$
Proof of Theorem 4.
We verify the corresponding conditions in Proposition 4 with Mathematica:
\(\blacksquare\)
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Ikeda, Y. Second-order quantum argument shifts in \(Ugl_d\). Theor Math Phys 220, 1294–1303 (2024). https://doi.org/10.1134/S004057792408004X
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DOI: https://doi.org/10.1134/S004057792408004X