Quantizations of cluster algebras

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1 Quantizations of cluster algebras Philipp Lampe Bielefeld University International Conference Mathematics Days in Sofia July 10, 2014, Sofia, Bulgaria Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

2 Introduction Fomin-Zelevinsky invented the notion of a cluster algebra to describe the dual of Lusztig s canonical basis. Later Berenstein-Zelevinsky studied quantizations of cluster algebras. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

3 Introduction Fomin-Zelevinsky invented the notion of a cluster algebra to describe the dual of Lusztig s canonical basis. Later Berenstein-Zelevinsky studied quantizations of cluster algebras. A cluster algebra is a commutative algebra defined by generators and relations. The generators are called cluster variables and can be grouped into several overlapping sets called clusters. Every cluster can be reached from an initial cluster by a sequence of mutations. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

4 Introduction Fomin-Zelevinsky invented the notion of a cluster algebra to describe the dual of Lusztig s canonical basis. Later Berenstein-Zelevinsky studied quantizations of cluster algebras. A cluster algebra is a commutative algebra defined by generators and relations. The generators are called cluster variables and can be grouped into several overlapping sets called clusters. Every cluster can be reached from an initial cluster by a sequence of mutations. In this talk, we wish to address the following questions: 1 What is a cluster algebra? 2 Where do the exchange relations come from? 3 Why and how do we quantize cluster algebras? Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

5 What is a cluster algebra? Let Q be a quiver with n vertices. (A quiver is another word for a directed graph.) We assume that Q contains neither loops and nor 2-cycles. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

6 What is a cluster algebra? Let Q be a quiver with n vertices. (A quiver is another word for a directed graph.) We assume that Q contains neither loops and nor 2-cycles Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

7 What is a cluster algebra? Let Q be a quiver with n vertices. (A quiver is another word for a directed graph.) We assume that Q contains neither loops and nor 2-cycles The construction of Fomin-Zelevinsky s cluster algebra A(Q, x) = A(Q): Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

8 What is a cluster algebra? Let Q be a quiver with n vertices. (A quiver is another word for a directed graph.) We assume that Q contains neither loops and nor 2-cycles. x 1 x 3 x 4 x 2 The construction of Fomin-Zelevinsky s cluster algebra A(Q, x) = A(Q): 1 Let x = (x 1, x 2,..., x n ) be a sequence of algebr. independent variables over a base field K which constitute an initial cluster. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

9 What is a cluster algebra? Let Q be a quiver with n vertices. (A quiver is another word for a directed graph.) We assume that Q contains neither loops and nor 2-cycles. x 1 x 3 x 4 x 2 The construction of Fomin-Zelevinsky s cluster algebra A(Q, x) = A(Q): 1 Let x = (x 1, x 2,..., x n ) be a sequence of algebr. independent variables over a base field K which constitute an initial cluster. 2 All variables obtained from the initial cluster by a sequence of mutations Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

10 What is a cluster algebra? Let Q be a quiver with n vertices. (A quiver is another word for a directed graph.) We assume that Q contains neither loops and nor 2-cycles. x 1 x 3 x 4 x 2 The construction of Fomin-Zelevinsky s cluster algebra A(Q, x) = A(Q): 1 Let x = (x 1, x 2,..., x n ) be a sequence of algebr. independent variables over a base field K which constitute an initial cluster. 2 All variables obtained from the initial cluster by a sequence of mutations Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

11 What is a cluster algebra? Let Q be a quiver with n vertices. (A quiver is another word for a directed graph.) We assume that Q contains neither loops and nor 2-cycles. x 1 x 1 x 2 +x 4 x 3 x 4 x 2 The construction of Fomin-Zelevinsky s cluster algebra A(Q, x) = A(Q): 1 Let x = (x 1, x 2,..., x n ) be a sequence of algebr. independent variables over a base field K which constitute an initial cluster. 2 All variables obtained from the initial cluster by a sequence of mutations Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

12 What is a cluster algebra? Let Q be a quiver with n vertices. (A quiver is another word for a directed graph.) We assume that Q contains neither loops and nor 2-cycles. x 1 x 1 x 2 +x 4 x 3 x 1 x 2 x 3 +x 1 x 2 +x 4 x 3 x 4 x 2 The construction of Fomin-Zelevinsky s cluster algebra A(Q, x) = A(Q): 1 Let x = (x 1, x 2,..., x n ) be a sequence of algebr. independent variables over a base field K which constitute an initial cluster. 2 All variables obtained from the initial cluster by a sequence of mutations Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

13 What is a cluster algebra? Let Q be a quiver with n vertices. (A quiver is another word for a directed graph.) We assume that Q contains neither loops and nor 2-cycles. x 1 x 1 x 2 +x 4 x 3 x 1 x 2 x 3 +x 1 x 2 +x 4 x 3 x 4 x 2 The construction of Fomin-Zelevinsky s cluster algebra A(Q, x) = A(Q): 1 Let x = (x 1, x 2,..., x n ) be a sequence of algebr. independent variables over a base field K which constitute an initial cluster. 2 All variables obtained from the initial cluster by a sequence of mutations generate a cluster algebra A(Q) K [x ±1 1, x ±1 2,..., x ±1 n ]. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

14 Basic notions Cluster algebras of finite type: The cluster algebras with only finitely many cluster variables have been classified by finite type root systems. More precisely, a cluster algebra is of finite type if and only if the mutation class of Q contains an orientation of a Dynkin diagram of type A, D, E. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

15 Basic notions Cluster algebras of finite type: The cluster algebras with only finitely many cluster variables have been classified by finite type root systems. More precisely, a cluster algebra is of finite type if and only if the mutation class of Q contains an orientation of a Dynkin diagram of type A, D, E. Freezing: Sometimes we freeze certain vertices of the quiver to obtain two kinds of vertices mutable and frozen vertices. Sequences of mutations at mutable vertices yield a smaller set C of cluster variables. In this case, we define the cluster algebra (without invertible coefficients) to be generated by C and the set of frozen variables. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

16 Basic notions Cluster algebras of finite type: The cluster algebras with only finitely many cluster variables have been classified by finite type root systems. More precisely, a cluster algebra is of finite type if and only if the mutation class of Q contains an orientation of a Dynkin diagram of type A, D, E. Freezing: Sometimes we freeze certain vertices of the quiver to obtain two kinds of vertices mutable and frozen vertices. Sequences of mutations at mutable vertices yield a smaller set C of cluster variables. In this case, we define the cluster algebra (without invertible coefficients) to be generated by C and the set of frozen variables. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

17 The exchange relations come from Lie Theory A motivating example is the Lie algebra sl 3 of traceless matrices: sl 3 (C) = {A Mat 3 3 (C) tr(a) = 0}. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

18 The exchange relations come from Lie Theory A motivating example is the Lie algebra sl 3 of traceless matrices: sl 3 (C) = {A Mat 3 3 (C) tr(a) = 0}. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

19 The exchange relations come from Lie Theory A motivating example is the Lie algebra sl 3 of traceless matrices: sl 3 (C) = {A Mat 3 3 (C) tr(a) = 0}. It s an eight-dimensional Lie algebra spanned by: The weight space V α1 = Ce 1 spanned by e 1 = Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

20 The exchange relations come from Lie Theory A motivating example is the Lie algebra sl 3 of traceless matrices: sl 3 (C) = {A Mat 3 3 (C) tr(a) = 0}. It s an eight-dimensional Lie algebra spanned by: The weight space V α2 = Ce 2 spanned by e 2 = Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

21 The exchange relations come from Lie Theory A motivating example is the Lie algebra sl 3 of traceless matrices: sl 3 (C) = {A Mat 3 3 (C) tr(a) = 0}. It s an eight-dimensional Lie algebra spanned by: The weight space V α1 +α 2 = Ce 12 spanned by e 12 = Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

22 The exchange relations come from Lie Theory A motivating example is the Lie algebra sl 3 of traceless matrices: sl 3 (C) = {A Mat 3 3 (C) tr(a) = 0}. It s an eight-dimensional Lie algebra spanned by: The weight space V α1 = Cf 1 spanned by f 1 = Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

23 The exchange relations come from Lie Theory A motivating example is the Lie algebra sl 3 of traceless matrices: sl 3 (C) = {A Mat 3 3 (C) tr(a) = 0}. It s an eight-dimensional Lie algebra spanned by: The weight space V α2 = Cf 2 spanned by f 2 = Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

24 The exchange relations come from Lie Theory A motivating example is the Lie algebra sl 3 of traceless matrices: sl 3 (C) = {A Mat 3 3 (C) tr(a) = 0}. It s an eight-dimensional Lie algebra spanned by: The weight space V α1 α 2 = Cf 12 spanned by f 12 = Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

25 The exchange relations come from Lie Theory A motivating example is the Lie algebra sl 3 of traceless matrices: sl 3 (C) = {A Mat 3 3 (C) tr(a) = 0}. It s an eight-dimensional Lie algebra spanned by: The Cartan subalgebra h = Ch 1 Ch 2 spanned by h 1 = 0 1 0, h 2 = Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

26 The universal enveloping algebra and its bases Representations of g correspond to modules over its universal enveloping algebra U(g). Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

27 The universal enveloping algebra and its bases Representations of g correspond to modules over its universal enveloping algebra U(g). It is generated by elements E 1, E 2, F 1, F 2, H 1, H 2 (corresponding to e 1, e 2, f 1, f 2, h 1, h 1 ) subject to certain relations such as the Serre relation E 2 1 E 2 2E 1 E 2 E 1 + E 2 E 2 1 = 0. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

28 The universal enveloping algebra and its bases Representations of g correspond to modules over its universal enveloping algebra U(g). It is generated by elements E 1, E 2, F 1, F 2, H 1, H 2 (corresponding to e 1, e 2, f 1, f 2, h 1, h 1 ) subject to certain relations such as the Serre relation E 2 1 E 2 2E 1 E 2 E 1 + E 2 E 2 1 = 0. The Lie algebra g = sl 3 admits a triangular decomposition g = n h n into strictly upper triangular, diagonal and strictly lower triangular matrices. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

29 The universal enveloping algebra and its bases Representations of g correspond to modules over its universal enveloping algebra U(g). It is generated by elements E 1, E 2, F 1, F 2, H 1, H 2 (corresponding to e 1, e 2, f 1, f 2, h 1, h 1 ) subject to certain relations such as the Serre relation E 2 1 E 2 2E 1 E 2 E 1 + E 2 E 2 1 = 0. The Lie algebra g = sl 3 admits a triangular decomposition g = n h n into strictly upper triangular, diagonal and strictly lower triangular matrices. The universal enveloping algebras U(g) and U(n) admit several Poincaré-Birkhoff-Witt bases: every ordered basis of g or n gives rise to a PBW basis of its universal enveloping algebra. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

30 The universal enveloping algebra and its bases Representations of g correspond to modules over its universal enveloping algebra U(g). It is generated by elements E 1, E 2, F 1, F 2, H 1, H 2 (corresponding to e 1, e 2, f 1, f 2, h 1, h 1 ) subject to certain relations such as the Serre relation E 2 1 E 2 2E 1 E 2 E 1 + E 2 E 2 1 = 0. The Lie algebra g = sl 3 admits a triangular decomposition g = n h n into strictly upper triangular, diagonal and strictly lower triangular matrices. The universal enveloping algebras U(g) and U(n) admit several Poincaré-Birkhoff-Witt bases: every ordered basis of g or n gives rise to a PBW basis of its universal enveloping algebra. For example, the basis (e 1, e 12, e 2 ) of n from above yields the basis P = {E a 1 (E 1 E 2 E 2 E 1 ) b E c 2 (a, b, c) N 3 } of U(n). Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

31 Lusztig s canonical basis and representations of g Lusztig constructed another basis of U(n), the canonical basis B, so that application of B to lowest vectors v 0, i.e., the sets {bv 0 b B, bv 0 = 0}, yield bases of the irreducible representations. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

32 Lusztig s canonical basis and representations of g Lusztig constructed another basis of U(n), the canonical basis B, so that application of B to lowest vectors v 0, i.e., the sets {bv 0 b B, bv 0 = 0}, yield bases of the irreducible representations. For g = sl 3, Lusztig s canonical basis is: B = {E a 1 E b 2 E c 1 a + c b} {E a 2 E b 1 E c 2 a + c b}. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

33 Lusztig s canonical basis and representations of g Lusztig constructed another basis of U(n), the canonical basis B, so that application of B to lowest vectors v 0, i.e., the sets {bv 0 b B, bv 0 = 0}, yield bases of the irreducible representations. For g = sl 3, Lusztig s canonical basis is: B = {E a 1 E b 2 E c 1 a + c b} {E a 2 E b 1 E c 2 a + c b}. For example, a basis of the natural representation of U(sl 3 ) is given by: Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

34 Lusztig s canonical basis and representations of g Lusztig constructed another basis of U(n), the canonical basis B, so that application of B to lowest vectors v 0, i.e., the sets {bv 0 b B, bv 0 = 0}, yield bases of the irreducible representations. For g = sl 3, Lusztig s canonical basis is: B = {E a 1 E b 2 E c 1 a + c b} {E a 2 E b 1 E c 2 a + c b}. For example, a basis of the natural representation of U(sl 3 ) is given by: v 3 Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

35 Lusztig s canonical basis and representations of g Lusztig constructed another basis of U(n), the canonical basis B, so that application of B to lowest vectors v 0, i.e., the sets {bv 0 b B, bv 0 = 0}, yield bases of the irreducible representations. For g = sl 3, Lusztig s canonical basis is: B = {E a 1 E b 2 E c 1 a + c b} {E a 2 E b 1 E c 2 a + c b}. For example, a basis of the natural representation of U(sl 3 ) is given by: v 2 E 2 v 3 Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

36 Lusztig s canonical basis and representations of g Lusztig constructed another basis of U(n), the canonical basis B, so that application of B to lowest vectors v 0, i.e., the sets {bv 0 b B, bv 0 = 0}, yield bases of the irreducible representations. For g = sl 3, Lusztig s canonical basis is: B = {E a 1 E b 2 E c 1 a + c b} {E a 2 E b 1 E c 2 a + c b}. For example, a basis of the natural representation of U(sl 3 ) is given by: v 2 E 1 v 1 E 2 v 3 Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

37 Lusztig s canonical basis and representations of sl 3 Recall Lusztig s canonical basis B = {E a 1 E b 2 E c 1 a + c b} {E a 2 E b 1 E c 2 a + c b}. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

38 Lusztig s canonical basis and representations of sl 3 Recall Lusztig s canonical basis B = {E1 a E2 b E1 c a + c b} {E2 a E1 b E2 c a + c b}. A basis of the adjoint representation U(sl 3 ) is given by: Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

39 Lusztig s canonical basis and representations of sl 3 Recall Lusztig s canonical basis B = {E1 a E2 b E1 c a + c b} {E2 a E1 b E2 c a + c b}. A basis of the adjoint representation U(sl 3 ) is given by: Ce 2 E 1 Ce 12 E 2 E 2 E 1 Cf 1 Ch 1 Ch 2 E 1 Ce 1 E 2 E 2 E 1 Cf 12 Cf 2 Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

40 Cluster algebras and the dual of Lusztig s canonical basis To construct the canonical basis Lusztig considers the quantized universal enveloping algebra U q (n). U(n) Quantize U q (n) Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

41 Cluster algebras and the dual of Lusztig s canonical basis To construct the canonical basis Lusztig considers the quantized universal enveloping algebra U q (n). In the example g = sl 3 (C) we abbreviate v 1 = E 1, v 2 = E 1 E 2 E 2 E 1 and v 3 = E 2 to obtain the following. U(n) v 2 v 1 = v 1 v 2, v 3 v 2 = v 2 v 3 v 3 v 1 = v 1 v 3 + v 2 Quantize U q (n) v 2 v 1 = q 1 v 1 v 2, v 3 v 2 = q 1 v 2 v 3 v 3 v 1 = qv 1 v 3 + qv 2 Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

42 Cluster algebras and the dual of Lusztig s canonical basis To construct the canonical basis Lusztig considers the quantized universal enveloping algebra U q (n). In the example g = sl 3 (C) we abbreviate v 1 = E 1, v 2 = E 1 E 2 E 2 E 1 and v 3 = E 2 to obtain the following. The quantum structure enables us to define the canonical basis B. B U(n) Quantize v 2 v 1 = v 1 v 2, v 3 v 2 = v 2 v 3 v 3 v 1 = v 1 v 3 + v 2 B q U q (n) v 2 v 1 = q 1 v 1 v 2, v 3 v 2 = q 1 v 2 v 3 v 3 v 1 = qv 1 v 3 + qv 2 Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

43 Cluster algebras and the dual of Lusztig s canonical basis To construct the canonical basis Lusztig considers the quantized universal enveloping algebra U q (n). In the example g = sl 3 (C) we abbreviate v 1 = E 1, v 2 = E 1 E 2 E 2 E 1 and v 3 = E 2 to obtain the following. The quantum structure enables us to define the canonical basis B. B U(n) Quantize v 2 v 1 = v 1 v 2, v 3 v 2 = v 2 v 3 v 3 v 1 = v 1 v 3 + v 2 B q U q (n) Dualize v 2 v 1 = q 1 v 1 v 2, v 3 v 2 = q 1 v 2 v 3 v 3 v 1 = qv 1 v 3 + qv 2 B q U q (n) gr u 2 u 1 = q 1 u 1 u 2, u 3 u 2 = q 1 u 2 u 3 u 3 u 1 = qu 1 u 3 + (q q 1 )u 2 Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

44 Cluster algebras and the dual of Lusztig s canonical basis To construct the canonical basis Lusztig considers the quantized universal enveloping algebra U q (n). In the example g = sl 3 (C) we abbreviate v 1 = E 1, v 2 = E 1 E 2 E 2 E 1 and v 3 = E 2 to obtain the following. The quantum structure enables us to define the canonical basis B. B U(n) Quantum cluster algebra with 2 clusters u 2 u u 2 1 B q U q (n) Quantize Dualize u 2 u u 2 3 when u 2 and u 2 correspond to E 2 E 1 q 1 E 1 E 2 and E 1 E 2 q 1 E 2 E 1. B q U q (n) gr u 2 u 1 = q 1 u 1 u 2, u 3 u 2 = q 1 u 2 u 3 u 3 u 1 = qu 1 u 3 + (q q 1 )u 2 Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

45 Example of type A 3 P1 Z1 P2 Y3 P3 23 P1 Z1 P1 Z1 P3 23 P2 Y2 P2 Z2 P2 13 P3 23 P3 Y3 P1 Y3 P1 12 P1 Z1 P1 Z1 P1 Y1 P3 Y1 P3 23 P2 Y2 P2 Y2 P2 Z2 P2 Z2 P2 13 P2 13 P3 23 P3 Z3 P3 Z3 P3 Y3 P1 Y3 P1 12 P1 12 P1 Y1 P3 Y1 P2 Y2 P2 Z2 P2 13 P3 Z3 P3 Z3 P1 12 P1 12 P2 Y1 P3 Z3 Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

46 Quantum cluster algebras To describe such phenomena, Berenstein-Zelevinsky introduced quantum cluster algebras as q-deformations which specialize to the ordinary cluster algebras in the limit q = 1. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

47 Quantum cluster algebras To describe such phenomena, Berenstein-Zelevinsky introduced quantum cluster algebras as q-deformations which specialize to the ordinary cluster algebras in the limit q = 1. Geiß-Leclerc-Schröer showed that the algebra U q (n) gr (for a Kac-Moody Lie algebra g = n h n ) carries indeed the structure of a quantum cluster algebra A q (Q, x) with mutable and frozen vertices. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

48 Quantum cluster algebras To describe such phenomena, Berenstein-Zelevinsky introduced quantum cluster algebras as q-deformations which specialize to the ordinary cluster algebras in the limit q = 1. Geiß-Leclerc-Schröer showed that the algebra U q (n) gr (for a Kac-Moody Lie algebra g = n h n ) carries indeed the structure of a quantum cluster algebra A q (Q, x) with mutable and frozen vertices. Denote by M = {y a 1 1 y a 2 2 y y m m : y cluster of A q (Q, x)} the set of quantum cluster monomials. A main conjecture asserts that all quantum cluster monomials are dual canonical basis elements: M B q. There are partial results by Kimura-Qin and L. But it is open in general! Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

49 In [Gellert-L.] Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12 Quantum cluster algebras To describe such phenomena, Berenstein-Zelevinsky introduced quantum cluster algebras as q-deformations which specialize to the ordinary cluster algebras in the limit q = 1. Denote by M = {y a 1 1 y a 2 2 y y m m : y cluster of A q (Q, x)} the set of quantum cluster monomials. A key feature in the definition of quantum cluster algebras is the q-commutativity between variables within the same quantum cluster: in all clusters y as above we have y i y j = q λ i,j y j y i for some integers λ i,j. In order to keep q-commutativity intact under mutation, [BZ] impose some compatibility relation between Λ the signed adjacency matrix B of Q, namely B T Λ is a diagonal matrix with positive diagonal entries. The very same compatibility condition also parametrizes compatible Poisson structures for cluster algebras, see Gekhtman-Shapiro-Vainshtein.

50 Quantum cluster algebras A key feature in the definition of quantum cluster algebras is the q-commutativity between variables within the same quantum cluster: in all clusters y as above we have y i y j = q λ i,j y j y i for some integers λ i,j. In order to keep q-commutativity intact under mutation, [BZ] impose some compatibility relation between Λ the signed adjacency matrix B of Q, namely B T Λ is a diagonal matrix with positive diagonal entries. The very same compatibility condition also parametrizes compatible Poisson structures for cluster algebras, see Gekhtman-Shapiro-Vainshtein. In [Gellert-L.] we... reinterpret what it means for B to be of full rank via Pfaffians and perfect matchings. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

51 Quantum cluster algebras A key feature in the definition of quantum cluster algebras is the q-commutativity between variables within the same quantum cluster: in all clusters y as above we have y i y j = q λ i,j y j y i for some integers λ i,j. In order to keep q-commutativity intact under mutation, [BZ] impose some compatibility relation between Λ the signed adjacency matrix B of Q, namely B T Λ is a diagonal matrix with positive diagonal entries. The very same compatibility condition also parametrizes compatible Poisson structures for cluster algebras, see Gekhtman-Shapiro-Vainshtein. In [Gellert-L.] we... give an elementary proof that there always exists a quantization when B has full rank. Note that [GSV] prove a similar statement in the language of Poisson structures. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

52 Quantum cluster algebras A key feature in the definition of quantum cluster algebras is the q-commutativity between variables within the same quantum cluster: in all clusters y as above we have y i y j = q λ i,j y j y i for some integers λ i,j. In order to keep q-commutativity intact under mutation, [BZ] impose some compatibility relation between Λ the signed adjacency matrix B of Q, namely B T Λ is a diagonal matrix with positive diagonal entries. The very same compatibility condition also parametrizes compatible Poisson structures for cluster algebras, see Gekhtman-Shapiro-Vainshtein. In [Gellert-L.]: When a quantization exists, it is not necessarily unique. This ambiguity we make more precise by constructing matrices by using particular minors, relating all such quantizations. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

53 Conclusion Some, but not all cluster algebras admit a quantization in the sense of Berenstein-Zelevinsky. When this happens, the quantization is not necessarily unique. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

54 Conclusion Some, but not all cluster algebras admit a quantization in the sense of Berenstein-Zelevinsky. When this happens, the quantization is not necessarily unique. The quantum structure is essential when we want to study the connection to Lusztig s canonical basis, which was Fomin and Zelevinsky main motivation to introduce cluster algebras. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

55 Conclusion Some, but not all cluster algebras admit a quantization in the sense of Berenstein-Zelevinsky. When this happens, the quantization is not necessarily unique. The quantum structure is essential when we want to study the connection to Lusztig s canonical basis, which was Fomin and Zelevinsky main motivation to introduce cluster algebras. Surprisingly, later on various authors found similar cluster structures in other branches of mathematics, namely Poisson geometry, representation theory of quivers, algebraic combinatorics, etc. Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

56 Conclusion Some, but not all cluster algebras admit a quantization in the sense of Berenstein-Zelevinsky. When this happens, the quantization is not necessarily unique. The quantum structure is essential when we want to study the connection to Lusztig s canonical basis, which was Fomin and Zelevinsky main motivation to introduce cluster algebras. Surprisingly, later on various authors found similar cluster structures in other branches of mathematics, namely Poisson geometry, representation theory of quivers, algebraic combinatorics, etc. Thank you so much for your attention! Ph. Lampe (Bielefeld) Quantum cluster algebras July / 12

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